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Versions: (draft-huelsing-cfrg-hash-sig-xmss) 00 01 02 03 04 05 06 07 08 09 10

Crypto Forum Research Group                                  A. Huelsing
Internet-Draft                                              TU Eindhoven
Intended status: Informational                                  D. Butin
Expires: October 10, 2015                                   TU Darmstadt
                                                               S. Gazdag
                                                               genua mbH
                                                             A. Mohaisen
                                                           Verisign Labs
                                                           April 8, 2015


                  XMSS: Extended Hash-Based Signatures
             draft-irtf-cfrg-xmss-hash-based-signatures-00

Abstract

   This note describes the eXtended Merkle Signature Scheme (XMSS), a
   hash-based digital signature system.  It follows existing
   descriptions in scientific literature.  The note specifies the WOTS+
   one-time signature scheme, a single-tree (XMSS) and a multi-tree
   variant (XMSS^MT) of XMSS.  Both variants use WOTS+ as a main
   building block.  XMSS provides cryptographic digital signatures
   without relying on the conjectured hardness of mathematical problems.
   Instead, it is proven that it only relies on the properties of
   cryptographic hash functions.  XMSS provides strong security
   guarantees and, besides some special instantiations, is even secure
   when the collision resistance of the underlying hash function is
   broken.  It is suitable for compact implementations, relatively
   simple to implement, and naturally resists side-channel attacks.
   Unlike most other signature systems, hash-based signatures withstand
   attacks using quantum computers.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at http://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on October 10, 2015.



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Copyright Notice

   Copyright (c) 2015 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
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   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Conventions Used In This Document . . . . . . . . . . . .   5
   2.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.1.  Data Types  . . . . . . . . . . . . . . . . . . . . . . .   5
     2.2.  Operators . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.3.  Functions . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.4.  Strings of Base-w Numbers . . . . . . . . . . . . . . . .   6
     2.5.  Member Functions  . . . . . . . . . . . . . . . . . . . .   7
   3.  Primitives  . . . . . . . . . . . . . . . . . . . . . . . . .   8
     3.1.  WOTS+ One-Time Signatures . . . . . . . . . . . . . . . .   8
       3.1.1.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . .   8
         3.1.1.1.  WOTS+ Hashing Functions . . . . . . . . . . . . .   9
       3.1.2.  WOTS+ Chaining Function . . . . . . . . . . . . . . .   9
       3.1.3.  WOTS+ Private Key . . . . . . . . . . . . . . . . . .   9
       3.1.4.  WOTS+ Public Key  . . . . . . . . . . . . . . . . . .  10
       3.1.5.  WOTS+ Signature Generation  . . . . . . . . . . . . .  10
       3.1.6.  WOTS+ Signature Verification  . . . . . . . . . . . .  11
       3.1.7.  Pseudorandom Key Generation . . . . . . . . . . . . .  12
   4.  Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . .  12
     4.1.  XMSS: eXtended Merkle Signature Scheme  . . . . . . . . .  13
       4.1.1.  XMSS Parameters . . . . . . . . . . . . . . . . . . .  13
       4.1.2.  XMSS Hash Functions . . . . . . . . . . . . . . . . .  14
       4.1.3.  XMSS Private Key  . . . . . . . . . . . . . . . . . .  14
       4.1.4.  L-Trees . . . . . . . . . . . . . . . . . . . . . . .  14
       4.1.5.  TreeHash  . . . . . . . . . . . . . . . . . . . . . .  15
       4.1.6.  XMSS Public Key . . . . . . . . . . . . . . . . . . .  15
       4.1.7.  XMSS Signature  . . . . . . . . . . . . . . . . . . .  16
       4.1.8.  XMSS Signature Generation . . . . . . . . . . . . . .  18
       4.1.9.  XMSS Signature Verification . . . . . . . . . . . . .  19
       4.1.10. Pseudorandom Key Generation . . . . . . . . . . . . .  20
       4.1.11. Free Index Handling and Partial Secret Keys . . . . .  21



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     4.2.  XMSS^MT: Multi-Tree XMSS  . . . . . . . . . . . . . . . .  21
       4.2.1.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . .  21
       4.2.2.  XMSS Algorithms Without Message Hash  . . . . . . . .  22
       4.2.3.  XMSS^MT Private Key . . . . . . . . . . . . . . . . .  22
       4.2.4.  XMSS^MT Public Key  . . . . . . . . . . . . . . . . .  22
       4.2.5.  XMSS^MT Signature . . . . . . . . . . . . . . . . . .  23
       4.2.6.  XMSS^MT Signature Generation  . . . . . . . . . . . .  24
       4.2.7.  XMSS^MT Signature Verification  . . . . . . . . . . .  25
       4.2.8.  Pseudorandom Key Generation . . . . . . . . . . . . .  26
       4.2.9.  Free Index Handling and Partial Secret Keys . . . . .  26
   5.  Parameter Sets  . . . . . . . . . . . . . . . . . . . . . . .  27
     5.1.  Zero Bitmasks . . . . . . . . . . . . . . . . . . . . . .  27
     5.2.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . . . .  28
     5.3.  XMSS Parameters . . . . . . . . . . . . . . . . . . . . .  29
       5.3.1.  XMSS Parameters . . . . . . . . . . . . . . . . . . .  29
         5.3.1.1.  XMSS Parameters with AES and SHA3 . . . . . . . .  29
         5.3.1.2.  XMSS Parameters with SHA3 . . . . . . . . . . . .  30
       5.3.2.  XMSS Parameters With Empty Bitmasks . . . . . . . . .  31
     5.4.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . . . .  32
       5.4.1.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . .  32
         5.4.1.1.  XMSS^MT Parameters with AES and SHA3  . . . . . .  32
         5.4.1.2.  XMSS^MT Parameters with SHA3  . . . . . . . . . .  33
       5.4.2.  XMSS^MT Parameters With Empty Bitmasks  . . . . . . .  35
   6.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .  38
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  38
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  49
     8.1.  Security Proofs . . . . . . . . . . . . . . . . . . . . .  50
     8.2.  Security Assumptions  . . . . . . . . . . . . . . . . . .  51
     8.3.  Post-Quantum Security . . . . . . . . . . . . . . . . . .  51
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  51
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  51
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  51
     10.2.  Informative References . . . . . . . . . . . . . . . . .  52
   Appendix A.  WOTS+ XDR Formats  . . . . . . . . . . . . . . . . .  53
   Appendix B.  XMSS XDR Formats . . . . . . . . . . . . . . . . . .  55
   Appendix C.  XMSS^MT XDR Formats  . . . . . . . . . . . . . . . .  65
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  87

1.  Introduction

   A (cryptographic) digital signature scheme provides asymmetric
   message authentication.  The key generation algorithm produces a key
   pair consisting of a private and a public key.  A message is signed
   using a private key to produce a signature.  A message/signature pair
   can be verified using a public key.  A One-Time Signature (OTS)
   scheme allows us to use a key pair to sign exactly one message
   securely.  A many-time signature system can be used to sign multiple
   messages.



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   One-Time Signature schemes, and Many-Time Signature (MTS) schemes
   composed of them, were proposed by Merkle in 1979 [Merkle79].  They
   were well-studied in the 1990s and have regained interest from 2006
   onwards because of their resistance against quantum-computer-aided
   attacks.  These kinds of signature schemes are called hash-based
   signature schemes as they are built out of a cryptographic hash
   function.  Hash-based signature schemes generally feature small
   private and public keys as well as fast signature generation and
   verification but large signatures and a relatively slow key
   generation.  In addition, they are suitable for compact
   implementations that benefit various applications and are naturally
   resistant to most kinds of side-channel attacks.

   Some progress has already been made toward standardizing and
   introducing hash signatures.  McGrew and Curcio have published an
   Internet-Draft [DC14] specifying the "textbook" Lamport-Diffie-
   Winternitz-Merkle (LDWM) scheme based on early publications.
   Independently, Buchmann, Dahmen and Huelsing have proposed XMSS
   [BDH11], the "eXtended Merkle Signature Scheme," offering better
   efficiency and a modern security proof.  Very recently, SPHINCS, a
   stateless hash-based signature scheme was introduced [BHH15], with
   the intent of being easier to deploy in current applications.  A
   reasonable next step toward introducing hash signatures would seem to
   complete the specifications of the basic algorithms - LDWM, XMSS,
   SPHINCS and/or variants [Kaliski15].

   The eXtended Merkle Signature Scheme (XMSS) [BDH11] is the latest
   hash-based signature scheme.  It has the smallest signatures out of
   such schemes and comes with a multi-tree variant that solves the
   problem of slow key generation.  Moreover, it can be shown that XMSS
   is secure, making only mild assumptions on the underlying hash
   function.  Especially, it is not required that the cryptographic hash
   function is collision-resistant for the security of XMSS.

   This note describes a single-tree and a multi-tree variant of the
   eXtended Merkle Signature Scheme (XMSS) [BDH11].  It also describes
   WOTS+, a variant of the Winternitz OTS scheme introduced in
   [Huelsing13] that is used by XMSS.  The schemes are described with
   enough specificity to ensure interoperability between
   implementations.











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   This note is structured as follows.  Notation is introduced in
   Section 2.  Section 3 describes the WOTS+ signature system.  Many
   time signature schemes are defined in Section 4: the eXtended Merkle
   Signature Scheme (XMSS) in Section 4.1, and its Multi-Tree variant
   (XMSS^MT) in Section 4.2.  Parameter sets are described in Section 5.
   Section 6 describes the rationale behind choices in this note.  The
   IANA registry for these signature systems is described in Section 7.
   Finally, security considerations are presented in Section 8.

1.1.  Conventions Used In This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in [RFC2119].

2.  Notation

2.1.  Data Types

   Bytes and byte strings are the fundamental data types.  A byte is a
   sequence of eight bits.  A single byte is denoted as a pair of
   hexadecimal digits with a leading "0x".  A byte string is an ordered
   sequence of zero or more bytes and is denoted as an ordered sequence
   of hexadecimal characters with a leading "0x".  For example, 0xe534f0
   is a byte string of length 3.  An array of byte strings is an
   ordered, indexed set starting with index 0 in which all byte strings
   have identical length.

2.2.  Operators

   When a and b are integers, mathematical operators are defined as
   follows:

      ^ : a ^ b denotes the result of a raised to the power of b.

      * : a * b denotes the product of a and b. This operator is
      sometimes used implicitly in the absence of ambiguity, as in usual
      mathematical notation.

      / : a / b denotes the quotient of a by b.

      % : a % b denotes the non-negative remainder of the integer
      division of a by b.

      + : a + b denotes the sum of a and b.

      - : a - b denotes the difference of a and b.




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   The standard order of operations is used when evaluating arithmetic
   expressions.

   Arrays are used in the common way, where the i^th element of an array
   A is denoted A[i].  Byte strings are treated as arrays of bytes where
   necessary: If X is a byte string, then X[i] denotes its i^th byte,
   where X[0] is the leftmost byte.  In addition, bytes(X, i, j) with i
   < j denotes the range of bytes from the i^th to the j^th byte in X,
   inclusively.  For example, if X = 0x01020304, then X[0] is 0x01 and
   bytes(X, 1, 2) is 0x0203.

   If A and B are byte strings of equal length, then:

      A AND B denotes the bitwise logical conjunction operation.

      A XOR B denotes the bitwise logical exclusive disjunction
      operation.

   When B is a byte and i is an integer, then B >> i denotes the logical
   right-shift operation.  Similarly, B << i denotes the logical left-
   shift operation.

   If X is a x-byte string and Y a y-byte string, then X || Y denotes
   the concatenation of X and Y, with X || Y =
   X[0]...X[x-1]Y[0]...Y[y-1].

2.3.  Functions

   If x is a non-negative real number, then we define the following
   functions:

      ceil(x) : returns the smallest integer greater or equal than x.

      floor(x) : returns the largest integer less or equal than x.

      lg(x) : returns the base-2 logarithm of x.

   If x, y, and z are real numbers, then we define the functions max(x,
   y) and max(x, y, z) which return the maximum value of the set {x, y}
   and {x, y, z}, respectively.

2.4.  Strings of Base-w Numbers

   A byte string can be considered as a string of base-w numbers, i.e.
   integers in the set {0, ... , w - 1}.  The correspondence is defined
   by the function base_w(X, w) as follows.  If X is a m-byte string, w
   is a member of the set {4, 8, 16}, then base_w(X, w) outputs a length
   ceil(8m/lg(w)) array of integers between 0 and w - 1.  In case lg(w)



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   does not divide 8 * m without a remainder, X is virtually padded with
   a sufficient amount of zero bits.

   Algorithm 1: base_w(X, w)

   i_byte = 0;
   i_bit = 0;
   for ( i=0; i < ceil(8m/lg(w)); i++ ){
     if( i_bit + lg(w) <= 8 ){
       basew[i] = ((X[i_byte] << i_bit) >> (8-lg(w))) AND (w-1);
       i_bit += lg(w);
       if ( i_bit == 8 ){
         i_bit = 0;
         i_byte = i_byte + 1;
       }
     } else {
       basew[i] = ((X[i_byte] << i_bit) >> (8-lg(w))) AND (w-1);
       i_byte = i_byte + 1;
       if ( i_byte < m ){
         basew[i] += (X[i_byte] >> (8-(i_bit + lg(w)-8))) AND (w-1);
         i_bit = i_bit + lg(w)-8;
       }
     }
   }
   return basew;

   For example, if X is 0x1234, then base_w(X, 8) returns the array {0,
   4, 4, 3, 2, 0}.

                      X (represented as bits)
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
         | 0| 0| 0| 1| 0| 0| 1| 0| 0| 0| 1| 1| 0| 1| 0| 0|
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

                      X (padded with zeros)
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
         | 0| 0| 0| 1| 0| 0| 1| 0| 0| 0| 1| 1| 0| 1| 0| 0| 0| 0|
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

                 X (represented as base-w numbers)
         +--------+--------+--------+--------+--------+--------+
         |    0   |    4   |    4   |    3   |    2   |    0   |
         +--------+--------+--------+--------+--------+--------+

2.5.  Member Functions

   To simplify algorithm descriptions, we assume the existence of member
   functions.  If a complex data structure like a public key PK contains



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   a value X then getX(PK) returns the value of X for this public key.
   Accordingly, setX(PK, X, Y) sets value X in PK to the value hold by
   Y.

3.  Primitives

3.1.  WOTS+ One-Time Signatures

   This section describes the WOTS+ one-time signature system, as
   defined in [Huelsing13].  WOTS+ is a one-time signature scheme; while
   a private key can be used to sign any message, each private key MUST
   be used only once to sign a single message.  In particular, if a
   secret key is used to sign two different messages, the scheme becomes
   insecure.

   The section starts with an explanation of parameters.  Afterwards,
   the so-called chaining function, which forms the main building block
   of the WOTS+ scheme, is explained.  It follows a description of the
   algorithms for key generation, signing and verification.  Finally,
   pseudorandom key generation is discussed.

3.1.1.  WOTS+ Parameters

   WOTS+ uses the parameters m, n, and w; they all take positive integer
   values.  These parameters are summarized as follows:

      m : the message length in bytes

      n : the length, in bytes, of a secret key, public key, or
      signature element

      w : the Winternitz parameter; it is a member of the set {4, 8, 16}

   The parameters are used to compute values l, l_1 and l_2:

      l : the number of n-byte string elements in a WOTS+ secret key,
      public key, and signature.  It is computed as l = l_1 + l_2, with
      l_1 = ceil(8m/lg(w)) and l_2 = floor(lg(l_1*(w-1))/lg(w)) + 1

   The value of n is determined by the cryptographic hash function used
   for WOTS+.  The hash function is chosen to ensure an appropriate
   level of security.  The value of m is often the length of a message
   digest.  The parameter w can be chosen from the set {4,8,16}. A
   larger value of w results in shorter signatures but slower overall
   signing operations; it has little effect on security.  Choices of w
   are limited to the values 4, 8 and 16 since these values yield
   optimal trade-offs.




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3.1.1.1.  WOTS+ Hashing Functions

   The WOTS+ algorithm uses a cryptographic hash function F.  F accepts
   and returns byte strings of length n. Security requirements on F are
   discussed in Section 8.

3.1.2.  WOTS+ Chaining Function

   The chaining function (Algorithm 2) computes an iteration of F on an
   n-byte input using a vector of n-byte strings called bitmasks.  In
   each iteration, a bitmask is first XORed to an intermediate result
   before it is processed by F. In the following, bm is an array of at
   least w-2 n-byte strings (that contains the bitmasks).  The chaining
   function takes as input an n-byte string X, a start index i, a number
   of steps s, and the bitmasks bm.  The chaining function returns as
   output the value obtained by iterating F for s times on input X,
   using the bitmasks from bm starting at index i.

   Algorithm 2: Chaining Function

   if s is equal to 0 then
     return X;
   end
   if (i+s) > w-1 then
     return NULL;
   end
   byte[n] tmp = chain(X, i, s-1, bm);
   tmp = F(tmp XOR bm[i+s-1]);
   return tmp;

3.1.3.  WOTS+ Private Key

   The private key in WOTS+, denoted by sk, is a length l array of
   n-byte strings.  This private key MUST be only used to sign exactly
   one message.  Each n-byte string MUST either be selected randomly
   from the uniform distribution or using a cryptographically secure
   pseudorandom procedure.  In the latter case, the security of the used
   procedure MUST at least match that of the WOTS+ parameters used.  For
   a further discussion on pseudorandom key generation see the end of
   this section.  The following pseudocode (Algorithm 3) describes an
   algorithm for generating sk.

   Algorithm 3: Generating a WOTS+ Private Key

   for ( i = 0; i < l; i = i + 1 ) {
     set sk[i] to a uniformly random n-byte string
   }
   return sk



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3.1.4.  WOTS+ Public Key

   A WOTS+ key pair defines a virtual structure that consists of l hash
   chains of length w. The l n-byte strings in the secret key each
   define the start node for one hash chain.  The public key consists of
   the end nodes of these hash chains.  Therefore, like the secret key,
   the public key is also a length l array of n-byte strings.  To
   compute the hash chain, the chaining function (Algorithm 2) is used.
   The bitmasks have to be provided by the calling algorithm.  The same
   bitmasks are used for all chains.  The following pseudocode
   (Algorithm 4) describes an algorithm for generating the public key
   pk, where sk is the private key.

   Algorithm 4 (WOTS_genPK): Generating a WOTS+ Public Key From a
   Private Key

   for ( i = 0; i < l; i = i + 1 ) {
     pk[i] = chain(sk[i], 0, w-1, bm);
   }
   return pk;

3.1.5.  WOTS+ Signature Generation

   A WOTS+ signature is a length l array of n-byte strings.  The WOTS+
   signature is generated by mapping a message to l integers between 0
   and w - 1.  To this end, the message is transformed into base w
   numbers using the base_w function defined in Section 2.4.  Next, a
   checksum is computed and appended to the transformed message as base
   w numbers using base_w().  Each of the base w integers is used to
   select a node from a different hash chain.  The signature is formed
   by concatenating the selected nodes.  The pseudocode for signature
   generation is shown below (Algorithm 5), where M is the message and
   sig is the resulting signature.

   Algorithm 5 (WOTS_sign): Generating a signature from a private key
   and a message















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   csum = 0;
   // convert message to base w
   msg = base_w(M,w)
   // compute checksum
   for ( i = 0; i < l_1; i = i + 1 ) {
     csum = csum + w - 1 - msg[i]
   }
   // Convert csum to base w
   msg = msg || base_w(csum, w);
   for ( i = 0; i < l; i = i + 1 ) {
     sig[i] = chain(sk[i], 0, msg[i], bm)
   }
   return sig

   The data format for a signature is given below.

   WOTS+ Signature

          +---------------------------------+
          |          algorithm OID          |
          +---------------------------------+
          |                                 |
          |           sig_ots[0]            |    n bytes
          |                                 |
          +---------------------------------+
          |                                 |
          ~              ....               ~
          |                                 |
          +---------------------------------+
          |                                 |
          |          sig_ots[l-1]           |    n bytes
          |                                 |
          +---------------------------------+

3.1.6.  WOTS+ Signature Verification

   In order to verify a signature sig on a message M, the verifier
   computes a WOTS+ public key value from the signature.  This can be
   done by "completing" the chain computations starting from the
   signature values, using the base-w values of the message hash and its
   checksum.  This step, called WOTS_pkFromSig, is described below in
   Algorithm 6.  The result of WOTS_pkFromSig is then compared to the
   given public key.  If the values are equal, the signature is
   accepted.  Otherwise, the signature is rejected.

   Algorithm 6 (WOTS_pkFromSig): Computing a WOTS+ public key from a
   message and its signature




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   csum = 0;
   // convert message to base w
   msg = base_w(M,w)
   // compute checksum
   for ( i = 0; i < l_1; i = i + 1 ) {
     csum = csum + w - 1 - msg[i]
   }
   // Convert csum to base w
   msg = msg || base_w(csum, w);
   for ( i = 0; i < l; i = i + 1 ) {
     tmp_pk[i] = chain(sig[i], msg[i], w-1-msg[i], bm)
   }
   return tmp_pk

   Note: XMSS uses WOTS_pkFromSig to compute a public key value and
   delays the comparison to a later point.

3.1.7.  Pseudorandom Key Generation

   An implementation MAY use a cryptographically secure pseudorandom
   method to generate the secret key from a single n-byte value.  For
   example, the method suggested in [BDH11] and explained below MAY be
   used.  Other methods MAY be used.  The choice of a pseudorandom
   method does not affect interoperability, but the cryptographic
   strength MUST match that of the used WOTS+ parameters.

   The advantage of generating the secret key elements from a random
   n-byte string is that only this n-byte string needs to be stored
   instead of the full secret key.  The key can be regenerated when
   needed.  The suggested method from [BDH11] uses a pseudorandom
   function G(K,M) that takes an n-byte key and an n-byte message.
   During key generation a uniformly random n-byte string S is sampled
   from a secure source of randomness.  The secret key elements are
   computed as sk[i] = G(S,i) whenever needed.  The second parameter of
   G is i, represented as n-byte string in the common way.  To implement
   G, an implementation MAY use the hash function F in PRF mode.  When
   WOTS+ is used within XMSS or XMSS^MT, an implementation SHOULD use
   PRF_m, taking the first n bytes from the output.

4.  Schemes

   In this section, the extended Merkle signature scheme (XMSS) is
   described using WOTS+.  XMSS comes in two flavours: First, a single-
   tree variant (XMSS) and second a multi-tree variant (XMSS^MT).  Both
   allow combining a large number of WOTS+ key pairs under a single
   small public key.  The main ingredient added is a binary hash tree
   construction.  XMSS uses a single hash tree while XMSS^MT uses a tree
   of XMSS key pairs.



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4.1.  XMSS: eXtended Merkle Signature Scheme

   XMSS is a method for signing a potentially large but fixed number of
   messages.  It is based on the Merkle signature scheme.  XMSS uses
   four cryptographic components: WOTS+ as OTS method, two additional
   cryptographic hash functions H and H_m, and a pseudorandom function
   PRF_m.  One of the main advantages of XMSS with WOTS+ is that it does
   not rely on the collision resistance of the used hash functions but
   on weaker properties.  Each XMSS public/private key pair is
   associated with a perfect binary tree, every node of which contains
   an n-byte value.  Each tree leaf contains a special tree hash of a
   WOTS+ public key value.  Each non-leaf tree node is computed by first
   concatenating the values of its child nodes, computing the XOR with a
   bitmask, and applying the hash function H to the result.  The value
   corresponding to the root of the XMSS tree forms the XMSS public key
   together with the bitmasks.

   To generate a key pair that can be used to sign 2^h messages, a tree
   of height h is used.  XMSS is a stateful signature scheme, meaning
   that the secret key changes after every signature.  To prevent one-
   time secret keys from being used twice, the WOTS+ key pairs are
   numbered from 0 to (2^h)-1 according to the related leaf, starting
   from index 0 for the leftmost leaf.  The secret key contains an index
   that is updated after every signature, such that it contains the
   index of the next unused WOTS+ key pair.

   A signature consists of the index of the used WOTS+ key pair, the
   WOTS+ signature on the message and the so-called authentication path.
   The latter is a vector of tree nodes that allow a verifier to compute
   a value for the root of the tree.  A verifier computes the root value
   and compares it to the respective value in the XMSS public key.  If
   they match, the signature is valid.  The XMSS secret key consists of
   all WOTS+ secret keys and the actual index.  To reduce storage, a
   pseudorandom key generation procedure, as described in [BDH11], MAY
   be used.  The security of the used method MUST at least match the
   security of the XMSS instance.

4.1.1.  XMSS Parameters

   XMSS has the following parameters:

      h : the height (number of levels - 1) of the tree

      n : the length in bytes of each node

      m : the length of the message digest

      w : the Winternitz parameter as defined for WOTS+ in Section 3.1



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   There are N = 2^h leaves in the tree.  XMSS uses num_bm = max{2 * (h
   + ceil(lg(l))), w - 2} bitmasks produced during key generation.

   For XMSS and XMSS^MT, secret and public keys are denoted by SK and
   PK.  For WOTS+, secret and public keys are denoted by sk and pk,
   respectively.  XMSS and XMSS^MT signatures are denoted by Sig.  WOTS+
   signatures are denoted by sig.

4.1.2.  XMSS Hash Functions

   Besides the cryptographic hash function F required by WOTS+, XMSS
   uses three more functions:

      A cryptographic hash function H.  H accepts byte strings of length
      (2 * n) and returns an n-byte string.

      A cryptographic hash function H_m.  H_m accepts byte strings of
      arbitrary length and returns an m-byte string.

      A pseudorandom function PRF_m.  PRF_m accepts byte strings of
      arbitrary length and an m-byte key and returns an m-byte string.

4.1.3.  XMSS Private Key

   An XMSS private key contains N = 2^h WOTS+ private keys, the leaf
   index idx of the next WOTS+ private key that has not yet been used
   and SK_PRF, an m-byte key for the PRF.  The leaf index idx is
   initialized to zero when the XMSS private key is created.  The PRF
   key SK_PRF MUST be sampled from a secure source of randomness that
   follows the uniform distribution.  The WOTS+ secret keys MUST be
   generated as described in Section 3.1.  To reduce the secret key
   size, a cryptographic pseudorandom method MAY be used as discussed at
   the end of this section.  For the following algorithm descriptions,
   the existence of a method getWOTS_SK(SK,i) is assumed.  This method
   takes as inputs an XMSS secret key SK and an integer i and outputs
   the i^th WOTS+ secret key of SK.

4.1.4.  L-Trees

   To compute the leaves of the binary hash tree, a so-called L-tree is
   used.  An L-tree is an unbalanced binary hash tree, distinct but
   similar to the main XMSS binary hash tree.  The algorithm ltree
   (Algorithm 7) takes as input a WOTS+ public key pk and compresses it
   to a single n-byte value pk[0].  The algorithm uses the first (2 *
   ceil( log(l) )) of the num_bm n-byte bitmasks bm.

   Algorithm 7: ltree




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   unsigned int l' = l
   unsigned int j = 0
   while ( l' > 1 ) {
     for ( i = 0; i < floor(l' / 2); i = i + 1 ) {
       pk[i] = H((pk[2i] XOR bm[j]) || (pk[2i + 1] XOR bm[j + 1]))
     }
     if ( l' is equal to 1 % 2 ) {
       pk[floor(l' / 2) + 1] = pk[l']
     }
     l' = ceil(l' / 2)
     j = j + 2
   }
   return pk[0]

4.1.5.  TreeHash

   For the computation of the internal n-byte nodes of a Merkle tree,
   the subroutine treeHash (Algorithm 8) accepts an XMSS secret key SK,
   an unsigned integer s (the start index), an unsigned integer h (the
   target node height) and the bitmasks bm.  The treeHash algorithm
   returns the root node of a tree of height h with the leftmost leaf
   being the hash of the WOTS+ pk with index s.  The treeHash algorithm
   uses a stack holding up to (h-1) n-byte strings, with the usual stack
   functions push() and pop().

   Algorithm 8: treeHash

   for ( i = 0; i < 2^h; i = i + 1 ) {
     pk = WOTS_genPK (getWOTS_SK(SK, s+i), bm)
     node = ltree(pk, bm)
     while ( Top node on Stack has same height h' as node ) {
       node = H((Stack.pop() XOR bm[2l + 2h']) ||
                (node XOR bm[2l + 2h' + 1]))
     }
     Stack.push(node)
   }
   return Stack.pop()

4.1.6.  XMSS Public Key

   The XMSS public key is computed as described in XMSS_genPK (Algorithm
   9).  The algorithm takes the num_bm n-byte bitmasks bm, the XMSS
   secret key SK, and the tree height h.  The XMSS public key PK
   consists of the root of the binary hash tree and the bitmasks bm.

   Algorithm 9: XMSS_genPK - Generate an XMSS public key from an XMSS
   private key




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   for ( i = 0; i < num_bm; i = i + 1 ) {
      set bm[i] to a uniformly random n-byte string
   }
   root = treeHash(SK, 0, h, bm)
   PK = root || bm
   return PK

   Public and private key generation MAY be interleaved to save space.
   Especially, when a pseudorandom method is used to generate the secret
   key, generation MAY be done when the respective WOTS+ key pair is
   needed by treeHash.

   The format of an XMSS public key is given below.

   XMSS Public Key

         +---------------------------------+
         |          algorithm OID          |
         +---------------------------------+
         |                                 |
         |            root node            |     n bytes
         |                                 |
         +---------------------------------+
         |                                 |
         |              bm[0]              |     n bytes
         |                                 |
         +---------------------------------+
         |                                 |
         ~              ....               ~
         |                                 |
         +---------------------------------+
         |                                 |
         |          bm[num_bm-1]           |     n bytes
         |                                 |
         +---------------------------------+

4.1.7.  XMSS Signature

   An XMSS signature is a (4 + m + (l + h) * n)-byte string consisting
   of

      the index idx_sig of the used WOTS+ key pair (4 bytes),

      a byte string r used for randomized hashing (m bytes),

      a WOTS+ signature sig_ots (l * n bytes),





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      the so called authentication path 'auth' for the leaf associated
      with the used WOTS+ key pair (h * n bytes).

   The authentication path is an array of h n-byte strings.  It contains
   the siblings of the nodes on the path from the used leaf to the root.
   It does not contain the nodes on the path itself.  These nodes are
   needed by a verifier to compute a root node for the tree from the
   WOTS+ public key.  A node Node is addressed by its position in the
   tree.  Node(x,y) denotes the x^th node on level y with x = 0 being
   the leftmost node on a level.  The leaves are on level 0, the root is
   on level h.  An authentication path contains exactly one node on
   every layer 0 <= x <= h-1.  For the i^th WOTS+ key pair, counting
   from zero, the j^th authentication path node is

      Node(j, floor(i / (2^j)) + 1) if floor(i / (2^j)) is even or

      Node(j, floor(i / (2^j)) - 1) if floor(i / (2^j)) is odd.

   Given an XMSS secret key SK and bitmasks bm, all nodes in a tree are
   determined.  Their value is defined in terms of treeHash (Algorithm
   8):

      Node(x,y) = treeHash(SK, x * 2^y, y, bm).

   The data format for a signature is given below.

   XMSS Signature
























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          +---------------------------------+
          |          algorithm OID          |
          +---------------------------------+
          |                                 |
          |          index idx_sig          |    4 bytes
          |                                 |
          +---------------------------------+
          |                                 |
          |          randomness r           |    m bytes
          |                                 |
          +---------------------------------+
          |                                 |
          |     WOTS+ signature sig_ots     |    l * n bytes
          |                                 |
          +---------------------------------+
          |                                 |
          |             auth[0]             |    n bytes
          |                                 |
          +---------------------------------+
          |                                 |
          ~              ....               ~
          |                                 |
          +---------------------------------+
          |                                 |
          |            auth[h-1]            |    n bytes
          |                                 |
          +---------------------------------+

4.1.8.  XMSS Signature Generation

   To compute the XMSS signature of a message M with an XMSS private
   key, the signer first computes a randomized message digest.  Then a
   WOTS+ signature of the message is computed using the next unused
   WOTS+ private key.  Next, the authentication path is computed.
   Finally, the secret key is updated, i.e.  idx is incremented.  An
   implementation MUST NOT output the signature before the updated
   private key.

   The node values of the authentication path MAY be computed in any
   way.  This computation is assumed to be performed by the subroutine
   buildAuth for the function XMSS_sign, as below.  The fastest
   alternative is to store all tree nodes and set the array in the
   signature by copying them, respectively.  The least storage-intensive
   alternative is to recompute all nodes for each signature online.
   There exist several algorithms in between, with different time/
   storage trade-offs.  For an overview see [BDS09].  Note that the
   details of this procedure are not relevant to interoperability; it is
   not necessary to know any of these details in order to perform the



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   signature verification operation.  As a consequence, buildAuth is not
   specified here.

   The algorithm XMSS_sign (Algorithm 10) described below calculates an
   updated secret key SK and a signature on a message M.  XMSS_sign
   takes as inputs a message M of an arbitrary length, an XMSS secret
   key SK and bitmasks bm.  It returns the byte string containing the
   concatenation of the updated secret key SK and the signature Sig.

   Algorithm 10: XMSS_sign - Generate an XMSS signature and update the
   XMSS secret key

   idx_sig = getIdx(SK)
   auth = buildAuth(SK, bm, idx_sig)
   byte[m] r = PRF_m(getSK_PRF(SK), M)
   byte[m] M' = H_m(r || M)
   sig_ots = WOTS_sign(getWOTS_SK(SK, idx_sig), M', bm)
   Sig = (idx_sig || r || sig_ots || auth)
   setIdx(SK, idx_sig + 1)
   return (SK || Sig)

4.1.9.  XMSS Signature Verification

   An XMSS signature is verified by first computing the message digest
   using randomness r and a message M.  Then the used WOTS+ public key
   pk_ots is computed from the WOTS+ signature using WOTS_pkFromSig.
   The WOTS+ public key in turn is used to compute the corresponding
   leaf using an L-tree.  The leaf, together with index idx_sig,
   authentication path auth and bitmasks bm is used to compute an
   alternative root value for the tree.  These first steps are done by
   XMSS_rootFromSig (Algorithm 11).  The verification succeeds if and
   only if the computed root value matches the one in the XMSS public
   key.  In any other case it MUST return fail.

   The main part of XMSS signature verification is done by the function
   XMSS_rootFromSig (Algorithm 11) described below.  XMSS_rootFromSig
   takes as inputs an XMSS signature Sig, a message M, and the bitmasks
   bm.  XMSS_rootFromSig returns an n-byte string holding the value of
   the root of a tree defined by the input data.

   Algorithm 11: XMSS_rootFromSig - Compute a root node using an XMSS
   signature, a message, and bitmasks bm









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   byte[m] M' = H_m(r || M)
   pk_ots = WOTS_pkFromSig(sig_ots, M', bm)
   byte[n][2] node
   node[0] = ltree(pk_ots, bm)
   for ( k = 1; k < h; k = k + 1 ) {
     if ( floor(i / (2^k)) % 2 is equal to 0 ) {
       node[1] = H((node[0] XOR bm[2l + 2k]) ||
                    (auth[k - 1] XOR bm[2l + 2k + 1]))
     } else {
       node[1] = H((auth[k - 1] XOR bm[2l + 2k]) ||
                    (node[0] XOR bm[2l + 2k + 1]))
     }
     node[0] = node[1]
   }
   return node[0]

   The full XMSS signature verification is depicted below for
   completeness.  XMSS^MT uses only XMSS_rootFromSig and delegates the
   comparison to a later comparison of data depending on its output.

   Algorithm 12: XMSS_verify - Verify an XMSS signature using an XMSS
   signature, the corresponding XMSS public key and a message

   byte[n] node = XMSS_rootFromSig(Sig, M, getBM(PK))
   if ( node is equal to root in PK ) {
     return true
   } else {
     return false
   }

4.1.10.  Pseudorandom Key Generation

   An implementation MAY use a cryptographically secure pseudorandom
   method to generate the XMSS secret key from a single n-byte value.
   For example, the method suggested in [BDH11] and explained below MAY
   be used.  Other methods MAY be used.  The choice of a pseudorandom
   method does not affect interoperability, but the cryptographic
   strength MUST match that of the used XMSS parameters.

   For XMSS a similar method than the one used for WOTS+ can be used.
   The suggested method from [BDH11] uses a pseudorandom function G(K,M)
   that takes an n-byte key and an n-byte message.  During key
   generation a uniformly random n-byte string S is sampled from a
   secure source of randomness.  This seed S is used to generate an
   n-byte value S_ots for each WOTS+ key pair.  This n-byte value can
   then be used to compute the respective WOTS+ secret key using the
   method described in Section 3.1.7.  The seeds for the WOTS+ key pairs
   are computed as S_ots[i] = G(S,i).  The second parameter of G is the



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   index i of the WOTS+ key pair, represented as n-byte string in the
   common way.  To implement G an implementation SHOULD use PRF_m,
   taking the first n bytes from the output.  An advantage of this
   method is that a WOTS+ key can be computed using only l+1 evaluations
   of G when S is given.

4.1.11.  Free Index Handling and Partial Secret Keys

   Some applications might require to work with partial secret keys or
   copies of secret keys.  Examples include delegation of signing rights
   / proxy signatures, and load balancing.  Such applications MAY use
   their own key format and MAY use a signing algorithm different from
   the one described above.  The index in partial secret keys or copies
   of a secret key MAY be manipulated as required by the applications.
   However, applications MUST establish means that guarantee that each
   index and thereby each WOTS+ key pair is used to sign only a single
   message.

4.2.  XMSS^MT: Multi-Tree XMSS

   XMSS^MT is a method for signing a large but fixed number of messages.
   It was first described in [HRB13].  It builds on XMSS.  XMSS^MT uses
   a tree of several layers of XMSS trees.  The trees on top and
   intermediate layers are used to sign the root nodes of the trees on
   the respective layer below.  Trees on the lowest layer are used to
   sign the actual messages.  All XMSS trees have equal height.

   Consider an XMSS^MT tree of total height h that has d layers of XMSS
   trees of height h / d.  Then layer d - 1 contains one XMSS tree,
   layer d - 2 contains 2^(h / d) XMSS trees, and so on.  Finally, layer
   0 contains 2^(h - h / d) XMSS trees.

4.2.1.  XMSS^MT Parameters

   In addition to all XMSS parameters, an XMSS^MT system requires the
   number of tree layers d, specified as an integer value that divides h
   without remainder.  The same tree height h / d and the same
   Winternitz parameter w are used for all tree layers.

   All the trees on higher layers sign root nodes of other trees which
   are n-byte strings.  Hence, no message compression is needed and
   WOTS+ is used to sign the root nodes themselves instead of their hash
   values.  Hence the WOTS+ message length for these layers is n not m.
   Accordingly, the values of l_1, l_2 and l change for these layers.
   The parameters l_1_n, l_2_n, and l_n denote the respective values
   computed using n as message length for WOTS+.





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4.2.2.  XMSS Algorithms Without Message Hash

   As all XMSS trees besides those on layer 0 are used to sign short
   fixed length messages, the initial message hash can be omitted.  In
   the description below XMSS_sign_wo_hash and XMSS_rootFromSig_wo_hash
   are versions of XMSS_sign and XMSS_rootFromSig, respectively, that
   omit the initial message hash.  They are obtained by setting M' = M
   in the above algorithms.  Accordingly, the evaluations of H_m and
   PRF_m SHOULD be omitted.  This also means that no randomization
   element r for the message hash is required.  XMSS signatures
   generated by XMSS_sign_wo_hash and verified by
   XMSS_rootFromSig_wo_hash MUST NOT contain a value r.

4.2.3.  XMSS^MT Private Key

   An XMSS^MT private key SK_MT consists of one reduced XMSS private key
   for each XMSS tree.  These reduced XMSS private keys contain no
   pseudorandom function key and no index.  Instead, SK_MT contains a
   single m-byte pseudorandom function key SK_PRF and a single (ceil(h /
   8))-byte index idx_MT.  The index is a global index over all WOTS+
   key pairs of all XMSS trees on layer 0.  It is initialized with 0.
   It stores the index of the last used WOTS+ key pair on the bottom
   layer, i.e. a number between 0 and 2^h - 1.

   The algorithm descriptions below uses a function getXMSS_SK(SK, x, y)
   that outputs the reduced secret key of the x^th XMSS tree on the y^th
   layer.

4.2.4.  XMSS^MT Public Key

   The XMSS^MT public key PK_MT contains the root of the single XMSS
   tree on layer d-1 and the bitmasks.  The same bitmasks are used for
   all XMSS tress.  Algorithm 13 shows pseudocode to generate PK_MT.
   First, num_bm = max{ 2 * (h / d + ceil(lg(l))), 2 * (h / d +
   ceil(lg(l_n))), w - 2 } n-byte bitmasks bm are chosen uniformly at
   random.  The n-byte root node of the top layer tree is computed using
   treeHash.  The algorithm XMSSMT_genPK takes the XMSS^MT secret key
   SK_MT as an input and outputs an XMSS^MT public key PK_MT.

   Algorithm 13: XMSSMT_genPK - Generate an XMSS^MT public key from an
   XMSS^MT private key

   for ( i = 0; i < num_bm; i = i + 1 ) {
      set bm[i] to a uniformly random n-byte string
   }
   root = treeHash(getXMSS_SK(SK_MT, 0, d - 1), 0, h / d, bm)
   PK_MT = root || bm
   return PK_MT



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   The format of an XMSS^MT public key is given below.

   XMSS^MT Public Key

         +---------------------------------+
         |          algorithm OID          |
         +---------------------------------+
         |                                 |
         |            root node            |     n bytes
         |                                 |
         +---------------------------------+
         |                                 |
         |              bm[0]              |     n bytes
         |                                 |
         +---------------------------------+
         |                                 |
         ~              ....               ~
         |                                 |
         +---------------------------------+
         |                                 |
         |          bm[num_bm-1]           |     n bytes
         |                                 |
         +---------------------------------+

4.2.5.  XMSS^MT Signature

   An XMSS^MT signature Sig_MT is a byte string of length (ceil(h / 8) +
   m + (h + l + (d - 1) * l_n) * n).  It consists of

      the index idx_sig of the used WOTS+ key pair on the bottom layer
      (ceil(h / 8) bytes),

      a byte string r used for randomized hashing (m bytes),

      one reduced XMSS signature ((h + l) * n bytes),

      d-1 reduced XMSS signatures with message length n ((h + l_n) * n
      bytes).

   The reduced XMSS signatures contain no index idx and no byte string
   r. They only contain a WOTS+ signature sig_ots and an authentication
   path auth.  The first reduced XMSS signature contains a WOTS+
   signature that consists of l n-byte elements.  The remaining reduced
   XMSS signatures contain a WOTS+ signature on an n-byte message and
   hence consist of l_n n-byte elements.

   The data format for a signature is given below.




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   XMSS^MT signature

          +---------------------------------+
          |          algorithm OID          |
          +---------------------------------+
          |                                 |
          |          index idx_sig          |    ceil(h / 8) bytes
          |                                 |
          +---------------------------------+
          |                                 |
          |          randomness r           |    m bytes
          |                                 |
          +---------------------------------+
          |                                 |
          |  (reduced) XMSS signature Sig   |    (h + l) * n bytes
          |        (bottom layer 0)         |
          |                                 |
          +---------------------------------+
          |                                 |
          |  (reduced) XMSS signature Sig   |    (h + l_n) * n bytes
          |            (layer 1)            |
          |                                 |
          +---------------------------------+
          |                                 |
          ~              ....               ~
          |                                 |
          +---------------------------------+
          |                                 |
          |  (reduced) XMSS signature Sig   |    (h + l_n) * n bytes
          |           (layer d-1)           |
          |                                 |
          +---------------------------------+

4.2.6.  XMSS^MT Signature Generation

   To compute the XMSS^MT signature Sig_MT of a message M using an
   XMSS^MT private key SK_MT and bitmasks bm, XMSSMT_sign (Algorithm 14)
   described below uses XMSS_sign and XMSS_sign_wo_hash as defined in
   Section 4.2.2.  First, the signature index is set to idx.  Next,
   PRF_m is used to compute a pseudorandom m-byte string r.  This m-byte
   string is then used to compute a randomized message digest of length
   m.  The message digest is signed using the WOTS+ key pair on the
   bottom layer with absolute index idx.  The authentication path for
   the WOTS+ key pair is computed as well as the root of the containing
   XMSS tree.  The root is signed by the parent XMSS tree.  This is
   repeated until the top tree is reached.





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   Algorithm 14: XMSSMT_sign - Generate an XMSS^MT signature and update
   the XMSS^MT secret key

   SK_PRF = getSK_PRF(SK_MT)
   idx_sig = getIdx(SK_MT)
   setIdx(SK_MT, idx_sig + 1)
   Sig_MT = idx_sig
   unsigned int idx_tree = (h - h / d) most significant bits of idx_sig
   unsigned int idx_leaf = (h / d) least significant bits of idx_sig
   SK = idx_leaf || SK_PRF || getXMSS_SK(SK_MT, idx_tree, 0)
   Sig_tmp = XMSS_sign(M, SK, bm)
   Sig_tmp = Sig_tmp without idx
   Sig_MT = Sig_MT || Sig_tmp
   for ( j = 1; j < d; j = j + 1 ) {
      root = treeHash(SK, 0, h / d, bm)
      idx_leaf = (h / d) least significant bits of idx_tree
      idx_tree = (h - j * (h / d)) most significant bytes of idx_tree
      SK = idx_leaf || SK_PRF || getXMSS_SK(SK_MT, idx_tree, j)
      Sig_tmp = XMSS_sign_wo_hash(root, SK, bm) with idx removed
      Sig_MT = Sig_MT || Sig_tmp
   }
   return SK_MT || Sig_MT

   Algorithm 14 is only one method to compute XMSS^MT signatures.
   Especially, there exist time-memory trade-offs that allow to reduce
   the signing time to less than the signing time of an XMSS scheme with
   tree height h / d. These trade-offs prevent certain values from being
   recomputed several times by keeping a state and distribute all
   computations over all signature generations.  Details can be found in
   [Huelsing13a].

4.2.7.  XMSS^MT Signature Verification

   XMSS^MT signature verification (Algorithm 15) can be summarized as d
   XMSS signature verifications with small changes.  First, only the
   message is hashed.  The remaining XMSS signatures are on the root
   nodes of trees which have a fixed length.  Second, instead of
   comparing the computed root node to a given value, a signature on the
   root is verified.  Only the root node of the top tree is compared to
   the value in the XMSS^MT public key.  XMSSMT_verify uses
   XMSS_rootFromSig and XMSS_rootFromSig_wo_hash.  XMSSMT_verify takes
   as inputs an XMSS^MT signature Sig^MT, a message M and a public key
   PK_MT.  It outputs a boolean.

   Algorithm 15: XMSSMT_verify - Verify an XMSS^MT signature Sig_MT on a
   message M using an XMSS^MT public key PK_MT





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   idx = getIdx(Sig_MT)
   unsigned int idx_leaf = (h / d) least significant bits of idx
   unsigned int idx_tree = (h - h / d) most significant bits of idx
   Sig' = leaf || setR(Sig_MT) || getXMSSSignature(Sig, 0)
   byte[n] node = XMSS_rootFromSig(Sig', M, getBm(PK_MT))
   for ( j = 1; j < d; j = j + 1 ) {
      idx_leaf = (h / d) least significant bytes of idx_tree
      idx_tree = (h - j * h / d) most significant bytes of idx_tree
      Sig' = idx_leaf || getXMSSSignature(Sig, j)
      node = XMSS_rootFromSig_wo_hash(Sig', node, getBm(PK_MT))
   }
   if ( node is equal to getRoot(PK_MT) ) {
     return true
   } else {
     return false
   }

4.2.8.  Pseudorandom Key Generation

   Like for XMSS, an implementation MAY use a cryptographically secure
   pseudorandom method to generate the XMSS^MT secret key from a single
   n-byte value.  For example, the method explained below MAY be used.
   Other methods MAY be used.  The choice of a pseudorandom method does
   not affect interoperability, but the cryptographic strength MUST
   match that of the used XMSS parameters.

   For XMSS^MT a method similar to that for XMSS and WOTS+ can be used.
   The method uses a pseudorandom function G(K,M) that takes an n-byte
   key and an n-byte message.  During key generation a uniformly random
   n-byte string S_MT is sampled from a secure source of randomness.
   This seed S_MT is used to generate one n-byte value S for each XMSS
   key pair.  This n-byte value can be used to compute the respective
   XMSS secret key using the method described in Section 4.1.10.  Let
   S[x][y] be the seed for the x^th XMSS secret key on layer y. The
   seeds are computed as S[x][y] = G(G(S, y), x).  The second parameter
   of G is the index x (resp. level y), represented as n-byte string in
   the common way.  To implement G an implementation SHOULD use PRF_m,
   taking the first n bytes from the output.

4.2.9.  Free Index Handling and Partial Secret Keys

   The content of Section 4.1.11 also applies to XMSS^MT.









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5.  Parameter Sets

   This note provides a first basic set of parameter sets which are
   assumed to cover most relevant applicants.  Parameter sets for three
   classical security levels are defined: 128, 256 and 512 bits.
   Function output sizes are n = 16, 32 and 64 bytes and m = 32, 64,
   respectively.  While m = n is used for n = 32 and n = 64, m = 32 is
   used for the n = 16 case.  Considering quantum-computer-aided
   attacks, these output sizes yield post-quantum security of 64, 128
   and 256 bits, respectively.  The n = 16 parameter sets are included
   to encourage adoption in the pre-quantum era as they lead to smaller
   signatures and faster runtimes than other parameter sets.  The n = 64
   parameter sets are provided to support post-quantum scenarios.

   For the n = 16 setting, this note only defines parameter sets with
   AES-based hash functions.  The reason is that they benefit from
   hardware acceleration on many modern platforms.  Let AES(K,M) denote
   evaluation of AES-128 with 128 bit key K and 128 bit message M.
   Define the 16-byte string IV = 0x0001020304050607080910111213141516.
   Then F and H are implemented as

      F(X) = AES(IV,X) XOR X

      H(X) = AES( AES(IV, X1) XOR X1, X2) XOR X2

   where X = X1 || X2, i.e. X1 denotes the most significant 16 bytes of
   X and X2 the least significant 16 bytes.  For these parameter sets
   H_m is implemented as SHA3-256 and PRF_m as SHA3-256 in PRF/MAC mode.

   For the n = m = 32 and n = m = 64 settings, all functions are
   implemented using SHA3-256 and SHA3-512, respectively.

5.1.  Zero Bitmasks


















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   For applications that require a very small public key this note
   additionally defines zero bitmasks parameter sets.  For these
   parameter sets the bitmasks are set to an all-zero string.  The XMSS
   and XMSS^MT public keys for these parameter sets contain no bitmasks.
   Instead, they only contain the single n-byte value holding the root
   node.  When handling zero bitmasks parameter sets, implementations
   MAY internally use an all-zero string as bitmasks and stick to the
   same algorithms as for the other parameter sets.  Implementations MAY
   omit the XOR with an all-zero bitmask.  Zero bitmasks parameter sets
   are only defined for n = 32 and n = 64, as formal security reductions
   require the used hash functions to be collision-resistant in this
   case.  Hence, the estimated classical security levels are 128 and 256
   bits for n = 32 and n = 64 with zero bitmasks, respectively.  The
   corresponding post-quantum security levels are approximately 85 and
   170 bits, respectively.

5.2.  WOTS+ Parameters

   To fully describe a WOTS+ signature method, the parameters m, n, and
   w, as well as the function F MUST be specified.  This section defines
   several WOTS+ signature systems, each of which is identified by a
   name.  Values for l are provided for convenience.

         +------------------------+--------+----+----+----+-----+
         | Name                   | F      | m  | n  | w  | l   |
         +------------------------+--------+----+----+----+-----+
         | WOTSP_AES128_M32_W4    | AES128 | 32 | 16 | 4  | 133 |
         |                        |        |    |    |    |     |
         | WOTSP_AES128_M32_W8    | AES128 | 32 | 16 | 8  | 90  |
         |                        |        |    |    |    |     |
         | WOTSP_AES128_M32_W16   | AES128 | 32 | 16 | 16 | 67  |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-256_M32_W4  | SHA3   | 32 | 32 | 4  | 133 |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-256_M32_W8  | SHA3   | 32 | 32 | 8  | 90  |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-256_M32_W16 | SHA3   | 32 | 32 | 16 | 67  |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-512_M64_W4  | SHA3   | 64 | 64 | 4  | 261 |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-512_M64_W8  | SHA3   | 64 | 64 | 8  | 175 |
         |                        |        |    |    |    |     |
         | WOTSP_SHA3-512_M64_W16 | SHA3   | 64 | 64 | 16 | 131 |
         +------------------------+--------+----+----+----+-----+

                                  Table 1





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   Here SHA3 denotes the NIST standard hash function, also known as
   Keccak [DRAFTFIPS202].  XDR formats for WOTS+ are listed in
   Appendix A.

5.3.  XMSS Parameters

   To fully describe an XMSS signature method, the parameters m, n, w,
   and h, as well as the functions F, H, H_m and PRF_m MUST be
   specified.  This section defines several XMSS signature systems, each
   of which is identified by a name.

   The XDR formats for XMSS are listed in Appendix B.

5.3.1.  XMSS Parameters

   We first define XMSS signature methods as described in Section 4.1.
   We define parameter sets that implement the functions using AES and
   SHA3 as described above as well as pure SHA3 parameter sets.

5.3.1.1.  XMSS Parameters with AES and SHA3

   The following XMSS signature methods implement the functions F, H,
   H_m and PRF_m using AES and SHA3 as described above.

          +--------------------------+----+----+----+-----+----+
          | Name                     | m  | n  | w  | l   | h  |
          +--------------------------+----+----+----+-----+----+
          | XMSS_AES128_M32_W4_H10   | 32 | 16 | 4  | 133 | 10 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W4_H16   | 32 | 16 | 4  | 133 | 16 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W4_H20   | 32 | 16 | 4  | 133 | 20 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W8_H10   | 32 | 16 | 8  | 90  | 10 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W8_H16   | 32 | 16 | 8  | 90  | 16 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W8_H20   | 32 | 16 | 8  | 90  | 20 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W16_H10  | 32 | 16 | 16 | 67  | 10 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W16_H16  | 32 | 16 | 16 | 67  | 16 |
          |                          |    |    |    |     |    |
          | XMSS_AES128_M32_W16_H20  | 32 | 16 | 16 | 67  | 20 |
          +--------------------------+----+----+----+-----+----+

                                  Table 2




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5.3.1.2.  XMSS Parameters with SHA3

   The following XMSS signature methods implement the functions F, H,
   H_m and PRF_m solely using SHA3 as described above.

         +----------------------------+----+----+----+-----+----+
         | Name                       | m  | n  | w  | l   | h  |
         +----------------------------+----+----+----+-----+----+
         | XMSS_SHA3-256_M32_W4_H10   | 32 | 32 | 4  | 133 | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W4_H16   | 32 | 32 | 4  | 133 | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W4_H20   | 32 | 32 | 4  | 133 | 20 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W8_H10   | 32 | 32 | 8  | 90  | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W8_H16   | 32 | 32 | 8  | 90  | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W8_H20   | 32 | 32 | 8  | 90  | 20 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W16_H10  | 32 | 32 | 16 | 67  | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W16_H16  | 32 | 32 | 16 | 67  | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-256_M32_W16_H20  | 32 | 32 | 16 | 67  | 20 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W4_H10   | 64 | 64 | 4  | 261 | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W4_H16   | 64 | 64 | 4  | 261 | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W4_H20   | 64 | 64 | 4  | 261 | 20 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W8_H10   | 64 | 64 | 8  | 175 | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W8_H16   | 64 | 64 | 8  | 175 | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W8_H20   | 64 | 64 | 8  | 175 | 20 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W16_H10  | 64 | 64 | 16 | 131 | 10 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W16_H16  | 64 | 64 | 16 | 131 | 16 |
         |                            |    |    |    |     |    |
         | XMSS_SHA3-512_M64_W16_H20  | 64 | 64 | 16 | 131 | 20 |
         +----------------------------+----+----+----+-----+----+

                                  Table 3





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5.3.2.  XMSS Parameters With Empty Bitmasks

   We now define XMSS signature methods for the zero bitmasks special
   case described in Section 5.1.  For this setting all signature
   methods implement the functions F, H, H_m and PRF_m solely using SHA3
   as described above.

        +------------------------------+----+----+----+-----+----+
        | Name                         | m  | n  | w  | l   | h  |
        +------------------------------+----+----+----+-----+----+
        | XMSS_SHA3-256_M32_W4_H10_z   | 32 | 32 | 4  | 133 | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W4_H16_z   | 32 | 32 | 4  | 133 | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W4_H20_z   | 32 | 32 | 4  | 133 | 20 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W8_H10_z   | 32 | 32 | 8  | 90  | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W8_H16_z   | 32 | 32 | 8  | 90  | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W8_H20_z   | 32 | 32 | 8  | 90  | 20 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W16_H10_z  | 32 | 32 | 16 | 67  | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W16_H16_z  | 32 | 32 | 16 | 67  | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-256_M32_W16_H20_z  | 32 | 32 | 16 | 67  | 20 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W4_H10_z   | 64 | 64 | 4  | 261 | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W4_H16_z   | 64 | 64 | 4  | 261 | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W4_H20_z   | 64 | 64 | 4  | 261 | 20 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W8_H10_z   | 64 | 64 | 8  | 175 | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W8_H16_z   | 64 | 64 | 8  | 175 | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W8_H20_z   | 64 | 64 | 8  | 175 | 20 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W16_H10_z  | 64 | 64 | 16 | 131 | 10 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W16_H16_z  | 64 | 64 | 16 | 131 | 16 |
        |                              |    |    |    |     |    |
        | XMSS_SHA3-512_M64_W16_H20_z  | 64 | 64 | 16 | 131 | 20 |
        +------------------------------+----+----+----+-----+----+

                                  Table 4



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5.4.  XMSS^MT Parameters

   To fully describe an XMSS^MT signature method, the parameters m, n,
   w, h, and d, as well as the functions F, H, H_m and PRF_m MUST be
   specified.  This section defines several XMSS^MT signature systems,
   each of which is identified by a name.

   XDR formats for XMSS^MT are listed in Appendix C.

5.4.1.  XMSS^MT Parameters

   We first define XMSS^MT signature methods as described in
   Section 4.2.  We define parameter sets that implement the functions
   using AES and SHA3 as described above as well as pure SHA3 parameter
   sets.

5.4.1.1.  XMSS^MT Parameters with AES and SHA3

   The following XMSS^MT signature methods implement the functions F, H,
   H_m and PRF_m using AES and SHA3 as described above.

     +-------------------------------+----+----+----+-----+----+----+
     | Name                          | m  | n  | w  | l   | h  | d  |
     +-------------------------------+----+----+----+-----+----+----+
     | XMSSMT_AES128_M32_W4_H20_D2   | 32 | 16 | 4  | 133 | 20 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H20_D4   | 32 | 16 | 4  | 133 | 20 | 4  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H40_D2   | 32 | 16 | 4  | 133 | 40 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H40_D4   | 32 | 16 | 4  | 133 | 40 | 4  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H40_D8   | 32 | 16 | 4  | 133 | 40 | 8  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H60_D3   | 32 | 16 | 4  | 133 | 60 | 3  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H60_D6   | 32 | 16 | 4  | 133 | 60 | 6  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W4_H60_D12  | 32 | 16 | 4  | 133 | 60 | 12 |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H20_D2   | 32 | 16 | 8  | 90  | 20 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H20_D4   | 32 | 16 | 8  | 90  | 20 | 4  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H40_D2   | 32 | 16 | 8  | 90  | 40 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H40_D4   | 32 | 16 | 8  | 90  | 40 | 4  |
     |                               |    |    |    |     |    |    |



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     | XMSSMT_AES128_M32_W8_H40_D8   | 32 | 16 | 8  | 90  | 40 | 8  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H60_D3   | 32 | 16 | 8  | 90  | 60 | 3  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H60_D6   | 32 | 16 | 8  | 90  | 60 | 6  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W8_H60_D12  | 32 | 16 | 8  | 90  | 60 | 12 |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H20_D2  | 32 | 16 | 16 | 67  | 20 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H20_D4  | 32 | 16 | 16 | 67  | 20 | 4  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H40_D2  | 32 | 16 | 16 | 67  | 40 | 2  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H40_D4  | 32 | 16 | 16 | 67  | 40 | 4  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H40_D8  | 32 | 16 | 16 | 67  | 40 | 8  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H60_D3  | 32 | 16 | 16 | 67  | 60 | 3  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H60_D6  | 32 | 16 | 16 | 67  | 60 | 6  |
     |                               |    |    |    |     |    |    |
     | XMSSMT_AES128_M32_W16_H60_D12 | 32 | 16 | 16 | 67  | 60 | 12 |
     +-------------------------------+----+----+----+-----+----+----+

                                  Table 5

5.4.1.2.  XMSS^MT Parameters with SHA3

   The following XMSS^MT signature methods implement the functions F, H,
   H_m and PRF_m solely using SHA3 as described above.

    +----------------------------------+----+----+----+-----+----+----+
    | Name                             | m  | n  | w  | l   | h  | d  |
    +----------------------------------+----+----+----+-----+----+----+
    | XMSSMT_SHA3-256_M32_W4_H20_D2    | 32 | 32 | 4  | 133 | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H20_D4    | 32 | 32 | 4  | 133 | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H40_D2    | 32 | 32 | 4  | 133 | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H40_D4    | 32 | 32 | 4  | 133 | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H40_D8    | 32 | 32 | 4  | 133 | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H60_D3    | 32 | 32 | 4  | 133 | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H60_D6    | 32 | 32 | 4  | 133 | 60 | 6  |



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    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W4_H60_D12   | 32 | 32 | 4  | 133 | 60 | 12 |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H20_D2    | 32 | 32 | 8  | 90  | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H20_D4    | 32 | 32 | 8  | 90  | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H40_D2    | 32 | 32 | 8  | 90  | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H40_D4    | 32 | 32 | 8  | 90  | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H40_D8    | 32 | 32 | 8  | 90  | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H60_D3    | 32 | 32 | 8  | 90  | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H60_D6    | 32 | 32 | 8  | 90  | 60 | 6  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W8_H60_D12   | 32 | 32 | 8  | 90  | 60 | 12 |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H20_D2   | 32 | 32 | 16 | 67  | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H20_D4   | 32 | 32 | 16 | 67  | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H40_D2   | 32 | 32 | 16 | 67  | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H40_D4   | 32 | 32 | 16 | 67  | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H40_D8   | 32 | 32 | 16 | 67  | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H60_D3   | 32 | 32 | 16 | 67  | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H60_D6   | 32 | 32 | 16 | 67  | 60 | 6  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-256_M32_W16_H60_D12  | 32 | 32 | 16 | 67  | 60 | 12 |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H20_D2    | 64 | 64 | 4  | 261 | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H20_D4    | 64 | 64 | 4  | 261 | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H40_D2    | 64 | 64 | 4  | 261 | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H40_D4    | 64 | 64 | 4  | 261 | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H40_D8    | 64 | 64 | 4  | 261 | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H60_D3    | 64 | 64 | 4  | 261 | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H60_D6    | 64 | 64 | 4  | 261 | 60 | 6  |



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    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W4_H60_D12   | 64 | 64 | 4  | 261 | 60 | 12 |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H20_D2    | 64 | 64 | 8  | 175 | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H20_D4    | 64 | 64 | 8  | 175 | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H40_D2    | 64 | 64 | 8  | 175 | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H40_D4    | 64 | 64 | 8  | 175 | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H40_D8    | 64 | 64 | 8  | 175 | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H60_D3    | 64 | 64 | 8  | 175 | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H60_D6    | 64 | 64 | 8  | 175 | 60 | 6  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W8_H60_D12   | 64 | 64 | 8  | 175 | 60 | 12 |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H20_D2   | 64 | 64 | 16 | 131 | 20 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H20_D4   | 64 | 64 | 16 | 131 | 20 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H40_D2   | 64 | 64 | 16 | 131 | 40 | 2  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H40_D4   | 64 | 64 | 16 | 131 | 40 | 4  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H40_D8   | 64 | 64 | 16 | 131 | 40 | 8  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H60_D3   | 64 | 64 | 16 | 131 | 60 | 3  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H60_D6   | 64 | 64 | 16 | 131 | 60 | 6  |
    |                                  |    |    |    |     |    |    |
    | XMSSMT_SHA3-512_M64_W16_H60_D12  | 64 | 64 | 16 | 131 | 60 | 12 |
    +----------------------------------+----+----+----+-----+----+----+

                                  Table 6

5.4.2.  XMSS^MT Parameters With Empty Bitmasks

   We now define XMSS^MT signature methods for the zero bitmasks special
   case described in Section 5.1.  For this setting all signature
   methods implement the functions F, H, H_m and PRF_m solely using SHA3
   as described above.

   +-----------------------------------+----+----+----+-----+----+----+
   | Name                              | m  | n  | w  | l   | h  | d  |
   +-----------------------------------+----+----+----+-----+----+----+



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   | XMSSMT_SHA3-256_M32_W4_H20_D2_z   | 32 | 32 | 4  | 133 | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H20_D4_z   | 32 | 32 | 4  | 133 | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H40_D2_z   | 32 | 32 | 4  | 133 | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H40_D4_z   | 32 | 32 | 4  | 133 | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H40_D8_z   | 32 | 32 | 4  | 133 | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H60_D3_z   | 32 | 32 | 4  | 133 | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H60_D6_z   | 32 | 32 | 4  | 133 | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W4_H60_D12_z  | 32 | 32 | 4  | 133 | 60 | 12 |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H20_D2_z   | 32 | 32 | 8  | 90  | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H20_D4_z   | 32 | 32 | 8  | 90  | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H40_D2_z   | 32 | 32 | 8  | 90  | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H40_D4_z   | 32 | 32 | 8  | 90  | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H40_D8_z   | 32 | 32 | 8  | 90  | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H60_D3_z   | 32 | 32 | 8  | 90  | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H60_D6_z   | 32 | 32 | 8  | 90  | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W8_H60_D12_z  | 32 | 32 | 16 | 67  | 60 | 12 |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H20_D2_z  | 32 | 32 | 16 | 67  | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H20_D4_z  | 32 | 32 | 16 | 67  | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H40_D2_z  | 32 | 32 | 16 | 67  | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H40_D4_z  | 32 | 32 | 16 | 67  | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H40_D8_z  | 32 | 32 | 16 | 67  | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H60_D3_z  | 32 | 32 | 16 | 67  | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H60_D6_z  | 32 | 32 | 16 | 67  | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-256_M32_W16_H60_D12_z | 32 | 32 | 16 | 67  | 60 | 12 |
   |                                   |    |    |    |     |    |    |



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   | XMSSMT_SHA3-512_M64_W4_H20_D2_z   | 64 | 64 | 4  | 261 | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H20_D4_z   | 64 | 64 | 4  | 261 | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H40_D2_z   | 64 | 64 | 4  | 261 | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H40_D4_z   | 64 | 64 | 4  | 261 | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H40_D8_z   | 64 | 64 | 4  | 261 | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H60_D3_z   | 64 | 64 | 4  | 261 | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H60_D6_z   | 64 | 64 | 4  | 261 | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W4_H60_D12_z  | 64 | 64 | 4  | 261 | 60 | 12 |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H20_D2_z   | 64 | 64 | 8  | 175 | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H20_D4_z   | 64 | 64 | 8  | 175 | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H40_D2_z   | 64 | 64 | 8  | 175 | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H40_D4_z   | 64 | 64 | 8  | 175 | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H40_D8_z   | 64 | 64 | 8  | 175 | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H60_D3_z   | 64 | 64 | 8  | 175 | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H60_D6_z   | 64 | 64 | 8  | 175 | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W8_H60_D12_z  | 64 | 64 | 8  | 175 | 60 | 12 |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H20_D2_z  | 64 | 64 | 16 | 131 | 20 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H20_D4_z  | 64 | 64 | 16 | 131 | 20 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H40_D2_z  | 64 | 64 | 16 | 131 | 40 | 2  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H40_D4_z  | 64 | 64 | 16 | 131 | 40 | 4  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H40_D8_z  | 64 | 64 | 16 | 131 | 40 | 8  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H60_D3_z  | 64 | 64 | 16 | 131 | 60 | 3  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H60_D6_z  | 64 | 64 | 16 | 131 | 60 | 6  |
   |                                   |    |    |    |     |    |    |
   | XMSSMT_SHA3-512_M64_W16_H60_D12_z | 64 | 64 | 16 | 131 | 60 | 12 |
   +-----------------------------------+----+----+----+-----+----+----+



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                                  Table 7

6.  Rationale

   The goal of this note is to describe the WOTS+, XMSS and XMSS^MT
   algorithms following the scientific literature.  Other signature
   methods are out of scope and may be an interesting follow-on work.
   The description is done in a modular way that allows to base a
   description of stateless hash-based signature algorithms like SPHINCS
   [BHH15] on it.

   The parameter w is constrained to powers of 2 to support simpler and
   more efficient implementations.  Furthermore, w is restricted to the
   set {4, 8, 16}. No bigger values are included since the decrease in
   signature size then becomes less significant.  The value w = 2 was
   not included since w = 4 leads to similar runtimes but a halved
   signature size.  This is the case because while chains get twice as
   long, thereby increasing runtime, the number of chains is roughly
   halved.  For instance, assuming m = n = 32, one obtains l = 38 for w
   = 2 and l = 19 for w = 4.

   The signature and public key formats are designed so that they are
   easy to parse.  Each format starts with a 32-bit enumeration value
   that indicates all of the details of the signature algorithm and
   hence defines all of the information that is needed in order to parse
   the format.

   The enumeration values used in this note are palindromes, which have
   the same byte representation in either host order or network order.
   This fact allows an implementation to omit the conversion between
   byte order for those enumerations.  Note however that the idx field
   used in XMSS and XMSS^MT signatures and secret keys must be properly
   converted to and from network byte order; this is the only field that
   requires such conversion.  There are 2^32 XDR enumeration values,
   2^16 of which are palindromes, which is adequate for the foreseeable
   future.  If there is a need for more assignments, non-palindromes can
   be assigned.

7.  IANA Considerations

   The Internet Assigned Numbers Authority (IANA) is requested to create
   three registries: one for WOTS+ signatures as defined in Section 3,
   one for XMSS signatures and one for XMSS^MT signatures; the latter
   two being defined in Section 4.  For the sake of clarity and
   convenience, the first sets of WOTS+, XMSS, and XMSS^MT parameter
   sets are defined in Section 5.  Additions to these registries require
   that a specification be documented in an RFC or another permanent and
   readily available reference in sufficient details to make



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   interoperability between independent implementations possible.  Each
   entry in the registry contains the following elements:

      a short name, such as "XMSS_SHA3-512_M64_W16_H20",

      a positive number, and

      a reference to a specification that completely defines the
      signature method test cases that can be used to verify the
      correctness of an implementation.

   Requests to add an entry to the registry MUST include the name and
   the reference.  The number is assigned by IANA.  These number
   assignments SHOULD use the smallest available palindromic number.
   Submitters SHOULD have their requests reviewed by the IRTF Crypto
   Forum Research Group (CFRG) at cfrg@ietf.org.  Interested applicants
   that are unfamiliar with IANA processes should visit http://
   www.iana.org.

   The numbers between 0xDDDDDDDD (decimal 3,722,304,989) and 0xFFFFFFFF
   (decimal 4,294,967,295) inclusive, will not be assigned by IANA, and
   are reserved for private use; no attempt will be made to prevent
   multiple sites from using the same value in different (and
   incompatible) ways [RFC2434].

   The WOTS+ registry is as follows.

      +-------------------------+-------------+--------------------+
      | Name                    |  Reference  | Numeric Identifier |
      +-------------------------+-------------+--------------------+
      | WOTSP_AES128_M32_W4     | Section 5.2 |     0x01000001     |
      |                         |             |                    |
      | WOTSP_AES128_M32_W8     | Section 5.2 |     0x02000002     |
      |                         |             |                    |
      | WOTSP_AES128_M32_W16    | Section 5.2 |     0x03000003     |
      |                         |             |                    |
      | WOTSP_SHA3-256_M32_W4   | Section 5.2 |     0x04000004     |
      |                         |             |                    |
      | WOTSP_SHA3-256_M32_W8   | Section 5.2 |     0x05000005     |
      |                         |             |                    |
      | WOTSP_SHA3-256_M32_W16  | Section 5.2 |     0x06000006     |
      |                         |             |                    |
      | WOTSP_SHA3-512_M64_W4   | Section 5.2 |     0x07000007     |
      |                         |             |                    |
      | WOTSP_SHA3-512_M64_W8   | Section 5.2 |     0x08000008     |
      |                         |             |                    |
      | WOTSP_SHA3-512_M64_W16  | Section 5.2 |     0x09000009     |
      +-------------------------+-------------+--------------------+



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                                  Table 8

   The XMSS registry is as follows.

    +------------------------------+-------------+--------------------+
    | Name                         |  Reference  | Numeric Identifier |
    +------------------------------+-------------+--------------------+
    | XMSS_SHA3-256_M32_W4_H10_Z   | Section 5.3 |     0x01000001     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W4_H16_Z   | Section 5.3 |     0x02000002     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W4_H20_Z   | Section 5.3 |     0x03000003     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H10_Z   | Section 5.3 |     0x04000004     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H16_Z   | Section 5.3 |     0x05000005     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H20_Z   | Section 5.3 |     0x06000006     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H10_Z  | Section 5.3 |     0x07000007     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H16_Z  | Section 5.3 |     0x08000008     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H20_Z  | Section 5.3 |     0x09000009     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H10_Z   | Section 5.3 |     0x0a00000a     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H16_Z   | Section 5.3 |     0x0b00000b     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H20_Z   | Section 5.3 |     0x0c00000c     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H10_Z   | Section 5.3 |     0x0d00000d     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H16_Z   | Section 5.3 |     0x0e00000e     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H20_Z   | Section 5.3 |     0x0f00000f     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H10_Z  | Section 5.3 |     0x01010101     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H16_Z  | Section 5.3 |     0x02010102     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H20_Z  | Section 5.3 |     0x03010103     |
    |                              |             |                    |
    | XMSS_AES128_M32_W4_H10       | Section 5.3 |     0x04010104     |
    |                              |             |                    |
    | XMSS_AES128_M32_W4_H16       | Section 5.3 |     0x05010105     |
    |                              |             |                    |
    | XMSS_AES128_M32_W4_H20       | Section 5.3 |     0x06010106     |



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    |                              |             |                    |
    | XMSS_AES128_M32_W8_H10       | Section 5.3 |     0x07010107     |
    |                              |             |                    |
    | XMSS_AES128_M32_W8_H16       | Section 5.3 |     0x08010108     |
    |                              |             |                    |
    | XMSS_AES128_M32_W8_H20       | Section 5.3 |     0x09010109     |
    |                              |             |                    |
    | XMSS_AES128_M32_W16_H10      | Section 5.3 |     0x0a01010a     |
    |                              |             |                    |
    | XMSS_AES128_M32_W16_H16      | Section 5.3 |     0x0b01010b     |
    |                              |             |                    |
    | XMSS_AES128_M32_W16_H20      | Section 5.3 |     0x0c01010c     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W4_H10     | Section 5.3 |     0x0d01010d     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W4_H16     | Section 5.3 |     0x0e01010e     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W4_H20     | Section 5.3 |     0x0f01010f     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H10     | Section 5.3 |     0x01020201     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H16     | Section 5.3 |     0x02020202     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W8_H20     | Section 5.3 |     0x03020203     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H10    | Section 5.3 |     0x04020204     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H16    | Section 5.3 |     0x05020205     |
    |                              |             |                    |
    | XMSS_SHA3-256_M32_W16_H20    | Section 5.3 |     0x06020206     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H10     | Section 5.3 |     0x07020207     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H16     | Section 5.3 |     0x08020208     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W4_H20     | Section 5.3 |     0x09020209     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H10     | Section 5.3 |     0x0a02020a     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H16     | Section 5.3 |     0x0b02020b     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W8_H20     | Section 5.3 |     0x0c02020c     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H10    | Section 5.3 |     0x0d02020d     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H16    | Section 5.3 |     0x0e02020e     |
    |                              |             |                    |
    | XMSS_SHA3-512_M64_W16_H20    | Section 5.3 |     0x0f02020f     |



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    +------------------------------+-------------+--------------------+

                                  Table 9

   The XMSS^MT registry is as follows.

   +---------------------------------------+------------+--------------+
   | Name                                  | Reference  |   Numeric    |
   |                                       |            |  Identifier  |
   +---------------------------------------+------------+--------------+
   | XMSSMT_SHA3-256_M32_W4_H20_D2_Z       |  Section   |  0x01000001  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H20_D4_Z       |  Section   |  0x02000002  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D2_Z       |  Section   |  0x03000003  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D4_Z       |  Section   |  0x04000004  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D8_Z       |  Section   |  0x05000005  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D3_Z       |  Section   |  0x06000006  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D6_Z       |  Section   |  0x07000007  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D12_Z      |  Section   |  0x08000008  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H20_D2_Z       |  Section   |  0x09000009  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H20_D4_Z       |  Section   |  0x0a00000a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D2_Z       |  Section   |  0x0b00000b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D4_Z       |  Section   |  0x0c00000c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D8_Z       |  Section   |  0x0d00000d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D3_Z       |  Section   |  0x0e00000e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D6_Z       |  Section   |  0x0f00000f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D12_Z      |  Section   |  0x00010100  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H20_D2_Z      |  Section   |  0x01010101  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H20_D4_Z      |  Section   |  0x02010102  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D2_Z      |  Section   |  0x03010103  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D4_Z      |  Section   |  0x04010104  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D8_Z      |  Section   |  0x05010105  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D3_Z      |  Section   |  0x06010106  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D6_Z      |  Section   |  0x07010107  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D12_Z     |  Section   |  0x08010108  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H20_D2_Z       |  Section   |  0x09010109  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H20_D4_Z       |  Section   |  0x0a01010a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D2_Z       |  Section   |  0x0b01010b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D4_Z       |  Section   |  0x0c01010c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D8_Z       |  Section   |  0x0d01010d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D3_Z       |  Section   |  0x0e01010e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D6_Z       |  Section   |  0x0f01010f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D12_Z      |  Section   |  0x00020200  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H20_D2_Z       |  Section   |  0x01020201  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H20_D4_Z       |  Section   |  0x02020202  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D2_Z       |  Section   |  0x03020203  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D4_Z       |  Section   |  0x04020204  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D8_Z       |  Section   |  0x05020205  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D3_Z       |  Section   |  0x06020206  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D6_Z       |  Section   |  0x07020207  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D12_Z      |  Section   |  0x08020208  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H20_D2_Z      |  Section   |  0x09020209  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H20_D4_Z      |  Section   |  0x0a02020a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D2_Z      |  Section   |  0x0b02020b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D4_Z      |  Section   |  0x0c02020c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D8_Z      |  Section   |  0x0d02020d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D3_Z      |  Section   |  0x0e02020e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D6_Z      |  Section   |  0x0f02020f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D12_Z     |  Section   |  0x00030300  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H20_D2           |  Section   |  0x01030301  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H20_D4           |  Section   |  0x02030302  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H40_D2           |  Section   |  0x03030303  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H40_D4           |  Section   |  0x04030304  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H40_D8           |  Section   |  0x05030305  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H60_D3           |  Section   |  0x06030306  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H60_D6           |  Section   |  0x07030307  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W4_H60_D12          |  Section   |  0x08030308  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H20_D2           |  Section   |  0x09030309  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H20_D4           |  Section   |  0x0a03030a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H40_D2           |  Section   |  0x0b03030b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H40_D4           |  Section   |  0x0c03030c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H40_D8           |  Section   |  0x0d03030d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H60_D3           |  Section   |  0x0e03030e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H60_D6           |  Section   |  0x0f03030f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W8_H60_D12          |  Section   |  0x00040400  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H20_D2          |  Section   |  0x01040401  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H20_D4          |  Section   |  0x02040402  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H40_D2          |  Section   |  0x03040403  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H40_D4          |  Section   |  0x04040404  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H40_D8          |  Section   |  0x05040405  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H60_D3          |  Section   |  0x06040406  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H60_D6          |  Section   |  0x07040407  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_AES128_M32_W16_H60_D12         |  Section   |  0x08040408  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H20_D2         |  Section   |  0x09040409  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H20_D4         |  Section   |  0x0a04040a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D2         |  Section   |  0x0b04040b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D4         |  Section   |  0x0c04040c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H40_D8         |  Section   |  0x0d04040d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D3         |  Section   |  0x0e04040e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D6         |  Section   |  0x0f04040f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W4_H60_D12        |  Section   |  0x00050500  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H20_D2         |  Section   |  0x01050501  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H20_D4         |  Section   |  0x02050502  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D2         |  Section   |  0x03050503  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D4         |  Section   |  0x04050504  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H40_D8         |  Section   |  0x05050505  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D3         |  Section   |  0x06050506  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D6         |  Section   |  0x07050507  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W8_H60_D12        |  Section   |  0x08050508  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H20_D2        |  Section   |  0x09050509  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H20_D4        |  Section   |  0x0a05050a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D2        |  Section   |  0x0b05050b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D4        |  Section   |  0x0c05050c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H40_D8        |  Section   |  0x0d05050d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D3        |  Section   |  0x0e05050e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D6        |  Section   |  0x0f05050f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-256_M32_W16_H60_D12       |  Section   |  0x00060600  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H20_D2         |  Section   |  0x01060601  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H20_D4         |  Section   |  0x02060602  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D2         |  Section   |  0x03060603  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D4         |  Section   |  0x04060604  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H40_D8         |  Section   |  0x05060605  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D3         |  Section   |  0x06060606  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D6         |  Section   |  0x07060607  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W4_H60_D12        |  Section   |  0x08060608  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H20_D2         |  Section   |  0x09060609  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H20_D4         |  Section   |  0x0a06060a  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D2         |  Section   |  0x0b06060b  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D4         |  Section   |  0x0c06060c  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H40_D8         |  Section   |  0x0d06060d  |
   |                                       |    5.4     |              |



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   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D3         |  Section   |  0x0e06060e  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D6         |  Section   |  0x0f06060f  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W8_H60_D12        |  Section   |  0x00070700  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H20_D2        |  Section   |  0x01070701  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H20_D4        |  Section   |  0x02070702  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D2        |  Section   |  0x03070703  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D4        |  Section   |  0x04070704  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H40_D8        |  Section   |  0x05070705  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D3        |  Section   |  0x06070706  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D6        |  Section   |  0x07070707  |
   |                                       |    5.4     |              |
   |                                       |            |              |
   | XMSSMT_SHA3-512_M64_W16_H60_D12       |  Section   |  0x08070708  |
   |                                       |    5.4     |              |
   +---------------------------------------+------------+--------------+

                                 Table 10

   An IANA registration of a signature system does not constitute an
   endorsement of that system or its security.

8.  Security Considerations










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   A signature system is considered secure if it prevents an attacker
   from forging a valid signature.  More specifically, consider a
   setting in which an attacker gets a public key and can learn
   signatures on arbitrary messages of his choice.  A signature system
   is secure if, even in this setting, the attacker can not produce a
   message signature pair of his choosing such that the verification
   algorithm accepts.

   Preventing an attacker from mounting an attack means that the attack
   is computationally too expensive to be carried out.  There exist
   various estimates when a computation is too expensive to be done.
   For that reason, this note only describes how expensive it is for an
   attacker to generate a forgery.  Parameters are accompanied by a bit
   security value.  The meaning of bit security is as follows.  A
   parameter set grants b bits of security if the best attack takes at
   least 2^(b-1) bit operations to achieve a success probability of 1/2.
   Hence, to mount a successful attack, an attacker needs to perform 2^b
   bit operations on average.  How the given values for bit security
   were estimated is described below.

8.1.  Security Proofs

   There exist formal security proofs for the schemes described here in
   the literature [Huelsing13a].  These proofs show that an attacker has
   to break at least one out of certain security properties of the used
   hash functions and PRFs to forge a signature.  The proofs in
   [Huelsing13a] do not consider the initial message compression.  For
   the scheme without initial message compression, these proofs show
   that an attacker has to break certain minimal security properties.
   In particular, it is not sufficient to break the collision resistance
   of the hash functions to generate a forgery.

   It is a folklore that one can securely combine a secure signature
   scheme for fixed length messages with an initial message digest.  It
   is easy to proof that an attacker either must break the security of
   the fixed-input-length signature scheme or the collision resistance
   of the used hash function.  XMSS and XMSS^MT use a known trick to
   prevent the applicability of collision attacks.  Namely, the schemes
   use a randomized message hash.  For technical reasons, it is not
   possible to formally prove that the resulting scheme is secure if the
   hash function is not collision-resistant but fulfills some weaker
   security properties.

   The given bit security values were estimated based on the complexity
   of the best known generic attacks against the required security
   properties of the used hash functions and PRFs.





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8.2.  Security Assumptions

   The security assumptions made to argue for the security of the
   described schemes are minimal.  Any signature algorithm that allows
   arbitrary size messages relies on the security of a cryptographic
   hash function.  For the schemes described here this is already
   sufficient to be secure.  In contrast, common signature schemes like
   RSA, DSA, and ECDSA additionally rely on the conjectured hardness of
   certain mathematical problems.

8.3.  Post-Quantum Security

   A post-quantum cryptosystem is a system that is secure against
   attackers with access to a reasonably sized quantum computer.  At the
   time of writing this note, whether or not it is feasible to build
   such machine is an open conjecture.  However, significant progress
   was made over the last few years in this regard.

   In contrast to RSA, DSA, and ECDSA, the described signature systems
   are post-quantum-secure if they are used with an appropriate
   cryptographic hash function.  In particular, for post-quantum
   security, the size of m and n must be twice the size required for
   classical security.  This is in order to protect against quantum
   square root attacks due to Grover's algorithm.  It has been shown
   that Grover's algorithm is optimal for finding preimages and
   collisions.

9.  Acknowledgements

   We would like to thank Burt Kaliski, and David McGrew for their help.

10.  References

10.1.  Normative References

   [DRAFTFIPS202]
              National Institute of Standards and Technology, "SHA-3
              Standard: Permutation-Based Hash and Extendable-Output
              Functions", Draft FIPS 202, 2014.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC2434]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", BCP 26, RFC 2434,
              October 1998.





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   [RFC4506]  Eisler, M., "XDR: External Data Representation Standard",
              STD 67, RFC 4506, May 2006.

10.2.  Informative References

   [BDH11]    Buchmann, J., Dahmen, E., and A. Huelsing, "XMSS - A
              Practical Forward Secure Signature Scheme Based on Minimal
              Security Assumptions", Lecture Notes in Computer Science
              volume 7071. Post-Quantum Cryptography, 2011.

   [BDS09]    Buchmann, J., Dahmen, E., and M. Szydlo, "Hash-based
              Digital Signature Schemes", Book chapter Post-Quantum
              Cryptography, Springer, 2009.

   [BHH15]    Bernstein, D., Hopwood, D., Huelsing, A., Lange, T.,
              Niederhagen, R., Papachristodoulou, L., Schneider, M.,
              Schwabe, P., and Z. Wilcox-O'Hearn, "SPHINCS: practical
              stateless hash-based signatures", To appear. Advances in
              Cryptology - EUROCRYPT, 2015.

   [DC14]     McGrew, D. and M. Curcio, "Hash-based signatures", draft-
              mcgrew-hash-sigs-02 (work in progress), July 2014.

   [HRB13]    Huelsing, A., Rausch, L., and J. Buchmann, "Optimal
              Parameters for XMSS^MT", Lecture Notes in Computer Science
              volume 8128. CD-ARES, 2013.

   [Huelsing13]
              Huelsing, A., "W-OTS+ - Shorter Signatures for Hash-Based
              Signature Schemes", Lecture Notes in Computer Science
              volume 7918. Progress in Cryptology - AFRICACRYPT, 2013.

   [Huelsing13a]
              Huelsing, A., "Practical Forward Secure Signatures using
              Minimal Security Assumptions", PhD thesis TU Darmstadt,
              2013.

   [Kaliski15]
              Kaliski, B., "Shoring up the Infrastructure: A Strategy
              for Standardizing Hash Signatures", Post Quantum NIST
              Workshop on Cybersecurity in a Post-Quantum World, 2015.

   [Merkle79]
              Merkle, R., "Secrecy, Authentication, and Public Key
              Systems", Stanford University Information Systems
              Laboratory Technical Report 1979-1, 1979.





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Appendix A.  WOTS+ XDR Formats

   The WOTS+ signature and public key formats are formally defined using
   XDR [RFC4506] in order to provide an unambiguous, machine readable
   definition.  Though XDR is used, these formats are simple and easy to
   parse without any special tools.  To avoid the need to convert to and
   from network / host byte order, the enumeration values are all
   palindromes.

   WOTS+ parameter sets are defined using XDR syntax as follows:


      /* ots_algorithm_type identifies a particular
         signature algorithm */

      enum ots_algorithm_type {
        wotsp_reserved         = 0x00000000,
        wotsp_aes128_m32_w4    = 0x01000001,
        wotsp_aes128_m32_w8    = 0x02000002,
        wotsp_aes128_m32_w16   = 0x03000003,
        wotsp_sha3-256_m32_w4  = 0x04000004,
        wotsp_sha3-256_m32_w8  = 0x05000005,
        wotsp_sha3-256_m32_w16 = 0x06000006,
        wotsp_sha3-512_m64_w4  = 0x07000007,
        wotsp_sha3-512_m64_w8  = 0x08000008,
        wotsp_sha3-512_m64_w16 = 0x09000009,
      };


   WOTS+ signatures are defined using XDR syntax as follows:





















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      /* Byte strings */

      typedef opaque bytestring32[32];
      typedef opaque bytestring64[64];

      union ots_signature switch (ots_algorithm_type type) {
        case wotsp_aes128_m32_w4:
        case wotsp_sha3-256_m32_w4:
          bytestring32 ots_sig_m32_l133[133];

        case wotsp_aes128_m32_w8:
        case wotsp_sha3-256_m32_w8:
          bytestring32 ots_sig_m32_l90[90];

        case wotsp_aes128_m32_w16:
        case wotsp_sha3-256_m32_w16:
          bytestring32 ots_sig_m32_l67[67];

        case wotsp_sha3-512_m64_w4:
          bytestring64 ots_sig_m64_l261[261];

        case wotsp_sha3-512_m64_w8:
          bytestring64 ots_sig_m64_l75[175];

        case wotsp_sha3-512_m64_w16:
          bytestring64 ots_sig_m64_l18[131];

        default:
          void;   /* error condition */
      };


   WOTS+ public keys are defined using XDR syntax as follows:


















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      union ots_pubkey switch (ots_algorithm_type type) {
        case wotsp_aes128_m32_w4:
        case wotsp_sha3-256_m32_w4:
          bytestring32 ots_pubk_m32_l133[133];

        case wotsp_aes128_m32_w8:
        case wotsp_sha3-256_m32_w8:
          bytestring32 ots_pubk_m32_l90[90];

        case wotsp_aes128_m32_w16:
        case wotsp_sha3-256_m32_w16:
          bytestring32 ots_pubk_m32_l67[67];

        case wotsp_sha3-512_m64_w4:
          bytestring64 ots_pubk_m64_l261[261];

        case wotsp_sha3-512_m64_w8:
          bytestring64 ots_pubk_m64_l75[175];

        case wotsp_sha3-512_m64_w16:
          bytestring64 ots_pubk_m64_l18[131];

        default:
          void;   /* error condition */
      };


Appendix B.  XMSS XDR Formats

   XMSS parameter sets are defined using XDR syntax as follows:


      /* Byte strings */

      typedef opaque bytestring4[4];
      typedef opaque bytestring16[16];

      /* Definition of parameter sets */

      enum xmss_algorithm_type {
        xmss_reserved                 = 0x00000000,

        /* Empty bitmasks */

        /* 128 bit classical security, 85 bit post-quantum security */

        xmss_sha3-256_m32_w4_h10_z  = 0x01000001,
        xmss_sha3-256_m32_w4_h16_z  = 0x02000002,



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        xmss_sha3-256_m32_w4_h20_z  = 0x03000003,

        xmss_sha3-256_m32_w8_h10_z  = 0x04000004,
        xmss_sha3-256_m32_w8_h16_z  = 0x05000005,
        xmss_sha3-256_m32_w8_h20_z  = 0x06000006,

        xmss_sha3-256_m32_w16_h10_z = 0x07000007,
        xmss_sha3-256_m32_w16_h16_z = 0x08000008,
        xmss_sha3-256_m32_w16_h20_z = 0x09000009,

        /* 256 bit classical security, 170 bit post-quantum security */

        xmss_sha3-512_m64_w4_h10_z  = 0x0a00000a,
        xmss_sha3-512_m64_w4_h16_z  = 0x0b00000b,
        xmss_sha3-512_m64_w4_h20_z  = 0x0c00000c,

        xmss_sha3-512_m64_w8_h10_z  = 0x0d00000d,
        xmss_sha3-512_m64_w8_h16_z  = 0x0e00000e,
        xmss_sha3-512_m64_w8_h20_z  = 0x0f00000f,

        xmss_sha3-512_m64_w16_h10_z = 0x01010101,
        xmss_sha3-512_m64_w16_h16_z = 0x02010102,
        xmss_sha3-512_m64_w16_h20_z = 0x03010103,

        /* Non-empty bitmasks */

        /* 128 bit classical security, 64 bit post-quantum security */

        xmss_aes128_m32_w4_h10        = 0x04010104,
        xmss_aes128_m32_w4_h16        = 0x05010105,
        xmss_aes128_m32_w4_h20        = 0x06010106,

        xmss_aes128_m32_w8_h10        = 0x07010107,
        xmss_aes128_m32_w8_h16        = 0x08010108,
        xmss_aes128_m32_w8_h20        = 0x09010109,

        xmss_aes128_m32_w16_h10       = 0x0a01010a,
        xmss_aes128_m32_w16_h16       = 0x0b01010b,
        xmss_aes128_m32_w16_h20       = 0x0c01010c,

        /* 256 bit classical security, 128 bit post-quantum security */

        xmss_sha3-256_m32_w4_h10      = 0x0d01010d,
        xmss_sha3-256_m32_w4_h16      = 0x0e01010e,
        xmss_sha3-256_m32_w4_h20      = 0x0f01010f,

        xmss_sha3-256_m32_w8_h10      = 0x01020201,
        xmss_sha3-256_m32_w8_h16      = 0x02020202,



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        xmss_sha3-256_m32_w8_h20      = 0x03020203,

        xmss_sha3-256_m32_w16_h10     = 0x04020204,
        xmss_sha3-256_m32_w16_h16     = 0x05020205,
        xmss_sha3-256_m32_w16_h20     = 0x06020206,

        /* 512 bit classical security, 256 bit post-quantum security */

        xmss_sha3-512_m64_w4_h10      = 0x07020207,
        xmss_sha3-512_m64_w4_h16      = 0x08020208,
        xmss_sha3-512_m64_w4_h20      = 0x09020209,

        xmss_sha3-512_m64_w8_h10      = 0x0a02020a,
        xmss_sha3-512_m64_w8_h16      = 0x0b02020b,
        xmss_sha3-512_m64_w8_h20      = 0x0c02020c,

        xmss_sha3-512_m64_w16_h10     = 0x0d02020d,
        xmss_sha3-512_m64_w16_h16     = 0x0e02020e,
        xmss_sha3-512_m64_w16_h20     = 0x0f02020f,
      };


   XMSS signatures are defined using XDR syntax as follows:


      /* Authentication path types */

      union xmss_path switch (xmss_algorithm_type type) {
        case xmss_sha3-256_m32_w4_h10_z:
        case xmss_sha3-256_m32_w8_h10_z:
        case xmss_sha3-256_m32_w16_h10_z:
        case xmss_sha3-256_m32_w4_h10:
        case xmss_sha3-256_m32_w8_h10:
        case xmss_sha3-256_m32_w16_h10:
          bytestring32 path_n32_t10[10];

        case xmss_sha3-256_m32_w4_h16_z:
        case xmss_sha3-256_m32_w8_h16_z:
        case xmss_sha3-256_m32_w16_h16_z:
        case xmss_sha3-256_m32_w4_h16:
        case xmss_sha3-256_m32_w8_h16:
        case xmss_sha3-256_m32_w16_h16:
          bytestring32 path_n32_t16[16];

        case xmss_sha3-256_m32_w4_h20_z:
        case xmss_sha3-256_m32_w8_h20_z:
        case xmss_sha3-256_m32_w16_h20_z:
        case xmss_sha3-256_m32_w4_h20:



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        case xmss_sha3-256_m32_w8_h20:
        case xmss_sha3-256_m32_w16_h20:
          bytestring32 path_n32_t20[20];

        case xmss_sha3-512_m64_w4_h10_z:
        case xmss_sha3-512_m64_w8_h10_z:
        case xmss_sha3-512_m64_w16_h10_z:
        case xmss_sha3-512_m64_w4_h10:
        case xmss_sha3-512_m64_w8_h10:
        case xmss_sha3-512_m64_w16_h10:
          bytestring64 path_n64_t10[10];

        case xmss_sha3-512_m64_w4_h16_z:
        case xmss_sha3-512_m64_w8_h16_z:
        case xmss_sha3-512_m64_w16_h16_z:
        case xmss_sha3-512_m64_w4_h16:
        case xmss_sha3-512_m64_w8_h16:
        case xmss_sha3-512_m64_w16_h16:
          bytestring64 path_n64_t16[16];

        case xmss_sha3-512_m64_w4_h20_z:
        case xmss_sha3-512_m64_w8_h20_z:
        case xmss_sha3-512_m64_w16_h20_z:
        case xmss_sha3-512_m64_w4_h20:
        case xmss_sha3-512_m64_w8_h20:
        case xmss_sha3-512_m64_w16_h20:
          bytestring64 path_n64_t20[20];

        case xmss_aes128_m32_w4_h10:
        case xmss_aes128_m32_w8_h10:
        case xmss_aes128_m32_w16_h10:
          bytestring16 path_n16_t10[10];

        case xmss_aes128_m32_w4_h16:
        case xmss_aes128_m32_w8_h16:
        case xmss_aes128_m32_w16_h16:
          bytestring16 path_n16_t16[16];

        case xmss_aes128_m32_w4_h20:
        case xmss_aes128_m32_w8_h20:
        case xmss_aes128_m32_w16_h20:
          bytestring16 path_n16_t20[20];

        default:
          void;     /* error condition */
      };

      /* Types for XMSS random strings */



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      union random_string_xmss switch (xmss_algorithm_type type) {
        case xmss_sha3-256_m32_w4_h10_z:
        case xmss_sha3-256_m32_w4_h16_z:
        case xmss_sha3-256_m32_w4_h20_z:
        case xmss_sha3-256_m32_w8_h10_z:
        case xmss_sha3-256_m32_w8_h16_z:
        case xmss_sha3-256_m32_w8_h20_z:
        case xmss_sha3-256_m32_w16_h10_z:
        case xmss_sha3-256_m32_w16_h16_z:
        case xmss_sha3-256_m32_w16_h20_z:
        case xmss_sha3-256_m32_w4_h10:
        case xmss_sha3-256_m32_w4_h16:
        case xmss_sha3-256_m32_w4_h20:
        case xmss_sha3-256_m32_w8_h10:
        case xmss_sha3-256_m32_w8_h16:
        case xmss_sha3-256_m32_w8_h20:
        case xmss_sha3-256_m32_w16_h10:
        case xmss_sha3-256_m32_w16_h16:
        case xmss_sha3-256_m32_w16_h20:
        case xmss_aes128_m32_w4_h10:
        case xmss_aes128_m32_w4_h16:
        case xmss_aes128_m32_w4_h20:
        case xmss_aes128_m32_w8_h10:
        case xmss_aes128_m32_w8_h16:
        case xmss_aes128_m32_w8_h20:
        case xmss_aes128_m32_w16_h10:
        case xmss_aes128_m32_w16_h16:
        case xmss_aes128_m32_w16_h20:
          bytestring32 rand_m32;

        case xmss_sha3-512_m64_w4_h10_z:
        case xmss_sha3-512_m64_w4_h16_z:
        case xmss_sha3-512_m64_w4_h20_z:
        case xmss_sha3-512_m64_w8_h10_z:
        case xmss_sha3-512_m64_w8_h16_z:
        case xmss_sha3-512_m64_w8_h20_z:
        case xmss_sha3-512_m64_w16_h10_z:
        case xmss_sha3-512_m64_w16_h16_z:
        case xmss_sha3-512_m64_w16_h20_z:
        case xmss_sha3-512_m64_w4_h10:
        case xmss_sha3-512_m64_w4_h16:
        case xmss_sha3-512_m64_w4_h20:
        case xmss_sha3-512_m64_w8_h10:
        case xmss_sha3-512_m64_w8_h16:
        case xmss_sha3-512_m64_w8_h20:
        case xmss_sha3-512_m64_w16_h10:
        case xmss_sha3-512_m64_w16_h16:
        case xmss_sha3-512_m64_w16_h20:



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          bytestring64 rand_m64;

        default:
          void;     /* error condition */
      };

      /* Corresponding WOTS+ type for given XMSS type */

      union xmss_ots_signature switch (xmss_algorithm_type type) {
        case xmss_sha3-256_m32_w4_h10_z:
        case xmss_sha3-256_m32_w4_h16_z:
        case xmss_sha3-256_m32_w4_h20_z:
          wotsp_sha3-256_m32_w4;

        case xmss_sha3-256_m32_w8_h10_z:
        case xmss_sha3-256_m32_w8_h16_z:
        case xmss_sha3-256_m32_w8_h20_z:
          wotsp_sha3-256_m32_w8;

        case xmss_sha3-256_m32_w16_h10_z:
        case xmss_sha3-256_m32_w16_h16_z:
        case xmss_sha3-256_m32_w16_h20_z:
          wotsp_sha3-256_m32_w16

        case xmss_sha3-512_m64_w4_h10_z:
        case xmss_sha3-512_m64_w4_h16_z:
        case xmss_sha3-512_m64_w4_h20_z:
          wotsp_sha3-512_m64_w4;

        case xmss_sha3-512_m64_w8_h10_z:
        case xmss_sha3-512_m64_w8_h16_z:
        case xmss_sha3-512_m64_w8_h20_z:
          wotsp_sha3-512_m64_w8;

        case xmss_sha3-512_m64_w16_h10_z:
        case xmss_sha3-512_m64_w16_h16_z:
        case xmss_sha3-512_m64_w16_h20_z:
          wotsp_sha3-512_m64_w16;

        case xmss_aes128_m32_w4_h10:
        case xmss_aes128_m32_w4_h16:
        case xmss_aes128_m32_w4_h20:
          wotsp_aes128_m32_w4;

        case xmss_aes128_m32_w8_h10:
        case xmss_aes128_m32_w8_h16:
        case xmss_aes128_m32_w8_h20:
          wotsp_aes128_m32_w8;



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        case xmss_aes128_m32_w16_h10:
        case xmss_aes128_m32_w16_h16:
        case xmss_aes128_m32_w16_h20:
          wotsp_aes128_m32_w16;

        case xmss_sha3-256_m32_w4_h10:
        case xmss_sha3-256_m32_w4_h16:
        case xmss_sha3-256_m32_w4_h20:
          wotsp_sha3-256_m32_w4;

        case xmss_sha3-256_m32_w8_h10:
        case xmss_sha3-256_m32_w8_h16:
        case xmss_sha3-256_m32_w8_h20:
          wotsp_sha3-256_m32_w8;

        case xmss_sha3-256_m32_w16_h10:
        case xmss_sha3-256_m32_w16_h16:
        case xmss_sha3-256_m32_w16_h20:
          wotsp_sha3-256_m32_w16;

        case xmss_sha3-512_m64_w4_h10:
        case xmss_sha3-512_m64_w4_h16:
        case xmss_sha3-512_m64_w4_h20:
          wotsp_sha3-512_m64_w4;

        case xmss_sha3-512_m64_w8_h10:
        case xmss_sha3-512_m64_w8_h16:
        case xmss_sha3-512_m64_w8_h20:
          wotsp_sha3-512_m64_w8;

        case xmss_sha3-512_m64_w16_h10:
        case xmss_sha3-512_m64_w16_h16:
        case xmss_sha3-512_m64_w16_h20:
          wotsp_sha3-512_m64_w16;

        default:
          void;     /* error condition */
      };

      /* XMSS signature structure */

      struct xmss_signature {
        /* WOTS+ key pair index */
        bytestring4 idx_sig;
        /* Random string for randomized hashing */
        random_string_xmss rand_string;
        /* WOTS+ signature */
        xmss_ots_signature sig_ots;



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        /* authentication path */
        xmss_path nodes;
      };


   When no bitmasks are used, XMSS public keys are defined using XDR
   syntax as follows:


      /* Types for XMSS root node */

      union xmss_root switch (xmss_algorithm_type type) {
        case xmss_sha3-256_m32_w4_h10_z:
        case xmss_sha3-256_m32_w4_h16_z:
        case xmss_sha3-256_m32_w4_h20_z:
        case xmss_sha3-256_m32_w8_h10_z:
        case xmss_sha3-256_m32_w16_h10_z:
        case xmss_sha3-256_m32_w8_h16_z:
        case xmss_sha3-256_m32_w16_h16_z:
        case xmss_sha3-256_m32_w8_h20_z:
        case xmss_sha3-256_m32_w16_h20_z:
          bytestring32 root_n32;

        case xmss_sha3-512_m64_w4_h10_z:
        case xmss_sha3-512_m64_w4_h16_z:
        case xmss_sha3-512_m64_w4_h20_z:
        case xmss_sha3-512_m64_w8_h10_z:
        case xmss_sha3-512_m64_w16_h10_z:
        case xmss_sha3-512_m64_w8_h16_z:
        case xmss_sha3-512_m64_w16_h16_z:
        case xmss_sha3-512_m64_w8_h20_z:
        case xmss_sha3-512_m64_w16_h20_z:
          bytestring64 root_n64;

        default:
          void;     /* error condition */
      };

      /* XMSS public key structure */

      struct xmss_public_key {
        xmss_root root;  /* Root node */
      };


   When bitmasks are used, XMSS public keys are defined using XDR syntax
   as follows:




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      /* Types for XMSS bitmasks */

      union xmss_bm switch (xmss_algorithm_type type) {
        case xmss_aes128_m32_w4_h10:
          bytestring16 bm_n16_bm36[36];

        case xmss_aes128_m32_w4_h16:
          bytestring16 bm_n16_bm48[48];

        case xmss_aes128_m32_w4_h20:
          bytestring16 bm_n16_bm56[56];

        case xmss_aes128_m32_w8_h10:
        case xmss_aes128_m32_w16_h10:
          bytestring16 bm_n16_bm34[34];

        case xmss_aes128_m32_w8_h16:
        case xmss_aes128_m32_w16_h16:
          bytestring16 bm_n16_bm46[46];

        case xmss_aes128_m32_w8_h20:
        case xmss_aes128_m32_w16_h20:
          bytestring16 bm_n16_bm54[54];

        case xmss_sha3-256_m32_w4_h10:
          bytestring32 bm_n32_bm36[36];

        case xmss_sha3-256_m32_w4_h16:
          bytestring32 bm_n32_bm48[48];

        case xmss_sha3-256_m32_w4_h20:
          bytestring32 bm_n32_bm56[56];

        case xmss_sha3-256_m32_w8_h10:
        case xmss_sha3-256_m32_w16_h10:
          bytestring32 bm_n32_bm34[34];

        case xmss_sha3-256_m32_w8_h16:
        case xmss_sha3-256_m32_w16_h16:
          bytestring32 bm_n32_bm46[46];

        case xmss_sha3-256_m32_w8_h20:
        case xmss_sha3-256_m32_w16_h20:
          bytestring32 bm_n32_bm54[54];

        case xmss_sha3-512_m64_w4_h10:
          bytestring64 bm_n64_bm38[38];




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        case xmss_sha3-512_m64_w4_h16:
          bytestring64 bm_n64_bm50[50];

        case xmss_sha3-512_m64_w4_h20:
          bytestring64 bm_n64_bm58[58];

        case xmss_sha3-512_m64_w8_h10:
        case xmss_sha3-512_m64_w16_h10:
          bytestring64 bm_n64_bm36[36];

        case xmss_sha3-512_m64_w8_h16:
        case xmss_sha3-512_m64_w16_h16:
          bytestring64 bm_n64_bm48[48];

        case xmss_sha3-512_m64_w8_h20:
        case xmss_sha3-512_m64_w16_h20:
          bytestring64 bm_n64_bm56[56];

        default:
          void;     /* error condition */
      };

      /* Types for XMSS root node */

      union xmss_root switch (xmss_algorithm_type type) {
        case xmss_aes128_m32_w4_h10:
        case xmss_aes128_m32_w4_h16:
        case xmss_aes128_m32_w4_h20:
        case xmss_aes128_m32_w8_h10:
        case xmss_aes128_m32_w16_h10:
        case xmss_aes128_m32_w8_h16:
        case xmss_aes128_m32_w16_h16:
        case xmss_aes128_m32_w8_h20:
        case xmss_aes128_m32_w16_h20:
          bytestring16 root_n16;

        case xmss_sha3-256_m32_w4_h10:
        case xmss_sha3-256_m32_w4_h16:
        case xmss_sha3-256_m32_w4_h20:
        case xmss_sha3-256_m32_w8_h10:
        case xmss_sha3-256_m32_w16_h10:
        case xmss_sha3-256_m32_w8_h16:
        case xmss_sha3-256_m32_w16_h16:
        case xmss_sha3-256_m32_w8_h20:
        case xmss_sha3-256_m32_w16_h20:
          bytestring32 root_n32;

        case xmss_sha3-512_m64_w4_h10:



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        case xmss_sha3-512_m64_w4_h16:
        case xmss_sha3-512_m64_w4_h20:
        case xmss_sha3-512_m64_w8_h10:
        case xmss_sha3-512_m64_w16_h10:
        case xmss_sha3-512_m64_w8_h16:
        case xmss_sha3-512_m64_w16_h16:
        case xmss_sha3-512_m64_w8_h20:
        case xmss_sha3-512_m64_w16_h20:
          bytestring64 root_n64;

        default:
          void;     /* error condition */
      };

      /* XMSS public key structure */

      struct xmss_public_key {
        xmss_bm bm;  /* Bitmasks */
        xmss_root root;  /* Root node */
      };


Appendix C.  XMSS^MT XDR Formats

   XMSS^MT parameter sets are defined using XDR syntax as follows:


      /* Byte strings */

      typedef opaque bytestring3[3];
      typedef opaque bytestring5[5];
      typedef opaque bytestring8[8];

      /* Definition of parameter sets */

      enum xmssmt_algorithm_type {
        xmssmt_reserved                 = 0x00000000,

        /* Empty bitmasks */

        /* 128 bit classical security, 85 bit post-quantum security */

        xmssmt_sha3-256_m32_w4_h20_d2_z   = 0x01000001,
        xmssmt_sha3-256_m32_w4_h20_d4_z   = 0x02000002,
        xmssmt_sha3-256_m32_w4_h40_d2_z   = 0x03000003,
        xmssmt_sha3-256_m32_w4_h40_d4_z   = 0x04000004,
        xmssmt_sha3-256_m32_w4_h40_d8_z   = 0x05000005,
        xmssmt_sha3-256_m32_w4_h60_d3_z   = 0x06000006,



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        xmssmt_sha3-256_m32_w4_h60_d6_z   = 0x07000007,
        xmssmt_sha3-256_m32_w4_h60_d12_z  = 0x08000008,

        xmssmt_sha3-256_m32_w8_h20_d2_z   = 0x09000009,
        xmssmt_sha3-256_m32_w8_h20_d4_z   = 0x0a00000a,
        xmssmt_sha3-256_m32_w8_h40_d2_z   = 0x0b00000b,
        xmssmt_sha3-256_m32_w8_h40_d4_z   = 0x0c00000c,
        xmssmt_sha3-256_m32_w8_h40_d8_z   = 0x0d00000d,
        xmssmt_sha3-256_m32_w8_h60_d3_z   = 0x0e00000e,
        xmssmt_sha3-256_m32_w8_h60_d6_z   = 0x0f00000f,
        xmssmt_sha3-256_m32_w8_h60_d12_z  = 0x00010100,

        xmssmt_sha3-256_m32_w16_h20_d2_z  = 0x01010101,
        xmssmt_sha3-256_m32_w16_h20_d4_z  = 0x02010102,
        xmssmt_sha3-256_m32_w16_h40_d2_z  = 0x03010103,
        xmssmt_sha3-256_m32_w16_h40_d4_z  = 0x04010104,
        xmssmt_sha3-256_m32_w16_h40_d8_z  = 0x05010105,
        xmssmt_sha3-256_m32_w16_h60_d3_z  = 0x06010106,
        xmssmt_sha3-256_m32_w16_h60_d6_z  = 0x07010107,
        xmssmt_sha3-256_m32_w16_h60_d12_z = 0x08010108,

        /* 256 bit classical security, 170 bit post-quantum security */

        xmssmt_sha3-512_m64_w4_h20_d2_z   = 0x09010109,
        xmssmt_sha3-512_m64_w4_h20_d4_z   = 0x0a01010a,
        xmssmt_sha3-512_m64_w4_h40_d2_z   = 0x0b01010b,
        xmssmt_sha3-512_m64_w4_h40_d4_z   = 0x0c01010c,
        xmssmt_sha3-512_m64_w4_h40_d8_z   = 0x0d01010d,
        xmssmt_sha3-512_m64_w4_h60_d3_z   = 0x0e01010e,
        xmssmt_sha3-512_m64_w4_h60_d6_z   = 0x0f01010f,
        xmssmt_sha3-512_m64_w4_h60_d12_z  = 0x00020200,

        xmssmt_sha3-512_m64_w8_h20_d2_z   = 0x01020201,
        xmssmt_sha3-512_m64_w8_h20_d4_z   = 0x02020202,
        xmssmt_sha3-512_m64_w8_h40_d2_z   = 0x03020203,
        xmssmt_sha3-512_m64_w8_h40_d4_z   = 0x04020204,
        xmssmt_sha3-512_m64_w8_h40_d8_z   = 0x05020205,
        xmssmt_sha3-512_m64_w8_h60_d3_z   = 0x06020206,
        xmssmt_sha3-512_m64_w8_h60_d6_z   = 0x07020207,
        xmssmt_sha3-512_m64_w8_h60_d12_z  = 0x08020208,

        xmssmt_sha3-512_m64_w16_h20_d2_z  = 0x09020209,
        xmssmt_sha3-512_m64_w16_h20_d4_z  = 0x0a02020a,
        xmssmt_sha3-512_m64_w16_h40_d2_z  = 0x0b02020b,
        xmssmt_sha3-512_m64_w16_h40_d4_z  = 0x0c02020c,
        xmssmt_sha3-512_m64_w16_h40_d8_z  = 0x0d02020d,
        xmssmt_sha3-512_m64_w16_h60_d3_z  = 0x0e02020e,
        xmssmt_sha3-512_m64_w16_h60_d6_z  = 0x0f02020f,



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        xmssmt_sha3-512_m64_w16_h60_d12_z = 0x00030300,

        /* Non-empty bitmasks */

        /* 128 bit classical security, 64 bit post-quantum security */

        xmssmt_aes128_m32_w4_h20_d2         = 0x01030301,
        xmssmt_aes128_m32_w4_h20_d4         = 0x02030302,
        xmssmt_aes128_m32_w4_h40_d2         = 0x03030303,
        xmssmt_aes128_m32_w4_h40_d4         = 0x04030304,
        xmssmt_aes128_m32_w4_h40_d8         = 0x05030305,
        xmssmt_aes128_m32_w4_h60_d3         = 0x06030306,
        xmssmt_aes128_m32_w4_h60_d6         = 0x07030307,
        xmssmt_aes128_m32_w4_h60_d12        = 0x08030308,

        xmssmt_aes128_m32_w8_h20_d2         = 0x09030309,
        xmssmt_aes128_m32_w8_h20_d4         = 0x0a03030a,
        xmssmt_aes128_m32_w8_h40_d2         = 0x0b03030b,
        xmssmt_aes128_m32_w8_h40_d4         = 0x0c03030c,
        xmssmt_aes128_m32_w8_h40_d8         = 0x0d03030d,
        xmssmt_aes128_m32_w8_h60_d3         = 0x0e03030e,
        xmssmt_aes128_m32_w8_h60_d6         = 0x0f03030f,
        xmssmt_aes128_m32_w8_h60_d12        = 0x00040400,

        xmssmt_aes128_m32_w16_h20_d2        = 0x01040401,
        xmssmt_aes128_m32_w16_h20_d4        = 0x02040402,
        xmssmt_aes128_m32_w16_h40_d2        = 0x03040403,
        xmssmt_aes128_m32_w16_h40_d4        = 0x04040404,
        xmssmt_aes128_m32_w16_h40_d8        = 0x05040405,
        xmssmt_aes128_m32_w16_h60_d3        = 0x06040406,
        xmssmt_aes128_m32_w16_h60_d6        = 0x07040407,
        xmssmt_aes128_m32_w16_h60_d12       = 0x08040408,

        /* 256 bit classical security, 128 bit post-quantum security */

        xmssmt_sha3-256_m32_w4_h20_d2       = 0x09040409,
        xmssmt_sha3-256_m32_w4_h20_d4       = 0x0a04040a,
        xmssmt_sha3-256_m32_w4_h40_d2       = 0x0b04040b,
        xmssmt_sha3-256_m32_w4_h40_d4       = 0x0c04040c,
        xmssmt_sha3-256_m32_w4_h40_d8       = 0x0d04040d,
        xmssmt_sha3-256_m32_w4_h60_d3       = 0x0e04040e,
        xmssmt_sha3-256_m32_w4_h60_d6       = 0x0f04040f,
        xmssmt_sha3-256_m32_w4_h60_d12      = 0x00050500,

        xmssmt_sha3-256_m32_w8_h20_d2       = 0x01050501,
        xmssmt_sha3-256_m32_w8_h20_d4       = 0x02050502,
        xmssmt_sha3-256_m32_w8_h40_d2       = 0x03050503,
        xmssmt_sha3-256_m32_w8_h40_d4       = 0x04050504,



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        xmssmt_sha3-256_m32_w8_h40_d8       = 0x05050505,
        xmssmt_sha3-256_m32_w8_h60_d3       = 0x06050506,
        xmssmt_sha3-256_m32_w8_h60_d6       = 0x07050507,
        xmssmt_sha3-256_m32_w8_h60_d12      = 0x08050508,

        xmssmt_sha3-256_m32_w16_h20_d2      = 0x09050509,
        xmssmt_sha3-256_m32_w16_h20_d4      = 0x0a05050a,
        xmssmt_sha3-256_m32_w16_h40_d2      = 0x0b05050b,
        xmssmt_sha3-256_m32_w16_h40_d4      = 0x0c05050c,
        xmssmt_sha3-256_m32_w16_h40_d8      = 0x0d05050d,
        xmssmt_sha3-256_m32_w16_h60_d3      = 0x0e05050e,
        xmssmt_sha3-256_m32_w16_h60_d6      = 0x0f05050f,
        xmssmt_sha3-256_m32_w16_h60_d12     = 0x00060600,

        /* 512 bit classical security, 256 bit post-quantum security */

        xmssmt_sha3-512_m64_w4_h20_d2       = 0x01060601,
        xmssmt_sha3-512_m64_w4_h20_d4       = 0x02060602,
        xmssmt_sha3-512_m64_w4_h40_d2       = 0x03060603,
        xmssmt_sha3-512_m64_w4_h40_d4       = 0x04060604,
        xmssmt_sha3-512_m64_w4_h40_d8       = 0x05060605,
        xmssmt_sha3-512_m64_w4_h60_d3       = 0x06060606,
        xmssmt_sha3-512_m64_w4_h60_d6       = 0x07060607,
        xmssmt_sha3-512_m64_w4_h60_d12      = 0x08060608,

        xmssmt_sha3-512_m64_w8_h20_d2       = 0x09060609,
        xmssmt_sha3-512_m64_w8_h20_d4       = 0x0a06060a,
        xmssmt_sha3-512_m64_w8_h40_d2       = 0x0b06060b,
        xmssmt_sha3-512_m64_w8_h40_d4       = 0x0c06060c,
        xmssmt_sha3-512_m64_w8_h40_d8       = 0x0d06060d,
        xmssmt_sha3-512_m64_w8_h60_d3       = 0x0e06060e,
        xmssmt_sha3-512_m64_w8_h60_d6       = 0x0f06060f,
        xmssmt_sha3-512_m64_w8_h60_d12      = 0x00070700,

        xmssmt_sha3-512_m64_w16_h20_d2      = 0x01070701,
        xmssmt_sha3-512_m64_w16_h20_d4      = 0x02070702,
        xmssmt_sha3-512_m64_w16_h40_d2      = 0x03070703,
        xmssmt_sha3-512_m64_w16_h40_d4      = 0x04070704,
        xmssmt_sha3-512_m64_w16_h40_d8      = 0x05070705,
        xmssmt_sha3-512_m64_w16_h60_d3      = 0x06070706,
        xmssmt_sha3-512_m64_w16_h60_d6      = 0x07070707,
        xmssmt_sha3-512_m64_w16_h60_d12     = 0x08070708,
      };


   XMSS^MT signatures are defined using XDR syntax as follows:





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      /* Type for XMSS^MT key pair index */
      /* Depends solely on h */

      union idx_sig_xmssmt switch (xmss_algorithm_type type) {
        case xmssmt_sha3-256_m32_w4_h20_d2_z:
        case xmssmt_sha3-256_m32_w4_h20_d4_z:
        case xmssmt_sha3-256_m32_w8_h20_d2_z:
        case xmssmt_sha3-256_m32_w8_h20_d4_z:
        case xmssmt_sha3-256_m32_w16_h20_d2_z:
        case xmssmt_sha3-256_m32_w16_h20_d4_z:
        case xmssmt_sha3-512_m64_w4_h20_d2_z:
        case xmssmt_sha3-512_m64_w4_h20_d4_z:
        case xmssmt_sha3-512_m64_w8_h20_d2_z:
        case xmssmt_sha3-512_m64_w8_h20_d4_z:
        case xmssmt_sha3-512_m64_w16_h20_d2_z:
        case xmssmt_sha3-512_m64_w16_h20_d4_z:
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w4_h20_d4:
        case xmssmt_aes128_m32_w8_h20_d2:
        case xmssmt_aes128_m32_w8_h20_d4:
        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_aes128_m32_w16_h20_d4:
        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w4_h20_d4:
        case xmssmt_sha3-256_m32_w8_h20_d2:
        case xmssmt_sha3-256_m32_w8_h20_d4:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w16_h20_d4:
        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w4_h20_d4:
        case xmssmt_sha3-512_m64_w8_h20_d2:
        case xmssmt_sha3-512_m64_w8_h20_d4:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-512_m64_w16_h20_d4:
          bytestring3 idx3;

        case xmssmt_sha3-256_m32_w4_h40_d2_z:
        case xmssmt_sha3-256_m32_w4_h40_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d8_z:
        case xmssmt_sha3-256_m32_w8_h40_d2_z:
        case xmssmt_sha3-256_m32_w8_h40_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d8_z:
        case xmssmt_sha3-256_m32_w16_h40_d2_z:
        case xmssmt_sha3-256_m32_w16_h40_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d8_z:
        case xmssmt_sha3-512_m64_w4_h40_d2_z:
        case xmssmt_sha3-512_m64_w4_h40_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d8_z:



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        case xmssmt_sha3-512_m64_w8_h40_d2_z:
        case xmssmt_sha3-512_m64_w8_h40_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d8_z:
        case xmssmt_sha3-512_m64_w16_h40_d2_z:
        case xmssmt_sha3-512_m64_w16_h40_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d8_z:
        case xmssmt_aes128_m32_w4_h40_d2:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w4_h40_d8:
        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w8_h40_d8:
        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_aes128_m32_w16_h40_d8:
        case xmssmt_sha3-256_m32_w4_h40_d2:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w4_h40_d8:
        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-256_m32_w8_h40_d8:
        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-512_m64_w4_h40_d8:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w16_h40_d8:
        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w8_h40_d8:
        case xmssmt_sha3-512_m64_w16_h40_d2:
        case xmssmt_sha3-512_m64_w16_h40_d4:
        case xmssmt_sha3-512_m64_w16_h40_d8:
          bytestring5 idx5;

        case xmssmt_sha3-256_m32_w4_h60_d3_z:
        case xmssmt_sha3-256_m32_w4_h60_d6_z:
        case xmssmt_sha3-256_m32_w4_h60_d12_z:
        case xmssmt_sha3-256_m32_w8_h60_d3_z:
        case xmssmt_sha3-256_m32_w8_h60_d6_z:
        case xmssmt_sha3-256_m32_w8_h60_d12_z:
        case xmssmt_sha3-256_m32_w16_h60_d3_z:
        case xmssmt_sha3-256_m32_w16_h60_d6_z:
        case xmssmt_sha3-256_m32_w16_h60_d12_z:
        case xmssmt_sha3-512_m64_w4_h60_d3_z:
        case xmssmt_sha3-512_m64_w4_h60_d6_z:
        case xmssmt_sha3-512_m64_w4_h60_d12_z:
        case xmssmt_sha3-512_m64_w8_h60_d3_z:



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        case xmssmt_sha3-512_m64_w8_h60_d6_z:
        case xmssmt_sha3-512_m64_w8_h60_d12_z:
        case xmssmt_sha3-512_m64_w16_h60_d3_z:
        case xmssmt_sha3-512_m64_w16_h60_d6_z:
        case xmssmt_sha3-512_m64_w16_h60_d12_z:
        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w4_h60_d6:
        case xmssmt_aes128_m32_w4_h60_d12:
        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w8_h60_d12:
        case xmssmt_aes128_m32_w16_h60_d3:
        case xmssmt_aes128_m32_w16_h60_d6:
        case xmssmt_aes128_m32_w16_h60_d12:
        case xmssmt_sha3-256_m32_w4_h60_d3:
        case xmssmt_sha3-256_m32_w4_h60_d6:
        case xmssmt_sha3-256_m32_w4_h60_d12:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w8_h60_d12:
        case xmssmt_sha3-256_m32_w16_h60_d3:
        case xmssmt_sha3-256_m32_w16_h60_d6:
        case xmssmt_sha3-256_m32_w16_h60_d12:
        case xmssmt_sha3-512_m64_w4_h60_d3:
        case xmssmt_sha3-512_m64_w4_h60_d6:
        case xmssmt_sha3-512_m64_w4_h60_d12:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w8_h60_d12:
        case xmssmt_sha3-512_m64_w16_h60_d3:
        case xmssmt_sha3-512_m64_w16_h60_d6:
        case xmssmt_sha3-512_m64_w16_h60_d12:
          bytestring8 idx8;

        default:
          void;     /* error condition */
      };

      union random_string_xmssmt switch (xmssmt_algorithm_type type) {
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w4_h20_d4:
        case xmssmt_aes128_m32_w4_h40_d2:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w4_h40_d8:
        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w4_h60_d6:
        case xmssmt_aes128_m32_w4_h60_d12:
        case xmssmt_aes128_m32_w8_h20_d2:



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        case xmssmt_aes128_m32_w8_h20_d4:
        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w8_h40_d8:
        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w8_h60_d12:
        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_aes128_m32_w16_h20_d4:
        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_aes128_m32_w16_h40_d8:
        case xmssmt_aes128_m32_w16_h60_d3:
        case xmssmt_aes128_m32_w16_h60_d6:
        case xmssmt_aes128_m32_w16_h60_d12:
        case xmssmt_sha3-256_m32_w4_h20_d2_z:
        case xmssmt_sha3-256_m32_w4_h20_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d2_z:
        case xmssmt_sha3-256_m32_w4_h40_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d8_z:
        case xmssmt_sha3-256_m32_w4_h60_d3_z:
        case xmssmt_sha3-256_m32_w4_h60_d6_z:
        case xmssmt_sha3-256_m32_w4_h60_d12_z:
        case xmssmt_sha3-256_m32_w8_h20_d2_z:
        case xmssmt_sha3-256_m32_w8_h20_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d2_z:
        case xmssmt_sha3-256_m32_w8_h40_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d8_z:
        case xmssmt_sha3-256_m32_w8_h60_d3_z:
        case xmssmt_sha3-256_m32_w8_h60_d6_z:
        case xmssmt_sha3-256_m32_w8_h60_d12_z:
        case xmssmt_sha3-256_m32_w16_h20_d2_z:
        case xmssmt_sha3-256_m32_w16_h20_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d2_z:
        case xmssmt_sha3-256_m32_w16_h40_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d8_z:
        case xmssmt_sha3-256_m32_w16_h60_d3_z:
        case xmssmt_sha3-256_m32_w16_h60_d6_z:
        case xmssmt_sha3-256_m32_w16_h60_d12_z:
        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w4_h20_d4:
        case xmssmt_sha3-256_m32_w4_h40_d2:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w4_h40_d8:
        case xmssmt_sha3-256_m32_w4_h60_d3:
        case xmssmt_sha3-256_m32_w4_h60_d6:
        case xmssmt_sha3-256_m32_w4_h60_d12:
        case xmssmt_sha3-256_m32_w8_h20_d2:



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        case xmssmt_sha3-256_m32_w8_h20_d4:
        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-256_m32_w8_h40_d8:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w8_h60_d12:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w16_h20_d4:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w16_h40_d8:
        case xmssmt_sha3-256_m32_w16_h60_d3:
        case xmssmt_sha3-256_m32_w16_h60_d6:
        case xmssmt_sha3-256_m32_w16_h60_d12:
          bytestring32 rand_m32;

        case xmssmt_sha3-512_m64_w4_h20_d2_z:
        case xmssmt_sha3-512_m64_w4_h20_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d2_z:
        case xmssmt_sha3-512_m64_w4_h40_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d8_z:
        case xmssmt_sha3-512_m64_w4_h60_d3_z:
        case xmssmt_sha3-512_m64_w4_h60_d6_z:
        case xmssmt_sha3-512_m64_w4_h60_d12_z:
        case xmssmt_sha3-512_m64_w8_h20_d2_z:
        case xmssmt_sha3-512_m64_w8_h20_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d2_z:
        case xmssmt_sha3-512_m64_w8_h40_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d8_z:
        case xmssmt_sha3-512_m64_w8_h60_d3_z:
        case xmssmt_sha3-512_m64_w8_h60_d6_z:
        case xmssmt_sha3-512_m64_w8_h60_d12_z:
        case xmssmt_sha3-512_m64_w16_h20_d2_z:
        case xmssmt_sha3-512_m64_w16_h20_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d2_z:
        case xmssmt_sha3-512_m64_w16_h40_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d8_z:
        case xmssmt_sha3-512_m64_w16_h60_d3_z:
        case xmssmt_sha3-512_m64_w16_h60_d6_z:
        case xmssmt_sha3-512_m64_w16_h60_d12_z:
        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w4_h20_d4:
        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-512_m64_w4_h40_d8:
        case xmssmt_sha3-512_m64_w4_h60_d3:
        case xmssmt_sha3-512_m64_w4_h60_d6:



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        case xmssmt_sha3-512_m64_w4_h60_d12:
        case xmssmt_sha3-512_m64_w8_h20_d2:
        case xmssmt_sha3-512_m64_w8_h20_d4:
        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w8_h40_d8:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w8_h60_d12:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-512_m64_w16_h20_d4:
        case xmssmt_sha3-512_m64_w16_h40_d2:
        case xmssmt_sha3-512_m64_w16_h40_d4:
        case xmssmt_sha3-512_m64_w16_h40_d8:
        case xmssmt_sha3-512_m64_w16_h60_d3:
        case xmssmt_sha3-512_m64_w16_h60_d6:
        case xmssmt_sha3-512_m64_w16_h60_d12:
          bytestring64 rand_m64;

        default:
          void;     /* error condition */
      };

      struct xmss_reduced_bottom {
        xmss_ots_signature sig_ots; /* WOTS+ signature */
        xmss_path nodes;            /* authentication path */
      };

      /* Type for individual reduced XMSS signatures on higher layers */

      union xmss_reduced_others (xmss_algorithm_type type) {
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w4_h20_d4:
          bytestring16 xmss_reduced_n16_t88[88];

        case xmssmt_aes128_m32_w4_h40_d2:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w4_h40_d8:
          bytestring16 xmss_reduced_n16_t108[108];

        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w4_h60_d6:
        case xmssmt_aes128_m32_w4_h60_d12:
          bytestring16 xmss_reduced_n16_t128[128];

        case xmssmt_aes128_m32_w8_h20_d2:
        case xmssmt_aes128_m32_w8_h20_d4:
          bytestring16 xmss_reduced_n16_t66[66];



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        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w8_h40_d8:
          bytestring16 xmss_reduced_n16_t86[86];

        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w8_h60_d12:
          bytestring16 xmss_reduced_n16_t106[106];

        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_aes128_m32_w16_h20_d4:
          bytestring16 xmss_reduced_n16_t55[55];

        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_aes128_m32_w16_h40_d8:
          bytestring16 xmss_reduced_n16_t75[75];

        case xmssmt_aes128_m32_w16_h60_d3:
        case xmssmt_aes128_m32_w16_h60_d6:
        case xmssmt_aes128_m32_w16_h60_d12:
          bytestring16 xmss_reduced_n16_t95[95];

        case xmssmt_sha3-256_m32_w4_h20_d2_z:
        case xmssmt_sha3-256_m32_w4_h20_d4_z:
        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w4_h20_d4:
          bytestring32 xmss_reduced_n32_t153[153];

        case xmssmt_sha3-256_m32_w4_h40_d2_z:
        case xmssmt_sha3-256_m32_w4_h40_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d8_z:
        case xmssmt_sha3-256_m32_w4_h40_d2:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w4_h40_d8:
          bytestring32 xmss_reduced_n32_t173[173];

        case xmssmt_sha3-256_m32_w4_h60_d3_z:
        case xmssmt_sha3-256_m32_w4_h60_d6_z:
        case xmssmt_sha3-256_m32_w4_h60_d12_z:
        case xmssmt_sha3-256_m32_w4_h60_d3:
        case xmssmt_sha3-256_m32_w4_h60_d6:
        case xmssmt_sha3-256_m32_w4_h60_d12:
          bytestring32 xmss_reduced_n32_t193[193];

        case xmssmt_sha3-256_m32_w8_h20_d2_z:
        case xmssmt_sha3-256_m32_w8_h20_d4_z:



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        case xmssmt_sha3-256_m32_w8_h20_d2:
        case xmssmt_sha3-256_m32_w8_h20_d4:
          bytestring32 xmss_reduced_n32_t110[110];

        case xmssmt_sha3-256_m32_w8_h40_d2_z:
        case xmssmt_sha3-256_m32_w8_h40_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d8_z:
        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-256_m32_w8_h40_d8:
          bytestring32 xmss_reduced_n32_t130[130];

        case xmssmt_sha3-256_m32_w8_h60_d3_z:
        case xmssmt_sha3-256_m32_w8_h60_d6_z:
        case xmssmt_sha3-256_m32_w8_h60_d12_z:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w8_h60_d12:
          bytestring32 xmss_reduced_n32_t150[150];

        case xmssmt_sha3-256_m32_w16_h20_d2_z:
        case xmssmt_sha3-256_m32_w16_h20_d4_z:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w16_h20_d4:
          bytestring32 xmss_reduced_n32_t87[87];

        case xmssmt_sha3-256_m32_w16_h40_d2_z:
        case xmssmt_sha3-256_m32_w16_h40_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d8_z:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w16_h40_d8:
          bytestring32 xmss_reduced_n32_t107[107];

        case xmssmt_sha3-256_m32_w16_h60_d3_z:
        case xmssmt_sha3-256_m32_w16_h60_d6_z:
        case xmssmt_sha3-256_m32_w16_h60_d12_z:
        case xmssmt_sha3-256_m32_w16_h60_d3:
        case xmssmt_sha3-256_m32_w16_h60_d6:
        case xmssmt_sha3-256_m32_w16_h60_d12:
          bytestring32 xmss_reduced_n32_t127[127];

        case xmssmt_sha3-512_m64_w4_h20_d2_z:
        case xmssmt_sha3-512_m64_w4_h20_d4_z:
        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w4_h20_d4:
          bytestring64 xmss_reduced_n64_t281[281];




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        case xmssmt_sha3-512_m64_w4_h40_d2_z:
        case xmssmt_sha3-512_m64_w4_h40_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d8_z:
        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-512_m64_w4_h40_d8:
          bytestring64 xmss_reduced_n64_t301[301];

        case xmssmt_sha3-512_m64_w4_h60_d3_z:
        case xmssmt_sha3-512_m64_w4_h60_d6_z:
        case xmssmt_sha3-512_m64_w4_h60_d12_z:
        case xmssmt_sha3-512_m64_w4_h60_d3:
        case xmssmt_sha3-512_m64_w4_h60_d6:
        case xmssmt_sha3-512_m64_w4_h60_d12:
          bytestring64 xmss_reduced_n64_t321[321];

        case xmssmt_sha3-512_m64_w8_h20_d2_z:
        case xmssmt_sha3-512_m64_w8_h20_d4_z:
          bytestring64 xmss_reduced_n64_t195[195];

        case xmssmt_sha3-512_m64_w8_h40_d2_z:
        case xmssmt_sha3-512_m64_w8_h40_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d8_z:
        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w8_h40_d8:
          bytestring64 xmss_reduced_n64_t215[215];

        case xmssmt_sha3-512_m64_w8_h60_d3_z:
        case xmssmt_sha3-512_m64_w8_h60_d6_z:
        case xmssmt_sha3-512_m64_w8_h60_d12_z:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w8_h60_d12:
          bytestring64 xmss_reduced_n64_t235[235];

        case xmssmt_sha3-512_m64_w16_h20_d2_z:
        case xmssmt_sha3-512_m64_w16_h20_d4_z:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-512_m64_w16_h20_d4:
          bytestring64 xmss_reduced_n64_t151[151];

        case xmssmt_sha3-512_m64_w16_h40_d2_z:
        case xmssmt_sha3-512_m64_w16_h40_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d8_z:
        case xmssmt_sha3-512_m64_w16_h40_d2:
        case xmssmt_sha3-512_m64_w16_h40_d4:
        case xmssmt_sha3-512_m64_w16_h40_d8:



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          bytestring64 xmss_reduced_n64_t171[171];

        case xmssmt_sha3-512_m64_w16_h60_d3_z:
        case xmssmt_sha3-512_m64_w16_h60_d6_z:
        case xmssmt_sha3-512_m64_w16_h60_d12_z:
        case xmssmt_sha3-512_m64_w16_h60_d3:
        case xmssmt_sha3-512_m64_w16_h60_d6:
        case xmssmt_sha3-512_m64_w16_h60_d12:
          bytestring64 xmss_reduced_n64_t191[191];

        default:
          void;     /* error condition */
      };

      /* xmss_reduced_array depends on d */

      union xmss_reduced_array (xmss_algorithm_type type) {
        case xmssmt_sha3-256_m32_w4_h20_d2_z:
        case xmssmt_sha3-256_m32_w8_h20_d2_z:
        case xmssmt_sha3-256_m32_w16_h20_d2_z:
        case xmssmt_sha3-512_m64_w4_h20_d2_z:
        case xmssmt_sha3-512_m64_w8_h20_d2_z:
        case xmssmt_sha3-512_m64_w16_h20_d2_z:
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w8_h20_d2:
        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w8_h20_d2:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w8_h20_d2:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-256_m32_w4_h40_d2_z:
        case xmssmt_sha3-256_m32_w8_h40_d2_z:
        case xmssmt_sha3-256_m32_w16_h40_d2_z:
        case xmssmt_sha3-512_m64_w4_h40_d2_z:
        case xmssmt_sha3-512_m64_w8_h40_d2_z:
        case xmssmt_sha3-512_m64_w16_h40_d2_z:
        case xmssmt_aes128_m32_w4_h40_d2:
        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w4_h40_d2:
        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w16_h40_d2:
          xmss_reduced_others xmss_red_arr_d2[1];



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        case xmssmt_sha3-256_m32_w4_h60_d3_z:
        case xmssmt_sha3-256_m32_w8_h60_d3_z:
        case xmssmt_sha3-256_m32_w16_h60_d3_z:
        case xmssmt_sha3-512_m64_w4_h60_d3_z:
        case xmssmt_sha3-512_m64_w8_h60_d3_z:
        case xmssmt_sha3-512_m64_w16_h60_d3_z:
        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w16_h60_d3:
        case xmssmt_sha3-256_m32_w4_h60_d3:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w16_h60_d3:
        case xmssmt_sha3-512_m64_w4_h60_d3:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w16_h60_d3:
          xmss_reduced_others xmss_red_arr_d3[2];

        case xmssmt_sha3-256_m32_w4_h20_d4_z:
        case xmssmt_sha3-256_m32_w8_h20_d4_z:
        case xmssmt_sha3-256_m32_w16_h20_d4_z:
        case xmssmt_sha3-512_m64_w4_h20_d4_z:
        case xmssmt_sha3-512_m64_w8_h20_d4_z:
        case xmssmt_sha3-512_m64_w16_h20_d4_z:
        case xmssmt_aes128_m32_w4_h20_d4:
        case xmssmt_aes128_m32_w8_h20_d4:
        case xmssmt_aes128_m32_w16_h20_d4:
        case xmssmt_sha3-256_m32_w4_h20_d4:
        case xmssmt_sha3-256_m32_w8_h20_d4:
        case xmssmt_sha3-256_m32_w16_h20_d4:
        case xmssmt_sha3-512_m64_w4_h20_d4:
        case xmssmt_sha3-512_m64_w8_h20_d4:
        case xmssmt_sha3-512_m64_w16_h20_d4:
        case xmssmt_sha3-256_m32_w4_h40_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d4_z:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w16_h40_d4:
          xmss_reduced_others xmss_red_arr_d4[3];



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        case xmssmt_sha3-256_m32_w4_h60_d6_z:
        case xmssmt_sha3-256_m32_w8_h60_d6_z:
        case xmssmt_sha3-256_m32_w16_h60_d6_z:
        case xmssmt_sha3-512_m64_w4_h60_d6_z:
        case xmssmt_sha3-512_m64_w8_h60_d6_z:
        case xmssmt_sha3-512_m64_w16_h60_d6_z:
        case xmssmt_aes128_m32_w4_h60_d6:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w16_h60_d6:
        case xmssmt_sha3-256_m32_w4_h60_d6:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w16_h60_d6:
        case xmssmt_sha3-512_m64_w4_h60_d6:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w16_h60_d6:
          xmss_reduced_others xmss_red_arr_d6[5];

        case xmssmt_sha3-256_m32_w4_h40_d8_z:
        case xmssmt_sha3-256_m32_w8_h40_d8_z:
        case xmssmt_sha3-256_m32_w16_h40_d8_z:
        case xmssmt_sha3-512_m64_w4_h40_d8_z:
        case xmssmt_sha3-512_m64_w8_h40_d8_z:
        case xmssmt_sha3-512_m64_w16_h40_d8_z:
        case xmssmt_aes128_m32_w4_h40_d8:
        case xmssmt_aes128_m32_w8_h40_d8:
        case xmssmt_aes128_m32_w16_h40_d8:
        case xmssmt_sha3-256_m32_w4_h40_d8:
        case xmssmt_sha3-256_m32_w8_h40_d8:
        case xmssmt_sha3-512_m64_w4_h40_d8:
        case xmssmt_sha3-256_m32_w16_h40_d8:
        case xmssmt_sha3-512_m64_w8_h40_d8:
        case xmssmt_sha3-512_m64_w16_h40_d8:
          xmss_reduced_others xmss_red_arr_d8[7];

        case xmssmt_sha3-256_m32_w4_h60_d12_z:
        case xmssmt_sha3-256_m32_w8_h60_d12_z:
        case xmssmt_sha3-256_m32_w16_h60_d12_z:
        case xmssmt_sha3-512_m64_w4_h60_d12_z:
        case xmssmt_sha3-512_m64_w8_h60_d12_z:
        case xmssmt_sha3-512_m64_w16_h60_d12_z:
        case xmssmt_aes128_m32_w4_h60_d12:
        case xmssmt_aes128_m32_w8_h60_d12:
        case xmssmt_aes128_m32_w16_h60_d12:
        case xmssmt_sha3-256_m32_w4_h60_d12:
        case xmssmt_sha3-256_m32_w8_h60_d12:
        case xmssmt_sha3-256_m32_w16_h60_d12:
        case xmssmt_sha3-512_m64_w4_h60_d12:
        case xmssmt_sha3-512_m64_w8_h60_d12:



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        case xmssmt_sha3-512_m64_w16_h60_d12:
          xmss_reduced_others xmss_red_arr_d12[11];

        default:
          void;     /* error condition */
      };

      /* XMSS^MT signature structure */

      struct xmssmt_signature {
        /* WOTS+ key pair index */
        idx_sig_xmssmt idx_sig;
        /* Random string for randomized hashing */
        random_string_xmssmt randomness;
        /* Reduced bottom layer XMSS signature */
        xmss_reduced_bottom;
        /* Array of reduced XMSS signatures with message length n */
        xmss_reduced_array;
      };


   When no bitmasks are used, XMSS^MT public keys are defined using XDR
   syntax as follows:


      /* Types for XMSS^MT root node */

      union xmssmt_root switch (xmssmt_algorithm_type type) {
        case xmssmt_sha3-256_m32_w4_h20_d2_z:
        case xmssmt_sha3-256_m32_w4_h20_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d2_z:
        case xmssmt_sha3-256_m32_w4_h40_d4_z:
        case xmssmt_sha3-256_m32_w4_h40_d8_z:
        case xmssmt_sha3-256_m32_w4_h60_d3_z:
        case xmssmt_sha3-256_m32_w4_h60_d6_z:
        case xmssmt_sha3-256_m32_w4_h60_d12_z:
        case xmssmt_sha3-256_m32_w8_h20_d2_z:
        case xmssmt_sha3-256_m32_w8_h20_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d2_z:
        case xmssmt_sha3-256_m32_w8_h40_d4_z:
        case xmssmt_sha3-256_m32_w8_h40_d8_z:
        case xmssmt_sha3-256_m32_w8_h60_d3_z:
        case xmssmt_sha3-256_m32_w8_h60_d6_z:
        case xmssmt_sha3-256_m32_w8_h60_d12_z:
        case xmssmt_sha3-256_m32_w16_h20_d2_z:
        case xmssmt_sha3-256_m32_w16_h20_d4_z:
        case xmssmt_sha3-256_m32_w16_h40_d2_z:
        case xmssmt_sha3-256_m32_w16_h40_d4_z:



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        case xmssmt_sha3-256_m32_w16_h40_d8_z:
        case xmssmt_sha3-256_m32_w16_h60_d3_z:
        case xmssmt_sha3-256_m32_w16_h60_d6_z:
        case xmssmt_sha3-256_m32_w16_h60_d12_z:
          bytestring32 root_n32;

        case xmssmt_sha3-512_m64_w4_h20_d2_z:
        case xmssmt_sha3-512_m64_w4_h20_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d2_z:
        case xmssmt_sha3-512_m64_w4_h40_d4_z:
        case xmssmt_sha3-512_m64_w4_h40_d8_z:
        case xmssmt_sha3-512_m64_w4_h60_d3_z:
        case xmssmt_sha3-512_m64_w4_h60_d6_z:
        case xmssmt_sha3-512_m64_w4_h60_d12_z:
        case xmssmt_sha3-512_m64_w8_h20_d2_z:
        case xmssmt_sha3-512_m64_w8_h20_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d2_z:
        case xmssmt_sha3-512_m64_w8_h40_d4_z:
        case xmssmt_sha3-512_m64_w8_h40_d8_z:
        case xmssmt_sha3-512_m64_w8_h60_d3_z:
        case xmssmt_sha3-512_m64_w8_h60_d6_z:
        case xmssmt_sha3-512_m64_w8_h60_d12_z:
        case xmssmt_sha3-512_m64_w16_h20_d2_z:
        case xmssmt_sha3-512_m64_w16_h20_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d2_z:
        case xmssmt_sha3-512_m64_w16_h40_d4_z:
        case xmssmt_sha3-512_m64_w16_h40_d8_z:
        case xmssmt_sha3-512_m64_w16_h60_d3_z:
        case xmssmt_sha3-512_m64_w16_h60_d6_z:
        case xmssmt_sha3-512_m64_w16_h60_d12_z:
          bytestring64 root_n64;

        default:
          void;     /* error condition */
      };

      /* XMSS^MT public key structure */

      struct xmssmt_public_key {
        xmssmt_root root;  /* Root node */
      };


   When bitmasks are used, XMSS^MT public keys are defined using XDR
   syntax as follows:


      /* Types for XMSS^MT bitmasks */



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      union xmssmt_bm switch (xmssmt_algorithm_type type) {
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w4_h60_d6:
          bytestring16 bm_n16_t36[36];

        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w4_h40_d2:
          bytestring16 bm_n16_t36[56];

        case xmssmt_aes128_m32_w4_h20_d4:
        case xmssmt_aes128_m32_w4_h40_d8:
        case xmssmt_aes128_m32_w4_h60_d12:
          bytestring16 bm_n16_t26[26];

        case xmssmt_aes128_m32_w8_h20_d2:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_aes128_m32_w16_h60_d6:
          bytestring16 bm_n16_t34[34];

        case xmssmt_aes128_m32_w8_h20_d4:
        case xmssmt_aes128_m32_w8_h40_d8:
        case xmssmt_aes128_m32_w8_h60_d12:
        case xmssmt_aes128_m32_w16_h20_d4:
        case xmssmt_aes128_m32_w16_h40_d8:
        case xmssmt_aes128_m32_w16_h60_d12:
          bytestring16 bm_n16_t24[24];

        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_aes128_m32_w16_h60_d3:
          bytestring16 bm_n16_t54[54];

        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w4_h60_d6:
          bytestring32 bm_n32_t36[36];

        case xmssmt_sha3-256_m32_w4_h20_d4:
        case xmssmt_sha3-256_m32_w4_h40_d8:
        case xmssmt_sha3-256_m32_w4_h60_d12:
          bytestring32 bm_n32_t26[26];

        case xmssmt_sha3-256_m32_w4_h40_d2:



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        case xmssmt_sha3-256_m32_w4_h60_d3:
          bytestring32 bm_n32_t56[56];

        case xmssmt_sha3-256_m32_w8_h20_d2:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w16_h60_d6:
          bytestring32 bm_n32_t34[34];

        case xmssmt_sha3-256_m32_w8_h20_d4:
        case xmssmt_sha3-256_m32_w8_h40_d8:
        case xmssmt_sha3-256_m32_w8_h60_d12:
        case xmssmt_sha3-256_m32_w16_h20_d4:
        case xmssmt_sha3-256_m32_w16_h40_d8:
        case xmssmt_sha3-256_m32_w16_h60_d12:
          bytestring32 bm_n32_t24[24];

        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w16_h60_d3:
          bytestring32 bm_n32_t54[54];

        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-512_m64_w4_h60_d6:
          bytestring64 bm_n64_t38[38];

        case xmssmt_sha3-512_m64_w4_h20_d4:
        case xmssmt_sha3-512_m64_w4_h40_d8:
        case xmssmt_sha3-512_m64_w4_h60_d12:
          bytestring64 bm_n64_t28[28];

        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-512_m64_w4_h60_d3:
          bytestring64 bm_n64_t58[58];

        case xmssmt_sha3-512_m64_w8_h20_d2:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-512_m64_w16_h40_d4:
        case xmssmt_sha3-512_m64_w16_h60_d6:
          bytestring64 bm_n64_t36[36];

        case xmssmt_sha3-512_m64_w8_h20_d4:



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        case xmssmt_sha3-512_m64_w8_h40_d8:
        case xmssmt_sha3-512_m64_w8_h60_d12:
        case xmssmt_sha3-512_m64_w16_h20_d4:
        case xmssmt_sha3-512_m64_w16_h40_d8:
        case xmssmt_sha3-512_m64_w16_h60_d12:
          bytestring64 bm_n64_t26[26];

        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w16_h40_d2:
        case xmssmt_sha3-512_m64_w16_h60_d3:
          bytestring64 bm_n64_t56[56];

        default:
          void;     /* error condition */
      };

      /* Types for XMSS^MT root node */

      union xmssmt_root switch (xmssmt_algorithm_type type) {
        case xmssmt_aes128_m32_w4_h20_d2:
        case xmssmt_aes128_m32_w4_h20_d4:
        case xmssmt_aes128_m32_w4_h40_d2:
        case xmssmt_aes128_m32_w4_h40_d4:
        case xmssmt_aes128_m32_w4_h40_d8:
        case xmssmt_aes128_m32_w4_h60_d3:
        case xmssmt_aes128_m32_w4_h60_d6:
        case xmssmt_aes128_m32_w4_h60_d12:
        case xmssmt_aes128_m32_w8_h20_d2:
        case xmssmt_aes128_m32_w8_h20_d4:
        case xmssmt_aes128_m32_w8_h40_d2:
        case xmssmt_aes128_m32_w8_h40_d4:
        case xmssmt_aes128_m32_w8_h40_d8:
        case xmssmt_aes128_m32_w8_h60_d3:
        case xmssmt_aes128_m32_w8_h60_d6:
        case xmssmt_aes128_m32_w8_h60_d12:
        case xmssmt_aes128_m32_w16_h20_d2:
        case xmssmt_aes128_m32_w16_h20_d4:
        case xmssmt_aes128_m32_w16_h40_d2:
        case xmssmt_aes128_m32_w16_h40_d4:
        case xmssmt_aes128_m32_w16_h40_d8:
        case xmssmt_aes128_m32_w16_h60_d3:
        case xmssmt_aes128_m32_w16_h60_d6:
        case xmssmt_aes128_m32_w16_h60_d12:
          bytestring16 root_n16;

        case xmssmt_sha3-256_m32_w4_h20_d2:
        case xmssmt_sha3-256_m32_w4_h20_d4:



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        case xmssmt_sha3-256_m32_w4_h40_d2:
        case xmssmt_sha3-256_m32_w4_h40_d4:
        case xmssmt_sha3-256_m32_w4_h40_d8:
        case xmssmt_sha3-256_m32_w4_h60_d3:
        case xmssmt_sha3-256_m32_w4_h60_d6:
        case xmssmt_sha3-256_m32_w4_h60_d12:
        case xmssmt_sha3-256_m32_w8_h20_d2:
        case xmssmt_sha3-256_m32_w8_h20_d4:
        case xmssmt_sha3-256_m32_w8_h40_d2:
        case xmssmt_sha3-256_m32_w8_h40_d4:
        case xmssmt_sha3-256_m32_w8_h40_d8:
        case xmssmt_sha3-256_m32_w8_h60_d3:
        case xmssmt_sha3-256_m32_w8_h60_d6:
        case xmssmt_sha3-256_m32_w8_h60_d12:
        case xmssmt_sha3-256_m32_w16_h20_d2:
        case xmssmt_sha3-256_m32_w16_h20_d4:
        case xmssmt_sha3-256_m32_w16_h40_d2:
        case xmssmt_sha3-256_m32_w16_h40_d4:
        case xmssmt_sha3-256_m32_w16_h40_d8:
        case xmssmt_sha3-256_m32_w16_h60_d3:
        case xmssmt_sha3-256_m32_w16_h60_d6:
        case xmssmt_sha3-256_m32_w16_h60_d12:
          bytestring32 root_n32;

        case xmssmt_sha3-512_m64_w4_h20_d2:
        case xmssmt_sha3-512_m64_w4_h20_d4:
        case xmssmt_sha3-512_m64_w4_h40_d2:
        case xmssmt_sha3-512_m64_w4_h40_d4:
        case xmssmt_sha3-512_m64_w4_h40_d8:
        case xmssmt_sha3-512_m64_w4_h60_d3:
        case xmssmt_sha3-512_m64_w4_h60_d6:
        case xmssmt_sha3-512_m64_w4_h60_d12:
        case xmssmt_sha3-512_m64_w8_h20_d2:
        case xmssmt_sha3-512_m64_w8_h20_d4:
        case xmssmt_sha3-512_m64_w8_h40_d2:
        case xmssmt_sha3-512_m64_w8_h40_d4:
        case xmssmt_sha3-512_m64_w8_h40_d8:
        case xmssmt_sha3-512_m64_w8_h60_d3:
        case xmssmt_sha3-512_m64_w8_h60_d6:
        case xmssmt_sha3-512_m64_w8_h60_d12:
        case xmssmt_sha3-512_m64_w16_h20_d2:
        case xmssmt_sha3-512_m64_w16_h20_d4:
        case xmssmt_sha3-512_m64_w16_h40_d2:
        case xmssmt_sha3-512_m64_w16_h40_d4:
        case xmssmt_sha3-512_m64_w16_h40_d8:
        case xmssmt_sha3-512_m64_w16_h60_d3:
        case xmssmt_sha3-512_m64_w16_h60_d6:
        case xmssmt_sha3-512_m64_w16_h60_d12:



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          bytestring64 root_n64;

        default:
          void;     /* error condition */
      };

      /* XMSS^MT public key structure */

      struct xmssmt_public_key {
        xmssmt_bm bm;  /* Bitmasks */
        xmssmt_root root;  /* Root node */
      };


Authors' Addresses

   Andreas Huelsing
   TU Eindhoven
   P.O. Box 513
   Eindhoven  5600 MB
   The Netherlands

   Email: a.t.huelsing@tue.nl


   Denis Butin
   TU Darmstadt
   Hochschulstrasse 10
   Darmstadt  64289
   Germany

   Email: dbutin@cdc.informatik.tu-darmstadt.de


   Stefan-Lukas Gazdag
   genua mbH
   Domagkstrasse 7
   Kirchheim bei Muenchen  85551
   Germany

   Email: stefan-lukas_gazdag@genua.eu










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   Aziz Mohaisen
   Verisign Labs
   12061 Bluemont Way
   Reston, VA  20190

   Phone: +1 703 948-3200
   Email: amohaisen@verisign.com












































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