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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 1, 2017                                    TU Darmstadt
                                                               S. Gazdag
                                                              genua GmbH
                                                             A. Mohaisen
                                                            SUNY Buffalo
                                                          March 30, 2017


                  XMSS: Extended Hash-Based Signatures
             draft-irtf-cfrg-xmss-hash-based-signatures-09

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 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 can 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 1, 2017.




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Copyright Notice

   Copyright (c) 2017 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
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  CFRG Note on Post-Quantum Cryptography  . . . . . . . . .   5
     1.2.  Conventions Used In This Document . . . . . . . . . . . .   5
   2.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.1.  Data Types  . . . . . . . . . . . . . . . . . . . . . . .   6
     2.2.  Functions . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.3.  Operators . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.4.  Integer to Byte Conversion  . . . . . . . . . . . . . . .   7
     2.5.  Hash Function Address Scheme  . . . . . . . . . . . . . .   7
     2.6.  Strings of Base w Numbers . . . . . . . . . . . . . . . .  10
     2.7.  Member Functions  . . . . . . . . . . . . . . . . . . . .  11
   3.  Primitives  . . . . . . . . . . . . . . . . . . . . . . . . .  12
     3.1.  WOTS+ One-Time Signatures . . . . . . . . . . . . . . . .  12
       3.1.1.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . .  12
         3.1.1.1.  WOTS+ Functions . . . . . . . . . . . . . . . . .  13
       3.1.2.  WOTS+ Chaining Function . . . . . . . . . . . . . . .  13
       3.1.3.  WOTS+ Private Key . . . . . . . . . . . . . . . . . .  14
       3.1.4.  WOTS+ Public Key  . . . . . . . . . . . . . . . . . .  15
       3.1.5.  WOTS+ Signature Generation  . . . . . . . . . . . . .  15
       3.1.6.  WOTS+ Signature Verification  . . . . . . . . . . . .  17
       3.1.7.  Pseudorandom Key Generation . . . . . . . . . . . . .  18
   4.  Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . .  18
     4.1.  XMSS: eXtended Merkle Signature Scheme  . . . . . . . . .  18
       4.1.1.  XMSS Parameters . . . . . . . . . . . . . . . . . . .  19
       4.1.2.  XMSS Hash Functions . . . . . . . . . . . . . . . . .  19
       4.1.3.  XMSS Private Key  . . . . . . . . . . . . . . . . . .  20
       4.1.4.  Randomized Tree Hashing . . . . . . . . . . . . . . .  20
       4.1.5.  L-Trees . . . . . . . . . . . . . . . . . . . . . . .  21
       4.1.6.  TreeHash  . . . . . . . . . . . . . . . . . . . . . .  22
       4.1.7.  XMSS Key Generation . . . . . . . . . . . . . . . . .  23
       4.1.8.  XMSS Signature  . . . . . . . . . . . . . . . . . . .  24



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       4.1.9.  XMSS Signature Generation . . . . . . . . . . . . . .  25
       4.1.10. XMSS Signature Verification . . . . . . . . . . . . .  27
       4.1.11. Pseudorandom Key Generation . . . . . . . . . . . . .  29
       4.1.12. Free Index Handling and Partial Private Keys  . . . .  30
     4.2.  XMSS^MT: Multi-Tree XMSS  . . . . . . . . . . . . . . . .  30
       4.2.1.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . .  30
       4.2.2.  XMSS^MT Key generation  . . . . . . . . . . . . . . .  30
       4.2.3.  XMSS^MT Signature . . . . . . . . . . . . . . . . . .  33
       4.2.4.  XMSS^MT Signature Generation  . . . . . . . . . . . .  34
       4.2.5.  XMSS^MT Signature Verification  . . . . . . . . . . .  36
       4.2.6.  Pseudorandom Key Generation . . . . . . . . . . . . .  37
       4.2.7.  Free Index Handling and Partial Private Keys  . . . .  38
   5.  Parameter Sets  . . . . . . . . . . . . . . . . . . . . . . .  38
     5.1.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . . . .  39
     5.2.  XMSS Parameters . . . . . . . . . . . . . . . . . . . . .  40
     5.3.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . . . .  41
   6.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .  43
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  44
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  48
     8.1.  Security Proofs . . . . . . . . . . . . . . . . . . . . .  48
     8.2.  Minimal Security Assumptions  . . . . . . . . . . . . . .  50
     8.3.  Post-Quantum Security . . . . . . . . . . . . . . . . . .  50
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  51
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  51
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  51
     10.2.  Informative References . . . . . . . . . . . . . . . . .  51
   Appendix A.  WOTS+ XDR Formats  . . . . . . . . . . . . . . . . .  53
   Appendix B.  XMSS XDR Formats . . . . . . . . . . . . . . . . . .  54
   Appendix C.  XMSS^MT XDR Formats  . . . . . . . . . . . . . . . .  59
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  66

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 using a key pair to sign exactly one message securely.
   A Many-Time Signature (MTS) system can be used to sign multiple
   messages.

   OTS schemes, and MTS schemes composed from them, were proposed by
   Merkle in 1979 [Merkle79].  They were well-studied in the 1990s and
   have regained interest from the mid 2000s 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



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   schemes generally feature small private and public keys as well as
   fast signature generation and verification but large signatures and
   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 introducing and
   standardizing hash-based signatures.  McGrew, Curcio, and Fluhrer
   have published an Internet-Draft [MCF17] specifying the Lamport-
   Diffie-Winternitz-Merkle (LDWM) scheme, also taking into account
   subsequent adaptations by Leighton and Micali.  Independently,
   Buchmann, Dahmen and Huelsing have proposed XMSS [BDH11], the
   eXtended Merkle Signature Scheme, offering better efficiency and a
   modern security proof.  Very recently, the stateless hash-based
   signature scheme SPHINCS was introduced [BHH15], with the intent of
   being easier to deploy in current applications.  A reasonable next
   step toward introducing hash-based signatures is 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
   stateful 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.
   Improvements upon XMSS, as described in [HRS16], are part of this
   note.

   This document describes a single-tree and a multi-tree variant of
   XMSS.  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.

   This document is structured as follows.  Notation is introduced in
   Section 2.  Section 3 describes the WOTS+ signature system.  MTS
   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.







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1.1.  CFRG Note on Post-Quantum Cryptography

   All post-quantum algorithms documented by CFRG are today considered
   ready for experimentation and further engineering development (e.g.
   to establish the impact of performance and sizes on IETF protocols).
   However, at the time of writing, we do not have significant
   deployment experience with such algorithms.

   Many of these algorithms come with specific restrictions, e.g.
   change of classical interface or less cryptanalysis of proposed
   parameters than established schemes.  CFRG has consensus that all
   documents describing post-quantum technologies include the above
   paragraph and a clear additional warning about any specific
   restrictions, especially as those might affect use or deployment of
   the specific scheme.  That guidance may be changed over time via
   document updates.

   Additionally, for XMSS:

   CFRG consensus is that we are confident in the cryptographic security
   of the signature schemes described in this document against quantum
   computers, given the current state of the research community's
   knowledge about quantum algorithms.  Indeed, we are confident that
   the security of a significant part of the Internet could be made
   dependent on the signature schemes defined in this document, if
   developers take care of the following.

   In contrast to traditional signature schemes, the signature schemes
   described in this document are stateful, meaning the secret key
   changes over time.  If a secret key state is used twice, no
   cryptographic security guarantees remain.  This is a new property
   that most developers will not be familiar with and requires careful
   handling of secret keys.  Developers should not use the schemes
   described here except in systems that prevent the reuse of secret key
   states.

1.2.  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








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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.  We assume big-endian representation for any
   data types or structures.

2.2.  Functions

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

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

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

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

2.3.  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 omitted in the absence of ambiguity, as in usual
      mathematical notation.

      / : a / b denotes the quotient of a by non-zero 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.

      ++ : a++ denotes incrementing a by 1, i.e., a = a + 1.

      -- : a-- denotes decrementing a by 1, i.e., a = a - 1.





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      << : a << b denotes a logical left shift with b being non-
      negative, i.e., a * 2^b.

      >> : a >> b denotes a logical right shift with b being non-
      negative, i.e. floor(a / 2^b).

   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.

   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 an 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].

2.4.  Integer to Byte Conversion

   If x and y are non-negative integers, we define Z = toByte(x, y) to
   be the y-byte string containing the binary representation of x in
   big-endian byte-order.

2.5.  Hash Function Address Scheme

   The schemes described in this document randomize each hash function
   call.  This means that aside from the initial message digest, for
   each hash function call a different key and different bitmask is
   used.  These values are pseudorandomly generated using a pseudorandom
   function that takes a key SEED and a 32-byte address ADRS as input
   and outputs an n-byte value, where n is the security parameter.  Here
   we explain the structure of address ADRS and propose setter methods
   to manipulate the address.  We explain the generation of the
   addresses in the following sections where they are used.





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   The schemes in the next two sections use two kinds of hash functions
   parameterized by security parameter n.  For the hash tree
   constructions, a hash function that maps an n-byte key and 2n-byte
   inputs to n-byte outputs is used.  To randomize this function, 3n
   bytes are needed - n bytes for the key and 2n bytes for a bitmask.
   For the OTS scheme constructions, a hash function that maps n-byte
   keys and n-byte inputs to n-byte outputs is used.  To randomize this
   function, 2n bytes are needed - n bytes for the key and n bytes for a
   bitmask.  Consequently, three addresses are needed for the first
   function and two addresses for the second one.

   There are three different types of addresses for the different use
   cases.  One type is used for the hashes in OTS schemes, one is used
   for hashes within the main Merkle tree construction, and one is used
   for hashes in the L-trees.  The latter is used to compress one-time
   public keys.  All these types share as much format as possible.  In
   the following we describe these types in detail.

   The structure of an address complies with word borders, with a word
   being 32 bits long in this context.  Only the tree address is too
   long to fit a single word but matches a double word.  An address is
   structured as follows.  It always starts with a layer address of one
   word in the most significant bits, followed by a tree address of two
   words.  Both addresses are needed for the multi-tree variant (see
   Section 4.2) and describe the position of a tree within a multi-tree.
   They are therefore set to zero in case of single-tree applications.
   For multi-tree hash-based signatures the layer address describes the
   height of a tree within the multi-tree starting from height zero for
   trees at the bottom layer.  The tree address describes the position
   of a tree within a layer of a multi-tree starting with index zero for
   the leftmost tree.  The next word defines the type of the address.
   It is set to 0 for an OTS address, to 1 for an L-tree address, and to
   2 for a hash tree address.  Whenever the type word of an address is
   changed, all following words should be initialized with 0 to prevent
   non-zero values in unused padding words.

   We first describe the OTS address case.  In this case, the type word
   is followed by an OTS address word that encodes the index of the OTS
   key pair within the tree.  The next word encodes the chain address
   followed by a word that encodes the address of the hash function call
   within the chain.  The last word, called keyAndMask, is used to
   generate two different addresses for one hash function call.  The
   word is set to zero to generate the key.  To generate the n-byte
   bitmask, the word is set to one.







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                       An OTS hash address
                     +------------------------+
                     | layer address  (32 bit)|
                     +------------------------+
                     | tree address   (64 bit)|
                     +------------------------+
                     | type = 0       (32 bit)|
                     +------------------------+
                     | OTS address    (32 bit)|
                     +------------------------+
                     | chain address  (32 bit)|
                     +------------------------+
                     | hash address   (32 bit)|
                     +------------------------+
                     | keyAndMask     (32 bit)|
                     +------------------------+

   We now discuss the L-tree case, which means that the type word is set
   to one.  In that case the type word is followed by an L-tree address
   word that encodes the index of the leaf computed with this L-tree.
   The next word encodes the height of the node inside the L-tree and
   the following word encodes the index of the node at that height,
   inside the L-tree.  This time, the last word, keyAndMask, is used to
   generate three different addresses for one function call.  The word
   is set to zero to generate the key.  To generate the most significant
   n bytes of the 2n-byte bitmask, the word is set to one.  The least
   significant bytes are generated using the address with the word set
   to two.

                          An L-tree address
                     +------------------------+
                     | layer address  (32 bit)|
                     +------------------------+
                     | tree address   (64 bit)|
                     +------------------------+
                     | type = 1       (32 bit)|
                     +------------------------+
                     | L-tree address (32 bit)|
                     +------------------------+
                     | tree height    (32 bit)|
                     +------------------------+
                     | tree index     (32 bit)|
                     +------------------------+
                     | keyAndMask     (32 bit)|
                     +------------------------+

   We now describe the remaining type for the main tree hash addresses.
   In this case the type word is set to two, followed by a zero padding



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   of one word.  The next word encodes the height of the tree node being
   input for the next computation, followed by a word that encodes the
   index of this node at that height.  As for the L-tree addresses, the
   last word, keyAndMask, is used to generate three different addresses
   for one function call.  The word is set to zero to generate the key.
   To generate the most significant n bytes of the 2n-byte bitmask, the
   word is set to one.  The least significant bytes are generated using
   the address with the word set to two.

                       A hash tree address
                     +------------------------+
                     | layer address  (32 bit)|
                     +------------------------+
                     | tree address   (64 bit)|
                     +------------------------+
                     | type = 2       (32 bit)|
                     +------------------------+
                     | Padding = 0    (32 bit)|
                     +------------------------+
                     | tree height    (32 bit)|
                     +------------------------+
                     | tree index     (32 bit)|
                     +------------------------+
                     | keyAndMask     (32 bit)|
                     +------------------------+

   All fields within these addresses encode unsigned integers.  When
   describing the generation of addresses we use setter methods that
   take positive integers and set the bits of a field to the binary
   representation of that integer of the length of the field.  We
   furthermore assume that the setType() method sets the four words
   following the type word to zero.

2.6.  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, out_len) as follows.  If X is a len_X-
   byte string, and w is a member of the set {4, 16}, then base_w(X, w,
   out_len) outputs an array of out_len integers between 0 and w - 1.
   The length out_len is REQUIRED to be less than or equal to 8 * len_X
   / lg(w).









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   Algorithm 1: base_w

     Input: len_X-byte string X, int w, output length out_len
     Output: out_len int array basew

       int in = 0;
       int out = 0;
       unsigned int total = 0;
       int bits = 0;
       int consumed;

       for ( consumed = 0; consumed < out_len; consumed++ ) {
           if ( bits == 0 ) {
               total = X[in];
               in++;
               bits += 8;
           }
           bits -= lg(w);
           basew[out] = (total >> bits) AND (w - 1);
           out++;
       }
       return basew;

   For example, if X is the (big-endian) byte string 0x1234, then
   base_w(X, 16, 4) returns the array a = {1, 2, 3, 4}.

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

                 X (represented as base 16 numbers)
         +-----------+-----------+-----------+-----------+
         |     1     |     2     |     3     |     4     |
         +-----------+-----------+-----------+-----------+

                          base_w(X, 16, 4)
         +-----------+-----------+-----------+-----------+
         |     1     |     2     |     3     |     4     |
         +-----------+-----------+-----------+-----------+
             a[0]        a[1]        a[2]        a[3]

2.7.  Member Functions

   To simplify algorithm descriptions, we assume the existence of member
   functions.  If a complex data structure like a public key PK contains
   a value X then getX(PK) returns the value of X for this public key.



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   Accordingly, setX(PK, X, Y) sets value X in PK to the value held by
   Y.  Since camelCase is used for member function names, a value z may
   be referred to as Z in the function name, e.g.  getZ.

3.  Primitives

3.1.  WOTS+ One-Time Signatures

   This section describes the WOTS+ OTS system, in a version similar to
   [Huelsing13].  WOTS+ is a OTS 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 private 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.  A description of the algorithms
   for key generation, signing and verification follows.  Finally,
   pseudorandom key generation is discussed.

3.1.1.  WOTS+ Parameters

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

      n : the message length as well as the length of a private key,
      public key, or signature element in bytes.

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

   The parameters are used to compute values len, len_1 and len_2:

      len : the number of n-byte string elements in a WOTS+ private key,
      public key, and signature.  It is computed as len = len_1 + len_2,
      with len_1 = ceil(8n / lg(w)) and len_2 = floor(lg(len_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 n is the input length that can be
   processed by the signing algorithm.  It is often the length of a
   message digest.  The parameter w can be chosen from the set {4, 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 and 16 since these values yield optimal
   trade-offs and easy implementation.





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   WOTS+ parameters are implicitly included in algorithm inputs as
   needed.

3.1.1.1.  WOTS+ Functions

   The WOTS+ algorithm uses a keyed cryptographic hash function F.  F
   accepts and returns byte strings of length n using keys of length n.
   More detail on specific instantiations can be found in Section 5.
   Security requirements on F are discussed in Section 8.  In addition,
   WOTS+ uses a pseudorandom function PRF.  PRF takes as input an n-byte
   key and a 32-byte index and generates pseudorandom outputs of length
   n.  More detail on specific instantiations can be found in Section 5.
   Security requirements on PRF 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 outputs of PRF.  It takes an OTS hash address as
   input.  This address will have the first six 32-bit words set to
   encode the address of this chain.  In each iteration, PRF is used to
   generate a key for F and a bitmask that is XORed to the intermediate
   result before it is processed by F.  In the following, ADRS is a
   32-byte OTS hash address as specified in Section 2.5 and SEED is an
   n-byte string.  To generate the keys and bitmasks, PRF is called with
   SEED as key and ADRS as input.  The chaining function takes as input
   an n-byte string X, a start index i, a number of steps s, as well as
   ADRS and SEED.  The chaining function returns as output the value
   obtained by iterating F for s times on input X, using the outputs of
   PRF.






















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   Algorithm 2: chain - Chaining Function

     Input: Input string X, start index i, number of steps s,
     seed SEED, address ADRS
     Output: value of F iterated s times on X

     if ( s == 0 ) {
       return X;
     }
     if ( (i + s) > w - 1 ) {
       return NULL;
     }
     byte[n] tmp = chain(X, i, s - 1, SEED, ADRS);

     ADRS.setHashAddress(i + s - 1);
     ADRS.setKeyAndMask(0);
     KEY = PRF(SEED, ADRS);
     ADRS.setKeyAndMask(1);
     BM = PRF(SEED, ADRS);

     tmp = F(KEY, tmp XOR BM);
     return tmp;

3.1.3.  WOTS+ Private Key

   The private key in WOTS+, denoted by sk (s for secret), is a length
   len array of n-byte strings.  This private key MUST be only used to
   sign at most 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
   Section 3.1.7.  The following pseudocode (Algorithm 3) describes an
   algorithm for generating sk.

   Algorithm 3: WOTS_genSK - Generating a WOTS+ Private Key

     Input: /
     Output: WOTS+ private key sk

     for ( i = 0; i < len; i++ ) {
       initialize sk[i] with 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 len
   hash chains of length w.  The len n-byte strings in the private 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
   private key, the public key is also a length len array of n-byte
   strings.  To compute the hash chain, the chaining function (Algorithm
   2) is used.  An OTS hash address ADRS and a seed SEED have to be
   provided by the calling algorithm.  This address will encode the
   address of the WOTS+ key pair within a greater structure.  Hence, a
   WOTS+ algorithm MUST NOT manipulate any other parts of ADRS than the
   last three 32-bit words.  Please note that the SEED used here is
   public information also available to a verifier.  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

     Input: WOTS+ private key sk, address ADRS, seed SEED
     Output: WOTS+ public key pk

     for ( i = 0; i < len; i++ ) {
       ADRS.setChainAddress(i);
       pk[i] = chain(sk[i], 0, w - 1, SEED, ADRS);
     }
     return pk;

3.1.5.  WOTS+ Signature Generation

   A WOTS+ signature is a length len array of n-byte strings.  The WOTS+
   signature is generated by mapping a message to len integers between 0
   and w - 1.  To this end, the message is transformed into len_1 base w
   numbers using the base_w function defined in Section 2.6.  Next, a
   checksum is computed and appended to the transformed message as len_2
   base w numbers using the base_w function.  Note that the checksum may
   reach a maximum value of len_1 * (w - 1) * 2^8 and therefore depends
   on the parameters n and w.  For the parameter sets given in Section 5
   a 32-bit unsigned integer is sufficient.  If other parameter settings
   are used the size of the variable MUST be sufficiently large.  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.
   An OTS hash address ADRS and a seed SEED have to be provided by the
   calling algorithm.  This address will encode the address of the WOTS+
   key pair within a greater structure.  Hence, a WOTS+ algorithm MUST
   NOT manipulate any other parts of ADRS than the last three 32-bit
   words.  Please note that the SEED used here is public information



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   also available to a verifier.  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

     Input: Message M, WOTS+ private key sk, address ADRS, seed SEED
     Output: WOTS+ signature sig

     csum = 0;

     // Convert message to base w
     msg = base_w(M, w, len_1);

     // Compute checksum
     for ( i = 0; i < len_1; i++ ) {
           csum = csum + w - 1 - msg[i];
     }

     // Convert csum to base w
     csum = csum << ( 8 - ( ( len_2 * lg(w) ) % 8 ));
     len_2_bytes = ceil( ( len_2 * lg(w) ) / 8 );
     msg = msg || base_w(toByte(csum, len_2_bytes), w, len_2);
     for ( i = 0; i < len; i++ ) {
          ADRS.setChainAddress(i);
          sig[i] = chain(sk[i], 0, msg[i], SEED, ADRS);
     }
     return sig;

   The data format for a signature is given below.

   WOTS+ Signature

             +---------------------------------+
             |                                 |
             |           sig_ots[0]            |    n bytes
             |                                 |
             +---------------------------------+
             |                                 |
             ~              ....               ~
             |                                 |
             +---------------------------------+
             |                                 |
             |          sig_ots[len - 1]       |    n bytes
             |                                 |
             +---------------------------------+




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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 MUST be rejected.  An OTS hash
   address ADRS and a seed SEED have to be provided by the calling
   algorithm.  This address will encode the address of the WOTS+ key
   pair within a greater structure.  Hence, a WOTS+ algorithm MUST NOT
   manipulate any other parts of ADRS than the last three 32-bit words.
   Please note that the SEED used here is public information also
   available to a verifier.

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

     Input: Message M, WOTS+ signature sig, address ADRS, seed SEED
     Output: 'Temporary' WOTS+ public key tmp_pk

     csum = 0;

     // Convert message to base w
     msg = base_w(M, w, len_1);

     // Compute checksum
     for ( i = 0; i < len_1; i++ ) {
           csum = csum + w - 1 - msg[i];
     }

     // Convert csum to base w
     csum = csum << ( 8 - ( ( len_2 * lg(w) ) % 8 ));
     len_2_bytes = ceil( ( len_2 * lg(w) ) / 8 );
     msg = msg || base_w(toByte(csum, len_2_bytes), w, len_2);
     for ( i = 0; i < len; i++ ) {
          ADRS.setChainAddress(i);
          tmp_pk[i] = chain(sig[i], msg[i], w - 1 - msg[i], SEED, ADRS);
     }
     return tmp_pk;

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






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3.1.7.  Pseudorandom Key Generation

   An implementation MAY use a cryptographically secure pseudorandom
   method to generate the private 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 private key elements from a random
   n-byte string is that only this n-byte string needs to be stored
   instead of the full private key.  The key can be regenerated when
   needed.  The suggested method from [BDH11] can be described using
   PRF.  During key generation a uniformly random n-byte string S is
   sampled from a secure source of randomness.  This string S is stored
   as private key.  The private key elements are computed as sk[i] =
   PRF(S, toByte(i, 32)) whenever needed.  Please note that this seed S
   MUST be different from the seed SEED used to randomize the hash
   function calls.  Also, this seed S MUST be kept secret.

4.  Schemes

   In this section, the eXtended Merkle Signature Scheme (XMSS) is
   described using WOTS+.  XMSS comes in two flavors: 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.

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_msg, and a pseudorandom function
   PRF.  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 keyed hash function H to the result.  The
   bitmasks and the keys for the hash function H are generated from a
   (public) seed that is part of the public key using the pseudorandom
   function PRF.  The value corresponding to the root of the XMSS tree
   forms the XMSS public key together with the seed.



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   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 private key changes with every signature generation.  To
   prevent one-time private 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 private key
   contains an index that is updated with every signature generation,
   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 starting from a WOTS+ signature.  A
   verifier computes the root value and compares it to the respective
   value in the XMSS public key.  If they match, the signature is
   declared valid.  The XMSS private key consists of all WOTS+ private
   keys and the current 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 the message digest as well as of each
      node

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

   There are 2^h leaves in the tree.

   For XMSS and XMSS^MT, private and public keys are denoted by SK (S
   for secret) and PK.  For WOTS+, private and public keys are denoted
   by sk (s for secret) and pk, respectively.  XMSS and XMSS^MT
   signatures are denoted by Sig.  WOTS+ signatures are denoted by sig.

   XMSS and XMSS^MT parameters are implicitly included in algorithm
   inputs as needed.

4.1.2.  XMSS Hash Functions

   Besides the cryptographic hash function F and the pseudorandom
   function PRF required by WOTS+, XMSS uses two more functions:





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

      A cryptographic hash function H_msg.  H_msg accepts 3n-byte keys
      and byte strings of arbitrary length and returns an n-byte string.

   More detail on specific instantiations can be found in Section 5.
   Security requirements on H and H_msg are discussed in Section 8.

4.1.3.  XMSS Private Key

   An XMSS private key SK contains 2^h WOTS+ private keys, the leaf
   index idx of the next WOTS+ private key that has not yet been used,
   SK_PRF, an n-byte key to generate pseudorandom values for randomized
   message hashing, the n-byte value root, which is the root node of the
   tree and SEED, the n-byte public seed used to pseudorandomly generate
   bitmasks and hash function keys.  Although root and SEED formally
   would be considered only part of the public key, they are needed e.g.
   for signature generation and hence are also required for functions
   that do not take the public key as input.

   The leaf index idx is initialized to zero when the XMSS private key
   is created.  The key SK_PRF MUST be sampled from a secure source of
   randomness that follows the uniform distribution.  The WOTS+ private
   keys MUST either be generated as described in Section 3.1 or, to
   reduce the private key size, a cryptographic pseudorandom method MUST
   be used as discussed in Section 4.1.11.  SEED is generated as a
   uniformly random n-byte string.  Although SEED is public, it is
   critical for security that it is generated using a good entropy
   source.  The root node is generated as described below in the section
   on key generation (Section 4.1.7).  That section also contains an
   example algorithm for combined private and public key generation.

   For the following algorithm descriptions, the existence of a method
   getWOTS_SK(SK, i) is assumed.  This method takes as inputs an XMSS
   private key SK and an integer i and outputs the i^th WOTS+ private
   key of SK.

4.1.4.  Randomized Tree Hashing

   To improve readability we introduce a function RAND_HASH(LEFT, RIGHT,
   SEED, ADRS) that does the randomized hashing in the tree.  It takes
   as input two n-byte values LEFT and RIGHT that represent the left and
   the right half of the hash function input, the seed SEED used as key
   for PRF and the address ADRS of this hash function call.  RAND_HASH
   first uses PRF with SEED and ADRS to generate a key KEY and n-byte
   bitmasks BM_0, BM_1.  Then it returns the randomized hash H(KEY,
   (LEFT XOR BM_0) || (RIGHT XOR BM_1)).



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   Algorithm 7: RAND_HASH

     Input:  n-byte value LEFT, n-byte value RIGHT, seed SEED,
             address ADRS
     Output: n-byte randomized hash

     ADRS.setKeyAndMask(0);
     KEY = PRF(SEED, ADRS);
     ADRS.setKeyAndMask(1);
     BM_0 = PRF(SEED, ADRS);
     ADRS.setKeyAndMask(2);
     BM_1 = PRF(SEED, ADRS);

     return H(KEY, (LEFT XOR BM_0) || (RIGHT XOR BM_1));

4.1.5.  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 8) takes as input a WOTS+ public key pk and compresses it
   to a single n-byte value pk[0].  Towards this end it also takes an
   L-tree address ADRS as input that encodes the address of the L-tree,
   and the seed SEED.

   Algorithm 8: ltree

     Input: WOTS+ public key pk, address ADRS, seed SEED
     Output: n-byte compressed public key value pk[0]

     unsigned int len' = len;
     ADRS.setTreeHeight(0);
     while ( len' > 1 ) {
       for ( i = 0; i < floor(len' / 2); i++ ) {
         ADRS.setTreeIndex(i);
         pk[i] = RAND_HASH(pk[2i], pk[2i + 1], SEED, ADRS);
       }
       if ( len' % 2 == 1 ) {
         pk[floor(len' / 2)] = pk[len' - 1];
       }
       len' = ceil(len' / 2);
       ADRS.setTreeHeight(ADRS.getTreeHeight() + 1);
     }
     return pk[0];







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4.1.6.  TreeHash

   For the computation of the internal n-byte nodes of a Merkle tree,
   the subroutine treeHash (Algorithm 9) accepts an XMSS private key SK
   (including seed SEED), an unsigned integer s (the start index), an
   unsigned integer t (the target node height), and an address ADRS that
   encodes the address of the containing tree.  For the height of a node
   within a tree counting starts with the leaves at height zero.  The
   treeHash algorithm returns the root node of a tree of height t with
   the leftmost leaf being the hash of the WOTS+ pk with index s.  It is
   REQUIRED that s % 2^t = 0, i.e. that the leaf at index s is a left
   most leaf of a sub-tree of height t.  Otherwise the hash-addressing
   scheme fails.  The treeHash algorithm described here uses a stack
   holding up to (t - 1) nodes, with the usual stack functions push()
   and pop().  We furthermore assume that the height of a node (an
   unsigned integer) is stored alongside a node's value (an n-byte
   string) on the stack.

   Algorithm 9: treeHash

     Input: XMSS private key SK, start index s, target node height t,
            address ADRS
     Output: n-byte root node - top node on Stack

     if( s % (1 << t) != 0 ) return -1;
     for ( i = 0; i < 2^t; i++ ) {
       SEED = getSEED(SK);
       ADRS.setType(0);   // Type = OTS hash address
       ADRS.setOTSAddress(s + i);
       pk = WOTS_genPK (getWOTS_SK(SK, s + i), SEED, ADRS);
       ADRS.setType(1);   // Type = L-tree address
       ADRS.setLTreeAddress(s + i);
       node = ltree(pk, SEED, ADRS);
       ADRS.setType(2);   // Type = hash tree address
       ADRS.setTreeHeight(0);
       ADRS.setTreeIndex(i + s);
       while ( Top node on Stack has same height t' as node ) {
          ADRS.setTreeIndex((ADRS.getTreeIndex() - 1) / 2);
          node = RAND_HASH(Stack.pop(), node, SEED, ADRS);
          ADRS.setTreeHeight(ADRS.getTreeHeight() + 1);
       }
       Stack.push(node);
     }
     return Stack.pop();







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4.1.7.  XMSS Key Generation

   The XMSS key pair is computed as described in XMSS_keyGen (Algorithm
   10).  The XMSS public key PK consists of the root of the binary hash
   tree and the seed SEED, both also stored in SK.  The root is computed
   using treeHash.  For XMSS, there is only a single main tree.  Hence,
   the used address is set to the all-zero string in the beginning.
   Note that we do not define any specific format or handling for the
   XMSS private key SK by introducing this algorithm.  It relates to
   requirements described earlier and simply shows a basic but very
   inefficient example to initialize a private key.

   Algorithm 10: XMSS_keyGen - Generate an XMSS key pair

     Input: /
     Output: XMSS private key SK, XMSS public key PK

     // Example initialization for SK-specific contents
     idx = 0;
     for ( i = 0; i < 2^h; i++ ) {
       wots_sk[i] = WOTS_genSK();
     }
     initialize SK_PRF with a uniformly random n-byte string;
     setSK_PRF(SK, SK_PRF);

     // Initialization for common contents
     initialize SEED with a uniformly random n-byte string;
     setSEED(SK, SEED);
     setWOTS_SK(SK, wots_sk));
     ADRS = toByte(0, 32);
     root = treeHash(SK, 0, h, ADRS);

     SK = idx || wots_sk || SK_PRF || root || SEED;
     PK = OID || root || SEED;
     return (SK || PK);

   The above is just an example algorithm.  It is strongly RECOMMENDED
   to use pseudorandom key generation to reduce the private key size.
   Public and private key generation MAY be interleaved to save space.
   Especially, when a pseudorandom method is used to generate the
   private 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.







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   XMSS Public Key

            +---------------------------------+
            |          algorithm OID          |
            +---------------------------------+
            |                                 |
            |            root node            |     n bytes
            |                                 |
            +---------------------------------+
            |                                 |
            |              SEED               |     n bytes
            |                                 |
            +---------------------------------+

4.1.8.  XMSS Signature

   An XMSS signature is a (4 + n + (len + 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 message hashing (n bytes),

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

      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 y^th node on level x with y = 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)) XOR 1)

   The computation of the authentication path is discussed in
   Section 4.1.9.

   The data format for a signature is given below.






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   XMSS Signature

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

4.1.9.  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 using a
   random value r, idx_sig, the index of the WOTS+ key pair to be used,
   and the root value from the public key as key.  Then a WOTS+
   signature of the message digest is computed using the next unused
   WOTS+ private key.  Next, the authentication path is computed.
   Finally, the private key is updated, i.e.  idx is incremented.  An
   implementation MUST NOT output the signature before the private key
   is updated.

   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 the respective nodes.  The least storage-
   intensive alternative is to recompute all nodes for each signature
   online using the treeHash algorithm (Algorithm 9).  There exist
   several algorithms in between, with different time/storage trade-



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   offs.  For an overview, see [BDS09].  A further approach can be found
   in [KMN14].  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 signature verification operation.  The
   following version of buildAuth is given for completeness.  It is a
   simple example for understanding, but extremely inefficient.  The use
   of one of the alternative algorithms is strongly RECOMMENDED.

   Given an XMSS private key SK, all nodes in a tree are determined.
   Their value is defined in terms of treeHash (Algorithm 9).  Hence,
   one can compute the authentication path as follows:

   (Example) buildAuth - Compute the authentication path for the i^th
   WOTS+ key pair

     Input: XMSS private key SK, WOTS+ key pair index i, ADRS
     Output: Authentication path auth

     for ( j = 0; j < h; j++ ) {
       k = floor(i / (2^j)) XOR 1;
       auth[j] = treeHash(SK, k * 2^j, j, ADRS);
     }

   We split the description of the signature generation into two main
   algorithms.  The first one, treeSig (Algorithm 11), generates the
   main part of an XMSS signature and is also used by the multi-tree
   version XMSS^MT.  XMSS_sign (Algorithm 12) calls treeSig but handles
   message compression before and the private key update afterwards.

   The algorithm treeSig (Algorithm 11) described below calculates the
   WOTS+ signature on an n-byte message and the corresponding
   authentication path.  treeSig takes as inputs an n-byte message M',
   an XMSS private key SK, and an address ADRS.  It returns the
   concatenation of the WOTS+ signature sig_ots and authentication path
   auth.
















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   Algorithm 11: treeSig - Generate a WOTS+ signature on a message with
   corresponding authentication path

     Input: n-byte message M', XMSS private key SK, ADRS
     Output: Concatenation of WOTS+ signature sig_ots and
             authentication path auth

     idx_sig = getIdx(SK);
     auth = buildAuth(SK, idx_sig, ADRS);
     ADRS.setType(0);   // Type = OTS hash address
     ADRS.setOTSAddress(idx_sig);
     sig_ots = WOTS_sign(getWOTS_SK(SK, idx_sig),
                         M', getSEED(SK), ADRS);
     Sig = sig_ots || auth;
     return Sig;

   The algorithm XMSS_sign (Algorithm 12) described below calculates an
   updated private key SK and a signature on a message M.  XMSS_sign
   takes as inputs a message M of arbitrary length, and an XMSS private
   key SK.  It returns the byte string containing the concatenation of
   the updated private key SK and the signature Sig.

   Algorithm 12: XMSS_sign - Generate an XMSS signature and update the
   XMSS private key

     Input: Message M, XMSS private key SK
     Output: Updated SK, XMSS signature Sig

     idx_sig = getIdx(SK);
     setIdx(SK, idx_sig + 1);
     ADRS = toByte(0, 32);
     byte[n] r = PRF(getSK_PRF(SK), toByte(idx_sig, 32));
     byte[n] M' = H_msg(r || getRoot(SK) || (toByte(idx_sig, n)), M);
     Sig = idx_sig || r || treeSig(M', SK, ADRS);
     return (SK || Sig);

4.1.10.  XMSS Signature Verification

   An XMSS signature is verified by first computing the message digest
   using randomness r, index idx_sig, the root from PK and 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 and authentication path auth is used to
   compute an alternative root value for the tree.  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.




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   As for signature generation, we split verification into two parts to
   allow for reuse in the XMSS^MT description.  The steps also needed
   for XMSS^MT are done by the function XMSS_rootFromSig (Algorithm 13).
   XMSS_verify (Algorithm 14) calls XMSS_rootFromSig as a subroutine and
   handles the XMSS-specific steps.

   The main part of XMSS signature verification is done by the function
   XMSS_rootFromSig (Algorithm 13) described below.  XMSS_rootFromSig
   takes as inputs an index idx_sig, a WOTS+ signature sig_ots, an
   authentication path auth, an n-byte message M', seed SEED, and
   address ADRS.  XMSS_rootFromSig returns an n-byte string holding the
   value of the root of a tree defined by the input data.

   Algorithm 13: XMSS_rootFromSig - Compute a root node from a tree
   signature

     Input: index idx_sig, WOTS+ signature sig_ots, authentication path
            auth, n-byte message M', seed SEED, address ADRS
     Output: n-byte root value node[0]

     ADRS.setType(0);   // Type = OTS hash address
     ADRS.setOTSAddress(idx_sig);
     pk_ots = WOTS_pkFromSig(sig_ots, M', SEED, ADRS);
     ADRS.setType(1);   // Type = L-tree address
     ADRS.setLTreeAddress(idx_sig);
     byte[n][2] node;
     node[0] = ltree(pk_ots, SEED, ADRS);
     ADRS.setType(2);   // Type = hash tree address
     ADRS.setTreeIndex(idx_sig);
     for ( k = 0; k < h; k++ ) {
       ADRS.setTreeHeight(k);
       if ( (floor(idx_sig / (2^k)) % 2) == 0 ) {
         ADRS.setTreeIndex(ADRS.getTreeIndex() / 2);
         node[1] = RAND_HASH(node[0], auth[k], SEED, ADRS);
       } else {
         ADRS.setTreeIndex((ADRS.getTreeIndex() - 1) / 2);
         node[1] = RAND_HASH(auth[k], node[0], SEED, ADRS);
       }
       node[0] = node[1];
     }
     return node[0];

   The full XMSS signature verification is depicted below (Algorithm
   14).  It handles message compression, delegates the root computation
   to XMSS_rootFromSig, and compares the result to the value in the
   public key.  XMSS_verify takes an XMSS signature Sig, a message M,
   and an XMSS public key PK.  XMSS_verify returns true if and only if




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   Sig is a valid signature on M under public key PK.  Otherwise, it
   returns false.

   Algorithm 14: XMSS_verify - Verify an XMSS signature using the
   corresponding XMSS public key and a message

     Input: XMSS signature Sig, message M, XMSS public key PK
     Output: Boolean

     ADRS = toByte(0, 32);
     byte[n] M' = H_msg(r || getRoot(PK) || (toByte(idx_sig, n)), M);

     byte[n] node = XMSS_rootFromSig(idx_sig, sig_ots, auth, M',
                                     getSEED(PK), ADRS);
     if ( node == getRoot(PK) ) {
       return true;
     } else {
       return false;
     }

4.1.11.  Pseudorandom Key Generation

   An implementation MAY use a cryptographically secure pseudorandom
   method to generate the XMSS private key from a single n-byte value.
   For example, the method suggested in [BDH11] and explained below MAY
   be used.  Other methods, such as the one in [HRS16], 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] can be described using PRF.  During
   key generation a uniformly random n-byte string S is sampled from a
   secure source of randomness.  This seed S MUST NOT be confused with
   the public seed SEED.  The seed S MUST be independent of SEED and as
   it is the main secret, it MUST be kept secret.  This seed S is used
   to generate an n-byte value S_ots for each WOTS+ key pair.  The
   n-byte value S_ots can then be used to compute the respective WOTS+
   private key using the method described in Section 3.1.7.  The seeds
   for the WOTS+ key pairs are computed as S_ots[i] = PRF(S, toByte(i,
   32)) where i is the index of the WOTS+ key pair.  An advantage of
   this method is that a WOTS+ key can be computed using only len + 1
   evaluations of PRF when S is given.








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4.1.12.  Free Index Handling and Partial Private Keys

   Some applications might require to work with partial private keys or
   copies of private 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 private keys or
   copies of a private 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, a so-called hypertree.  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.

4.2.2.  XMSS^MT Key generation

   An XMSS^MT private key SK_MT (S for secret) consists of one reduced
   XMSS private key for each XMSS tree.  These reduced XMSS private keys
   just contain the WOTS+ private keys corresponding to that XMSS key
   pair and no pseudorandom function key, no index, no public seed, no
   root node.  Instead, SK_MT contains a single n-byte pseudorandom
   function key SK_PRF, a single (ceil(h / 8))-byte index idx_MT, a
   single n-byte seed SEED, and a single root value root which is the



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   root of the single tree on the top layer.  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 reduced XMSS private keys MUST either be generated as described
   in Section 4.1.3 or using a cryptographic pseudorandom method as
   discussed in Section 4.2.6.  As for XMSS, the PRF key SK_PRF MUST be
   sampled from a secure source of randomness that follows the uniform
   distribution.  SEED is generated as a uniformly random n-byte string.
   Although SEED is public, it is critical for security that it is
   generated using a good entropy source.  The root is the root node of
   the single XMSS tree on the top layer.  Its computation is explained
   below.  As for XMSS, root and SEED are public information and would
   classically be considered part of the public key.  However, as both
   are needed for signing, which only takes the private key, they are
   also part of SK_MT.

   This document does not define any specific format for the XMSS^MT
   private key SK_MT as it is not required for interoperability.  The
   algorithm descriptions below use a function getXMSS_SK(SK, x, y) that
   outputs the reduced private key of the x^th XMSS tree on the y^th
   layer.

   The XMSS^MT public key PK_MT contains the root of the single XMSS
   tree on layer d - 1 and the seed SEED.  These are the same values as
   in the private key SK_MT.  The pseudorandom function PRF keyed with
   SEED is used to generate the bitmasks and keys for all XMSS trees.
   XMSSMT_keyGen (Algorithm 15) shows example pseudocode to generate
   SK_MT and PK_MT.  The n-byte root node of the top layer tree is
   computed using treeHash.  The algorithm XMSSMT_keyGen outputs an
   XMSS^MT private key SK_MT and an XMSS^MT public key PK_MT.  The
   algorithm below gives an example of how the reduced XMSS private keys
   can be generated.  However, any of the above mentioned ways is
   acceptable as long as the cryptographic strength of the used method
   matches or superseeds that of the used XMSS^MT parameter set.















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   Algorithm 15: XMSSMT_keyGen - Generate an XMSS^MT key pair

     Input: /
     Output: XMSS^MT private key SK_MT, XMSS^MT public key PK_MT

     // Example initialization
     idx_MT = 0;
     setIdx(SK_MT, idx_MT);
     initialize SK_PRF with a uniformly random n-byte string;
     setSK_PRF(SK_MT, SK_PRF);
     initialize SEED with a uniformly random n-byte string;
     setSEED(SK_MT, SEED);

     // Generate reduced XMSS private keys
     ADRS = toByte(0, 32);
     for ( layer = 0; layer < d; layer++ ) {
        ADRS.setLayerAddress(layer);
        for ( tree = 0; tree <
              (1 << ((d - 1 - layer) * (h / d)));
              tree++ ) {
           ADRS.setTreeAddress(tree);
           for ( i = 0; i < 2^(h / d); i++ ) {
             wots_sk[i] = WOTS_genSK();
           }
           setXMSS_SK(SK_MT, wots_sk, tree, layer);
        }
     }

     SK = getXMSS_SK(SK_MT, 0, d - 1);
     setSEED(SK, SEED);
     root = treeHash(SK, 0, h / d, ADRS);
     setRoot(SK_MT, root);

     PK_MT = OID || root || SEED;
     return (SK_MT || PK_MT);

   The above is just an example algorithm.  It is strongly RECOMMENDED
   to use pseudorandom key generation to reduce the private key size.
   Public and private key generation MAY be interleaved to save space.
   Especially, when a pseudorandom method is used to generate the
   private key, generation MAY be delayed to the point when the
   respective WOTS+ key pair is needed by another algorithm.

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







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   XMSS^MT Public Key

            +---------------------------------+
            |          algorithm OID          |
            +---------------------------------+
            |                                 |
            |            root node            |     n bytes
            |                                 |
            +---------------------------------+
            |                                 |
            |              SEED               |     n bytes
            |                                 |
            +---------------------------------+

4.2.3.  XMSS^MT Signature

   An XMSS^MT signature Sig_MT is a byte string of length (ceil(h / 8) +
   n + (h + d * len) * 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 message hashing (n bytes),

      d reduced XMSS signatures ((h / d + len) * n bytes each).

   The reduced XMSS signatures only contain a WOTS+ signature sig_ots
   and an authentication path auth.  They contain no index idx and no
   byte string r.

   The data format for a signature is given below.




















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   XMSS^MT signature

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

4.2.4.  XMSS^MT Signature Generation

   To compute the XMSS^MT signature Sig_MT of a message M using an
   XMSS^MT private key SK_MT, XMSSMT_sign (Algorithm 16) described below
   uses treeSig as defined in Section 4.1.9.  First, the signature index
   is set to idx_sig.  Next, PRF is used to compute a pseudorandom
   n-byte string r.  This n-byte string, idx_sig, and the root node from
   PK_MT are then used to compute a randomized message digest of length
   n.  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 16: XMSSMT_sign - Generate an XMSS^MT signature and update
   the XMSS^MT private key

     Input: Message M, XMSS^MT private key SK_MT
     Output: Updated SK_MT, signature Sig_MT

     // Init
     ADRS = toByte(0, 32);
     SEED = getSEED(SK_MT);
     SK_PRF = getSK_PRF(SK_MT);
     idx_sig = getIdx(SK_MT);

     // Update SK_MT
     setIdx(SK_MT, idx_sig + 1);

     // Message compression
     byte[n] r = PRF(SK_PRF, toByte(idx_sig, 32));
     byte[n] M' = H_msg(r || getRoot(SK_MT) || (toByte(idx_sig, n)), M);

     // Sign
     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 || getXMSS_SK(SK_MT, idx_tree, 0) || SK_PRF
           || toByte(0, n) || SEED;
     ADRS.setLayerAddress(0);
     ADRS.setTreeAddress(idx_tree);
     Sig_tmp = treeSig(M', SK, ADRS);
     Sig_MT = Sig_MT || r || Sig_tmp;
     for ( j = 1; j < d; j++ ) {
        root = treeHash(SK, 0, h / d, ADRS);
        idx_leaf = (h / d) least significant bits of idx_tree;
        idx_tree = (h - j * (h / d)) most significant bits of idx_tree;
        SK = idx_leaf || getXMSS_SK(SK_MT, idx_tree, j) || SK_PRF
               || toByte(0, n) || SEED;
        ADRS.setLayerAddress(j);
        ADRS.setTreeAddress(idx_tree);
        Sig_tmp = treeSig(root, SK, ADRS);
        Sig_MT = Sig_MT || Sig_tmp;
     }
     return SK_MT || Sig_MT;

   Algorithm 16 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



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   computations over all signature generations.  Details can be found in
   [Huelsing13a].

4.2.5.  XMSS^MT Signature Verification

   XMSS^MT signature verification (Algorithm 17) can be summarized as d
   XMSS signature verifications with small changes.  First, the message
   is hashed.  The XMSS signatures are then all on n-byte values.
   Second, instead of comparing the computed root node to a given value,
   a signature on this root node 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.  The function
   getXMSSSignature(Sig_MT, i) returns the ith reduced XMSS signature
   from the XMSS^MT signature Sig_MT.  XMSSMT_verify takes as inputs an
   XMSS^MT signature Sig_MT, a message M and a public key PK_MT.
   XMSSMT_verify returns true if and only if Sig_MT is a valid signature
   on M under public key PK_MT.  Otherwise, it returns false.


































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   Algorithm 17: XMSSMT_verify - Verify an XMSS^MT signature Sig_MT on a
   message M using an XMSS^MT public key PK_MT

     Input: XMSS^MT signature Sig_MT, message M,
            XMSS^MT public key PK_MT
     Output: Boolean

     idx_sig = getIdx(Sig_MT);
     SEED = getSEED(PK_MT);
     ADRS = toByte(0, 32);

     byte[n] M' = H_msg(getR(Sig_MT) || getRoot(PK_MT)
                        || (toByte(idx_sig, n)), M);

     unsigned int idx_leaf
                   = (h / d) least significant bits of idx_sig;
     unsigned int idx_tree
                   = (h - h / d) most significant bits of idx_sig;
     Sig' = getXMSSSignature(Sig_MT, 0);
     ADRS.setLayerAddress(0);
     ADRS.setTreeAddress(idx_tree);
     byte[n] node = XMSS_rootFromSig(idx_leaf, getSig_ots(Sig'),
                                      getAuth(Sig'), M', SEED, ADRS);
     for ( j = 1; j < d; j++ ) {
        idx_leaf = (h / d) least significant bits of idx_tree;
        idx_tree = (h - j * h / d) most significant bits of idx_tree;
        Sig' = getXMSSSignature(Sig_MT, j);
        ADRS.setLayerAddress(j);
        ADRS.setTreeAddress(idx_tree);
        node = XMSS_rootFromSig(idx_leaf, getSig_ots(Sig'),
                              getAuth(Sig'), node, SEED, ADRS);
     }
     if ( node == getRoot(PK_MT) ) {
       return true;
     } else {
       return false;
     }

4.2.6.  Pseudorandom Key Generation

   Like for XMSS, an implementation MAY use a cryptographically secure
   pseudorandom method to generate the XMSS^MT private key from a single
   n-byte value.  For example, the method explained below MAY be used.
   Other methods, such as the one in [HRS16], MAY be used.  The choice
   of a pseudorandom method does not affect interoperability, but the
   cryptographic strength MUST match that of the used XMSS^MT
   parameters.




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   For XMSS^MT a method similar to that for XMSS and WOTS+ can be used.
   The method uses PRF.  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 private
   key using the method described in Section 4.1.11.  Let S[x][y] be the
   seed for the x^th XMSS private key on layer y.  The seeds are
   computed as S[x][y] = PRF(PRF(S, toByte(y, 32)), toByte(x, 32)).

4.2.7.  Free Index Handling and Partial Private Keys

   The content of Section 4.1.12 also applies to XMSS^MT.

5.  Parameter Sets

   This section provides a basic set of parameter sets which are assumed
   to cover most relevant applications.  Parameter sets for two
   classical security levels are defined.  Parameters with n = 32
   provide a classical security level of 256 bits.  Parameters with n =
   64 provide a classical security level of 512 bits.  Considering
   quantum-computer-aided attacks, these output sizes yield post-quantum
   security of 128 and 256 bits, respectively.

   For the n = 32 and n = 64 settings, we give parameters that use
   SHA2-256, SHA2-512 as defined in [FIPS180], and the SHA3/Keccak-based
   extendable output functions SHAKE-128, SHAKE-256 as defined in
   [FIPS202].  The parameter sets using SHA2-256 are mandatory for
   deployment and therefore MUST be provided by any implementation.  The
   remaining parameter sets specified in this document are OPTIONAL.

   SHA2 does not provide a keyed-mode itself.  To implement the keyed
   hash functions the following is used for SHA2 with n = 32:

      F: SHA2-256(toByte(0, 32) || KEY || M),

      H: SHA2-256(toByte(1, 32) || KEY || M),

      H_msg: SHA2-256(toByte(2, 32) || KEY || M),

      PRF: SHA2-256(toByte(3, 32) || KEY || M).

   Accordingly, for SHA2 with n = 64 we use:

      F: SHA2-512(toByte(0, 64) || KEY || M),

      H: SHA2-512(toByte(1, 64) || KEY || M),

      H_msg: SHA2-512(toByte(2, 64) || KEY || M),



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      PRF: SHA2-512(toByte(3, 64) || KEY || M).

   The n-byte padding is used for two reasons.  First, it is necessary
   that the internal compression function takes 2n-byte blocks but keys
   are n and 3n bytes long.  Second, the padding is used to achieve
   independence of the different function families.  Finally, for the
   PRF no full-fledged HMAC is needed as the message length is fixed,
   meaning that standard length extension attacks are not a concern
   here.  For that reason, the simpler construction above suffices.

   Similar constructions are used with SHA3.  To implement the keyed
   hash functions the following is used for SHA3 with n = 32:

      F: SHAKE128(toByte(0, 32) || KEY || M, 256),

      H: SHAKE128(toByte(1, 32) || KEY || M, 256),

      H_msg: SHAKE128(toByte(2, 32) || KEY || M, 256),

      PRF: SHAKE128(toByte(3, 32) || KEY || M, 256).

   Accordingly, for SHA3 with n = 64 we use:

      F: SHAKE256(toByte(0, 64) || KEY || M, 512),

      H: SHAKE256(toByte(1, 64) || KEY || M, 512),

      H_msg: SHAKE256(toByte(2, 64) || KEY || M, 512),

      PRF: SHAKE256(toByte(3, 64) || KEY || M, 512).

   As for SHA2, an initial n-byte identifier is used to achieve
   independence of the different function families.  While a shorter
   identifier could be used in case of SHA3, we use n bytes for
   consistency with the SHA2 implementations.

5.1.  WOTS+ Parameters

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









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             +--------------------+----------+----+----+-----+
             | Name               | F / PRF  | n  | w  | len |
             +--------------------+----------+----+----+-----+
             | REQUIRED:          |          |    |    |     |
             |                    |          |    |    |     |
             | WOTSP_SHA2-256_W16 | SHA2-256 | 32 | 16 | 67  |
             |                    |          |    |    |     |
             | OPTIONAL:          |          |    |    |     |
             |                    |          |    |    |     |
             | WOTSP_SHA2-512_W16 | SHA2-512 | 64 | 16 | 131 |
             |                    |          |    |    |     |
             | WOTSP_SHAKE128_W16 | SHAKE128 | 32 | 16 | 67  |
             |                    |          |    |    |     |
             | WOTSP_SHAKE256_W16 | SHAKE256 | 64 | 16 | 131 |
             +--------------------+----------+----+----+-----+

                                  Table 1

   The implementation of the single functions is done as described
   above.  XDR formats for WOTS+ are listed in Appendix A.

5.2.  XMSS Parameters

   To fully describe an XMSS signature method, the parameters n, w, and
   h, as well as the functions F, H, H_msg, and PRF MUST be specified.
   This section defines different XMSS signature systems, each of which
   is identified by a name.  We define parameter sets that implement the
   functions using SHA2 and SHA3 for n = 32 and n = 64 as described
   above.






















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        +-----------------------+-----------+----+----+-----+----+
        | Name                  | Functions | n  | w  | len | h  |
        +-----------------------+-----------+----+----+-----+----+
        | REQUIRED:             |           |    |    |     |    |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-256_W16_H10 | SHA2-256  | 32 | 16 | 67  | 10 |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-256_W16_H16 | SHA2-256  | 32 | 16 | 67  | 16 |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-256_W16_H20 | SHA2-256  | 32 | 16 | 67  | 20 |
        |                       |           |    |    |     |    |
        | OPTIONAL:             |           |    |    |     |    |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-512_W16_H10 | SHA2-512  | 64 | 16 | 131 | 10 |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-512_W16_H16 | SHA2-512  | 64 | 16 | 131 | 16 |
        |                       |           |    |    |     |    |
        | XMSS_SHA2-512_W16_H20 | SHA2-512  | 64 | 16 | 131 | 20 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE128_W16_H10 | SHAKE128  | 32 | 16 | 67  | 10 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE128_W16_H16 | SHAKE128  | 32 | 16 | 67  | 16 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE128_W16_H20 | SHAKE128  | 32 | 16 | 67  | 20 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE256_W16_H10 | SHAKE256  | 64 | 16 | 131 | 10 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE256_W16_H16 | SHAKE256  | 64 | 16 | 131 | 16 |
        |                       |           |    |    |     |    |
        | XMSS_SHAKE256_W16_H20 | SHAKE256  | 64 | 16 | 131 | 20 |
        +-----------------------+-----------+----+----+-----+----+

                                  Table 2

   The XDR formats for XMSS are listed in Appendix B.

5.3.  XMSS^MT Parameters

   To fully describe an XMSS^MT signature method, the parameters n, w,
   h, and d, as well as the functions F, H, H_msg, and PRF MUST be
   specified.  This section defines several XMSS^MT signature systems,
   each of which is identified by a name.  We define parameter sets that
   implement the functions using SHA2 and SHA3 for n = 32 and n = 64 as
   described above.

   +-----------------------------+-----------+----+----+-----+----+----+
   | Name                        | Functions | n  | w  | len | h  | d  |
   +-----------------------------+-----------+----+----+-----+----+----+



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   | REQUIRED:                   |           |    |    |     |    |    |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H20_D2  | SHA2-256  | 32 | 16 | 67  | 20 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H20_D4  | SHA2-256  | 32 | 16 | 67  | 20 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H40_D2  | SHA2-256  | 32 | 16 | 67  | 40 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H40_D4  | SHA2-256  | 32 | 16 | 67  | 40 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H40_D8  | SHA2-256  | 32 | 16 | 67  | 40 | 8  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H60_D3  | SHA2-256  | 32 | 16 | 67  | 60 | 3  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H60_D6  | SHA2-256  | 32 | 16 | 67  | 60 | 6  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-256_W16_H60_D12 | SHA2-256  | 32 | 16 | 67  | 60 | 12 |
   |                             |           |    |    |     |    |    |
   | OPTIONAL:                   |           |    |    |     |    |    |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H20_D2  | SHA2-512  | 64 | 16 | 131 | 20 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H20_D4  | SHA2-512  | 64 | 16 | 131 | 20 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H40_D2  | SHA2-512  | 64 | 16 | 131 | 40 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H40_D4  | SHA2-512  | 64 | 16 | 131 | 40 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H40_D8  | SHA2-512  | 64 | 16 | 131 | 40 | 8  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H60_D3  | SHA2-512  | 64 | 16 | 131 | 60 | 3  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H60_D6  | SHA2-512  | 64 | 16 | 131 | 60 | 6  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHA2-512_W16_H60_D12 | SHA2-512  | 64 | 16 | 131 | 60 | 12 |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H20_D2  | SHAKE128  | 32 | 16 | 67  | 20 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H20_D4  | SHAKE128  | 32 | 16 | 67  | 20 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H40_D2  | SHAKE128  | 32 | 16 | 67  | 40 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H40_D4  | SHAKE128  | 32 | 16 | 67  | 40 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H40_D8  | SHAKE128  | 32 | 16 | 67  | 40 | 8  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H60_D3  | SHAKE128  | 32 | 16 | 67  | 60 | 3  |
   |                             |           |    |    |     |    |    |



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   | XMSSMT_SHAKE128_W16_H60_D6  | SHAKE128  | 32 | 16 | 67  | 60 | 6  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE128_W16_H60_D12 | SHAKE128  | 32 | 16 | 67  | 60 | 12 |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H20_D2  | SHAKE256  | 64 | 16 | 131 | 20 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H20_D4  | SHAKE256  | 64 | 16 | 131 | 20 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H40_D2  | SHAKE256  | 64 | 16 | 131 | 40 | 2  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H40_D4  | SHAKE256  | 64 | 16 | 131 | 40 | 4  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H40_D8  | SHAKE256  | 64 | 16 | 131 | 40 | 8  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H60_D3  | SHAKE256  | 64 | 16 | 131 | 60 | 3  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H60_D6  | SHAKE256  | 64 | 16 | 131 | 60 | 6  |
   |                             |           |    |    |     |    |    |
   | XMSSMT_SHAKE256_W16_H60_D12 | SHAKE256  | 64 | 16 | 131 | 60 | 12 |
   +-----------------------------+-----------+----+----+-----+----+----+

                                  Table 3

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

6.  Rationale

   The goal of this note is to describe the WOTS+, XMSS and XMSS^MT
   algorithms following the scientific literature.  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.

   This note slightly deviates from the scientific literature using a
   tweak that prevents multi-user / multi-target attacks against H_msg.
   To this end, the public key as well as the index of the used one-time
   key pair become part of the hash function key.  Thereby we achieve a
   domain separation that forces an attacker to decide which hash value
   to attack.

   For the generation of the randomness used for randomized message
   hashing, we apply a PRF, keyed with a secret value, to the index of
   the used one-time key pair instead of the message.  The reason is
   that this requires to process the message only once instead of twice.
   For long messages this improves speed and simplifies implementations
   on resource constrained devices that cannot hold the entire message
   in storage.





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   We give one mandatory set of parameters using SHA2-256.  The reasons
   are twofold.  On the one hand, SHA2-256 is part of most cryptographic
   libraries.  On the other hand, a 256-bit hash function leads to
   parameters that provide 128 bit of security even against quantum-
   computer-aided attacks.  A post-quantum security level of 256 bit
   seems overly conservative.  However, to prepare for possible
   cryptanalytic breakthroughs, we also provide OPTIONAL parameter sets
   using the less widely supported SHA2-512, SHAKE-256, and SHAKE-512
   functions.

   We suggest the value w = 16 for the Winternitz parameter.  No bigger
   values are included since the decrease in signature size then becomes
   less significant.  Furthermore, the value w = 16 considerably
   simplifies the implementations of some of the algorithms.  Please
   note that we do allow w = 4, but limit the specified parameter sets
   to w = 16 for efficiency reasons.

   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 private 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 further 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 detail to make
   interoperability between independent implementations possible.  Each
   entry in the registry contains the following elements:

      a short name, such as "XMSS_SHA2-256_W16_H20",



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      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 [RFC5226].

   The WOTS+ registry is as follows.

        +---------------------+-------------+--------------------+
        | Name                |  Reference  | Numeric Identifier |
        +---------------------+-------------+--------------------+
        | WOTSP_SHA2-256_W16  | Section 5.1 |     0x01000001     |
        |                     |             |                    |
        | WOTSP_SHA2-512_W16  | Section 5.1 |     0x02000002     |
        |                     |             |                    |
        | WOTSP_SHAKE128_W16  | Section 5.1 |     0x03000003     |
        |                     |             |                    |
        | WOTSP_SHAKE256_W16  | Section 5.1 |     0x04000004     |
        +---------------------+-------------+--------------------+

                                  Table 4

   The XMSS registry is as follows.














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       +------------------------+-------------+--------------------+
       | Name                   |  Reference  | Numeric Identifier |
       +------------------------+-------------+--------------------+
       | XMSS_SHA2-256_W16_H10  | Section 5.2 |     0x01000001     |
       |                        |             |                    |
       | XMSS_SHA2-256_W16_H16  | Section 5.2 |     0x02000002     |
       |                        |             |                    |
       | XMSS_SHA2-256_W16_H20  | Section 5.2 |     0x03000003     |
       |                        |             |                    |
       | XMSS_SHA2-512_W16_H10  | Section 5.2 |     0x04000004     |
       |                        |             |                    |
       | XMSS_SHA2-512_W16_H16  | Section 5.2 |     0x05000005     |
       |                        |             |                    |
       | XMSS_SHA2-512_W16_H20  | Section 5.2 |     0x06000006     |
       |                        |             |                    |
       | XMSS_SHAKE128_W16_H10  | Section 5.2 |     0x07000007     |
       |                        |             |                    |
       | XMSS_SHAKE128_W16_H16  | Section 5.2 |     0x08000008     |
       |                        |             |                    |
       | XMSS_SHAKE128_W16_H20  | Section 5.2 |     0x09000009     |
       |                        |             |                    |
       | XMSS_SHAKE256_W16_H10  | Section 5.2 |     0x0a00000a     |
       |                        |             |                    |
       | XMSS_SHAKE256_W16_H16  | Section 5.2 |     0x0b00000b     |
       |                        |             |                    |
       | XMSS_SHAKE256_W16_H20  | Section 5.2 |     0x0c00000c     |
       +------------------------+-------------+--------------------+

                                  Table 5

   The XMSS^MT registry is as follows.

    +-----------------------------+-------------+--------------------+
    | Name                        |  Reference  | Numeric Identifier |
    +-----------------------------+-------------+--------------------+
    | XMSSMT_SHA2-256_W16_H20_D2  | Section 5.3 |     0x01000001     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H20_D4  | Section 5.3 |     0x02000002     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H40_D2  | Section 5.3 |     0x03000003     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H40_D4  | Section 5.3 |     0x04000004     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H40_D8  | Section 5.3 |     0x05000005     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H60_D3  | Section 5.3 |     0x06000006     |
    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H60_D6  | Section 5.3 |     0x07000007     |



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    |                             |             |                    |
    | XMSSMT_SHA2-256_W16_H60_D12 | Section 5.3 |     0x08000008     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H20_D2  | Section 5.3 |     0x09000009     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H20_D4  | Section 5.3 |     0x0a00000a     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H40_D2  | Section 5.3 |     0x0b00000b     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H40_D4  | Section 5.3 |     0x0c00000c     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H40_D8  | Section 5.3 |     0x0d00000d     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H60_D3  | Section 5.3 |     0x0e00000e     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H60_D6  | Section 5.3 |     0x0f00000f     |
    |                             |             |                    |
    | XMSSMT_SHA2-512_W16_H60_D12 | Section 5.3 |     0x01010101     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H20_D2  | Section 5.3 |     0x02010102     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H20_D4  | Section 5.3 |     0x03010103     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H40_D2  | Section 5.3 |     0x04010104     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H40_D4  | Section 5.3 |     0x05010105     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H40_D8  | Section 5.3 |     0x06010106     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H60_D3  | Section 5.3 |     0x07010107     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H60_D6  | Section 5.3 |     0x08010108     |
    |                             |             |                    |
    | XMSSMT_SHAKE128_W16_H60_D12 | Section 5.3 |     0x09010109     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H20_D2  | Section 5.3 |     0x0a01010a     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H20_D4  | Section 5.3 |     0x0b01010b     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H40_D2  | Section 5.3 |     0x0c01010c     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H40_D4  | Section 5.3 |     0x0d01010d     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H40_D8  | Section 5.3 |     0x0e01010e     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H60_D3  | Section 5.3 |     0x0f01010f     |
    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H60_D6  | Section 5.3 |     0x01020201     |



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    |                             |             |                    |
    | XMSSMT_SHAKE256_W16_H60_D12 | Section 5.3 |     0x02020202     |
    +-----------------------------+-------------+--------------------+

                                  Table 6

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

8.  Security Considerations

   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
   new 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 for 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.  The given values for bit
   security were estimated according to [HRS16].

8.1.  Security Proofs

   A full security proof for all schemes described in this document can
   be found in [HRS16].  This proof shows 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 in any of the described
   schemes.  The proof in [HRS16] considers a different initial message
   compression than the randomized hashing used here.  We comment on
   this below.  For the original schemes, 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.

   More specifically, the requirements on the used functions are that F
   and H are post-quantum multi-function multi-target second-preimage
   resistant keyed functions, F fulfills an additional statistical
   requirement that roughly says that most images have at least two



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   preimages, PRF is a post-quantum pseudorandom function, H_msg is a
   post-quantum multi-target extended target collision resistant keyed
   hash function.  For detailed definitions of these properties see
   [HRS16].  To give some intuition: Multi-function multi-target second
   preimage resistance is an extension of second preimage resistance to
   keyed hash functions, covering the case where an adversary succeeds
   if it finds a second preimage for one out of many values.  The same
   holds for multi-target extended target collision resistance which
   just lacks the multi-function identifier as target collision
   resistance already considers keyed hash functions.  The proof in
   [HRS16] splits PRF into two functions.  When PRF is used for
   pseudorandom key generation or generation of randomness for
   randomized message hashing it is still considered a pseudorandom
   function.  Whenever PRF is used to generate bitmasks and hash
   function keys it is modeled as a random oracle.  This is due to
   technical reasons in the proof and an implementation using a
   pseudorandom function is secure.

   The proof in [HRS16] considers classical randomized hashing for the
   initial message compression, i.e., H(r, M) instead of H(r ||
   getRoot(PK) || index, M).  This classical randomized hashing allows
   to get a security reduction from extended target collision resistance
   [HRS16], a property that is conjectured to be strictly weaker than
   collision resistance.  However, it turns out that in this case, an
   attacker could still launch a multi-target attack even against
   multiple users at the same time.  The reason is that the adversary
   attacking u users at the same time learns u * 2^h randomized hashes
   H(r_i_j || M_i_j) with signature index i in [0, 2^h - 1] and user
   index j in [0, u].  It suffices to find a single pair (r*, M*) such
   that H(r* || M*) = H(r_i_u || M_i_u) for one out of the u * 2^h
   learned hashes.  Hence, an attacker can do a brute force search in
   time 2^n / u * 2^h instead of 2^n.

   The indexed randomized hashing H(r || getRoot(PK) || toByte(idx, n),
   M) used in this work makes the hash function calls position- and
   user-dependent.  This thwarts the above attack because each hash
   function evaluation during an attack can only target one of the
   learned randomized hash values.  More specifically, an attacker now
   has to decide which index idx and which root value to use for each
   query.  If one assumes that the used hash function is a random
   function it can be shown that a multi user existential forgery attack
   that targets this message compression has a complexity of 2^n hash
   function calls.

   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 and pseudorandom functions assuming
   conventional and quantum adversaries.  At the time of writing,



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   generic attacks are the best known attacks for the parameters
   suggested in the classical setting.  Also in the quantum setting
   there are no dedicated attacks known that perform better than generic
   attacks.  Nevertheless, the topic of quantum cryptanalysis of hash
   functions is not as well understood as in the classical setting.

8.2.  Minimal 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, either on collision resistance or on extended target
   collision resistance if randomized hashing is used for message
   compression.  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 a machine is an open conjecture.  However, significant progress
   was made over the last few years in this regard.  Hence, we consider
   it a matter of risk assessment to prepare for this case.

   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 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 in [HRS16] that
   variants of Grover's algorithm are the optimal generic attacks
   against the security properties of hash functions required for the
   described scheme.

   As stated above, we only consider generic attacks here, as
   cryptographic hash functions should be deprecated as soon as there
   exist dedicated attacks that perform significantly better.  This also
   applies for the quantum setting.  As soon as there exist dedicated
   quantum attacks against the used hash function that perform
   significantly better than the described generic attacks these hash
   functions should not be used anymore for the described schemes or the
   computation of the security level has to be redone.






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9.  Acknowledgements

   We would like to thank Johannes Braun, Peter Campbell, Scott Fluhrer,
   Burt Kaliski, Adam Langley, David McGrew, Rafael Misoczki, Sean
   Parkinson, Joost Rijneveld, Sebastian Roland, and the Keccak team for
   their help and comments.

10.  References

10.1.  Normative References

   [FIPS180]  National Institute of Standards and Technology, "Secure
              Hash Standard (SHS)", FIPS 180-4, 2012.

   [FIPS202]  National Institute of Standards and Technology, "SHA-3
              Standard: Permutation-Based Hash and Extendable-Output
              Functions", FIPS 202, 2015.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

   [RFC4506]  Eisler, M., Ed., "XDR: External Data Representation
              Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May
              2006, <http://www.rfc-editor.org/info/rfc4506>.

   [RFC5226]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", BCP 26, RFC 5226,
              DOI 10.17487/RFC5226, May 2008,
              <http://www.rfc-editor.org/info/rfc5226>.

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.









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   [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", Lecture Notes in
              Computer Science volume 9056. Advances in Cryptology -
              EUROCRYPT, 2015.

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

   [HRS16]    Huelsing, A., Rijneveld, J., and F. Song, "Mitigating
              Multi-Target Attacks in Hash-based Signatures", Lecture
              Notes in Computer Science volume 9614. Public-Key
              Cryptography - PKC 2016, 2016.

   [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,
              <http://tuprints.ulb.tu-darmstadt.de/3651/1/Thesis.pdf>.

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

   [KMN14]    Knecht, M., Meier, W., and C. Nicola, "A space- and time-
              efficient Implementation of the Merkle Tree Traversal
              Algorithm", Computing Research Repository
              (CoRR). arXiv:1409.4081, 2014.

   [MCF17]    McGrew, D., Curcio, M., and S. Fluhrer, "Hash-Based
              Signatures", Work in Progress, draft-mcgrew-hash-sigs-06,
              March 2017, <http://www.ietf.org/archive/id/
              draft-mcgrew-hash-sigs-06.txt>.

   [Merkle79]
              Merkle, R., "Secrecy, Authentication, and Public Key
              Systems", Stanford University Information Systems
              Laboratory Technical Report 1979-1, 1979,
              <http://www.merkle.com/papers/Thesis1979.pdf>.




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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.  Note that this representation includes all optional
   parameter sets.  The same applies for the XMSS and XMSS^MT formats
   below.

   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_sha2-256_w16 = 0x01000001,
        wotsp_sha2-512_w16 = 0x02000002,
        wotsp_shake128_w16 = 0x03000003,
        wotsp_shake256_w16 = 0x04000004,
      };


   WOTS+ signatures are defined using XDR syntax as follows:


      /* Byte strings */

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

      union ots_signature switch (ots_algorithm_type type) {
        case wotsp_sha2-256_w16:
        case wotsp_shake128_w16:
          bytestring32 ots_sig_n32_len67[67];

        case wotsp_sha2-512_w16:
        case wotsp_shake256_w16:
          bytestring64 ots_sig_n64_len18[131];

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





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


      union ots_pubkey switch (ots_algorithm_type type) {
        case wotsp_sha2-256_w16:
        case wotsp_shake128_w16:
          bytestring32 ots_pubk_n32_len67[67];

        case wotsp_sha2-512_w16:
        case wotsp_shake256_w16:
          bytestring64 ots_pubk_n64_len18[131];

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


Appendix B.  XMSS XDR Formats

































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   XMSS parameter sets are defined using XDR syntax as follows:


      /* Byte strings */

      typedef opaque bytestring4[4];

      /* Definition of parameter sets */

      enum xmss_algorithm_type {
        xmss_reserved         = 0x00000000,

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

        xmss_sha2-256_w16_h10 = 0x01000001,
        xmss_sha2-256_w16_h16 = 0x02000002,
        xmss_sha2-256_w16_h20 = 0x03000003,

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

        xmss_sha2-512_w16_h10 = 0x04000004,
        xmss_sha2-512_w16_h16 = 0x05000005,
        xmss_sha2-512_w16_h20 = 0x06000006,

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

        xmss_shake128_w16_h10 = 0x07000007,
        xmss_shake128_w16_h16 = 0x08000008,
        xmss_shake128_w16_h20 = 0x09000009,

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

        xmss_shake256_w16_h10 = 0x0a00000a,
        xmss_shake256_w16_h16 = 0x0b00000b,
        xmss_shake256_w16_h20 = 0x0c00000c,
      };


   XMSS signatures are defined using XDR syntax as follows:


      /* Authentication path types */

      union xmss_path switch (xmss_algorithm_type type) {
        case xmss_sha2-256_w16_h10:
        case xmss_shake128_w16_h10:
          bytestring32 path_n32_t10[10];




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        case xmss_sha2-256_w16_h16:
        case xmss_shake128_w16_h16:
          bytestring32 path_n32_t16[16];

        case xmss_sha2-256_w16_h20:
        case xmss_shake128_w16_h20:
          bytestring32 path_n32_t20[20];

        case xmss_sha2-512_w16_h10:
        case xmss_shake256_w16_h10:
          bytestring64 path_n64_t10[10];

        case xmss_sha2-512_w16_h16:
        case xmss_shake256_w16_h16:
          bytestring64 path_n64_t16[16];

        case xmss_sha2-512_w16_h20:
        case xmss_shake256_w16_h20:
          bytestring64 path_n64_t20[20];

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

      /* Types for XMSS random strings */

      union random_string_xmss switch (xmss_algorithm_type type) {
        case xmss_sha2-256_w16_h10:
        case xmss_sha2-256_w16_h16:
        case xmss_sha2-256_w16_h20:
        case xmss_shake128_w16_h10:
        case xmss_shake128_w16_h16:
        case xmss_shake128_w16_h20:
          bytestring32 rand_n32;

        case xmss_sha2-512_w16_h10:
        case xmss_sha2-512_w16_h16:
        case xmss_sha2-512_w16_h20:
        case xmss_shake256_w16_h10:
        case xmss_shake256_w16_h16:
        case xmss_shake256_w16_h20:
          bytestring64 rand_n64;

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

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



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      union xmss_ots_signature switch (xmss_algorithm_type type) {
        case xmss_sha2-256_w16_h10:
        case xmss_sha2-256_w16_h16:
        case xmss_sha2-256_w16_h20:
          wotsp_sha2-256_w16;

        case xmss_sha2-512_w16_h10:
        case xmss_sha2-512_w16_h16:
        case xmss_sha2-512_w16_h20:
          wotsp_sha2-512_w16;

        case xmss_shake128_w16_h10:
        case xmss_shake128_w16_h16:
        case xmss_shake128_w16_h20:
          wotsp_shake128_w16;

        case xmss_shake256_w16_h10:
        case xmss_shake256_w16_h16:
        case xmss_shake256_w16_h20:
          wotsp_shake256_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;
        /* authentication path */
        xmss_path nodes;
      };


   XMSS public keys are defined using XDR syntax as follows:


      /* Types for bitmask seed */

      union seed switch (xmss_algorithm_type type) {
        case xmss_sha2-256_w16_h10:
        case xmss_sha2-256_w16_h16:
        case xmss_sha2-256_w16_h20:



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        case xmss_shake128_w16_h10:
        case xmss_shake128_w16_h16:
        case xmss_shake128_w16_h20:
          bytestring32 seed_n32;

        case xmss_sha2-512_w16_h10:
        case xmss_sha2-512_w16_h16:
        case xmss_sha2-512_w16_h20:
        case xmss_shake256_w16_h10:
        case xmss_shake256_w16_h16:
        case xmss_shake256_w16_h20:
          bytestring64 seed_n64;

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

      /* Types for XMSS root node */

      union xmss_root switch (xmss_algorithm_type type) {
        case xmss_sha2-256_w16_h10:
        case xmss_sha2-256_w16_h16:
        case xmss_sha2-256_w16_h20:
        case xmss_shake128_w16_h10:
        case xmss_shake128_w16_h16:
        case xmss_shake128_w16_h20:
          bytestring32 root_n32;

        case xmss_sha2-512_w16_h10:
        case xmss_sha2-512_w16_h16:
        case xmss_sha2-512_w16_h20:
        case xmss_shake256_w16_h10:
        case xmss_shake256_w16_h16:
        case xmss_shake256_w16_h20:
          bytestring64 root_n64;

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

      /* XMSS public key structure */

      struct xmss_public_key {
        xmss_root root;  /* Root node */
        seed SEED;  /* Seed for bitmasks */
      };





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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,

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

        xmssmt_sha2-256_w16_h20_d2  = 0x01000001,
        xmssmt_sha2-256_w16_h20_d4  = 0x02000002,
        xmssmt_sha2-256_w16_h40_d2  = 0x03000003,
        xmssmt_sha2-256_w16_h40_d4  = 0x04000004,
        xmssmt_sha2-256_w16_h40_d8  = 0x05000005,
        xmssmt_sha2-256_w16_h60_d3  = 0x06000006,
        xmssmt_sha2-256_w16_h60_d6  = 0x07000007,
        xmssmt_sha2-256_w16_h60_d12 = 0x08000008,

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

        xmssmt_sha2-512_w16_h20_d2  = 0x09000009,
        xmssmt_sha2-512_w16_h20_d4  = 0x0a00000a,
        xmssmt_sha2-512_w16_h40_d2  = 0x0b00000b,
        xmssmt_sha2-512_w16_h40_d4  = 0x0c00000c,
        xmssmt_sha2-512_w16_h40_d8  = 0x0d00000d,
        xmssmt_sha2-512_w16_h60_d3  = 0x0e00000e,
        xmssmt_sha2-512_w16_h60_d6  = 0x0f00000f,
        xmssmt_sha2-512_w16_h60_d12 = 0x01010101,

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

        xmssmt_shake128_w16_h20_d2  = 0x02010102,
        xmssmt_shake128_w16_h20_d4  = 0x03010103,
        xmssmt_shake128_w16_h40_d2  = 0x04010104,
        xmssmt_shake128_w16_h40_d4  = 0x05010105,
        xmssmt_shake128_w16_h40_d8  = 0x06010106,
        xmssmt_shake128_w16_h60_d3  = 0x07010107,
        xmssmt_shake128_w16_h60_d6  = 0x08010108,
        xmssmt_shake128_w16_h60_d12 = 0x09010109,



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

        xmssmt_shake256_w16_h20_d2  = 0x0a01010a,
        xmssmt_shake256_w16_h20_d4  = 0x0b01010b,
        xmssmt_shake256_w16_h40_d2  = 0x0c01010c,
        xmssmt_shake256_w16_h40_d4  = 0x0d01010d,
        xmssmt_shake256_w16_h40_d8  = 0x0e01010e,
        xmssmt_shake256_w16_h60_d3  = 0x0f01010f,
        xmssmt_shake256_w16_h60_d6  = 0x01020201,
        xmssmt_shake256_w16_h60_d12 = 0x02020202,
      };


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


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

      union idx_sig_xmssmt switch (xmss_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake256_w16_h20_d4:
          bytestring3 idx3;

        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_sha2-512_w16_h40_d8:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake128_w16_h40_d8:
        case xmssmt_shake256_w16_h40_d2:
        case xmssmt_shake256_w16_h40_d4:
        case xmssmt_shake256_w16_h40_d8:
          bytestring5 idx5;

        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_sha2-512_w16_h60_d3:



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        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_shake128_w16_h60_d3:
        case xmssmt_shake128_w16_h60_d6:
        case xmssmt_shake128_w16_h60_d12:
        case xmssmt_shake256_w16_h60_d3:
        case xmssmt_shake256_w16_h60_d6:
        case xmssmt_shake256_w16_h60_d12:
          bytestring8 idx8;

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

      union random_string_xmssmt switch (xmssmt_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake128_w16_h40_d8:
        case xmssmt_shake128_w16_h60_d3:
        case xmssmt_shake128_w16_h60_d6:
        case xmssmt_shake128_w16_h60_d12:
          bytestring32 rand_n32;

        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_sha2-512_w16_h40_d8:
        case xmssmt_sha2-512_w16_h60_d3:
        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake256_w16_h20_d4:
        case xmssmt_shake256_w16_h40_d2:
        case xmssmt_shake256_w16_h40_d4:
        case xmssmt_shake256_w16_h40_d8:
        case xmssmt_shake256_w16_h60_d3:
        case xmssmt_shake256_w16_h60_d6:



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        case xmssmt_shake256_w16_h60_d12:
          bytestring64 rand_n64;

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

      /* Type for reduced XMSS signatures */

      union xmss_reduced (xmss_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake128_w16_h60_d6:
          bytestring32 xmss_reduced_n32_t77[77];

        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake128_w16_h40_d8:
        case xmssmt_shake128_w16_h60_d12:
          bytestring32 xmss_reduced_n32_t72[72];

        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake128_w16_h60_d3:
          bytestring32 xmss_reduced_n32_t87[87];

        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake256_w16_h40_d4:
        case xmssmt_shake256_w16_h60_d6:
          bytestring64 xmss_reduced_n32_t141[141];

        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_sha2-512_w16_h40_d8:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_shake256_w16_h20_d4:
        case xmssmt_shake256_w16_h40_d8:
        case xmssmt_shake256_w16_h60_d12:
          bytestring64 xmss_reduced_n32_t136[136];




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        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_sha2-512_w16_h60_d3:
        case xmssmt_shake256_w16_h40_d2:
        case xmssmt_shake256_w16_h60_d3:
          bytestring64 xmss_reduced_n32_t151[151];

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

      /* xmss_reduced_array depends on d */

      union xmss_reduced_array (xmss_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake256_w16_h40_d2:
          xmss_reduced xmss_red_arr_d2[2];

        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_sha2-512_w16_h60_d3:
        case xmssmt_shake128_w16_h60_d3:
        case xmssmt_shake256_w16_h60_d3:
          xmss_reduced xmss_red_arr_d3[3];

        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake256_w16_h20_d4:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake256_w16_h40_d4:
          xmss_reduced xmss_red_arr_d4[4];

        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_shake128_w16_h60_d6:
        case xmssmt_shake256_w16_h60_d6:
          xmss_reduced xmss_red_arr_d6[6];

        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-512_w16_h40_d8:
        case xmssmt_shake128_w16_h40_d8:



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        case xmssmt_shake256_w16_h40_d8:
          xmss_reduced xmss_red_arr_d8[8];

        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_shake128_w16_h60_d12:
        case xmssmt_shake256_w16_h60_d12:
          xmss_reduced xmss_red_arr_d12[12];

        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;
        /* Array of d reduced XMSS signatures */
        xmss_reduced_array;
      };


   XMSS^MT public keys are defined using XDR syntax as follows:


      /* Types for bitmask seed */

      union seed switch (xmssmt_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake128_w16_h60_d6:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake128_w16_h40_d8:
        case xmssmt_shake128_w16_h60_d12:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake128_w16_h60_d3:
          bytestring32 seed_n32;



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        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_sha2-512_w16_h40_d8:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_sha2-512_w16_h60_d3:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake256_w16_h40_d4:
        case xmssmt_shake256_w16_h60_d6:
        case xmssmt_shake256_w16_h20_d4:
        case xmssmt_shake256_w16_h40_d8:
        case xmssmt_shake256_w16_h60_d12:
        case xmssmt_shake256_w16_h40_d2:
        case xmssmt_shake256_w16_h60_d3:
          bytestring64 seed_n64;

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

      /* Types for XMSS^MT root node */

      union xmssmt_root switch (xmssmt_algorithm_type type) {
        case xmssmt_sha2-256_w16_h20_d2:
        case xmssmt_sha2-256_w16_h20_d4:
        case xmssmt_sha2-256_w16_h40_d2:
        case xmssmt_sha2-256_w16_h40_d4:
        case xmssmt_sha2-256_w16_h40_d8:
        case xmssmt_sha2-256_w16_h60_d3:
        case xmssmt_sha2-256_w16_h60_d6:
        case xmssmt_sha2-256_w16_h60_d12:
        case xmssmt_shake128_w16_h20_d2:
        case xmssmt_shake128_w16_h20_d4:
        case xmssmt_shake128_w16_h40_d2:
        case xmssmt_shake128_w16_h40_d4:
        case xmssmt_shake128_w16_h40_d8:
        case xmssmt_shake128_w16_h60_d3:
        case xmssmt_shake128_w16_h60_d6:
        case xmssmt_shake128_w16_h60_d12:
          bytestring32 root_n32;

        case xmssmt_sha2-512_w16_h20_d2:
        case xmssmt_sha2-512_w16_h20_d4:
        case xmssmt_sha2-512_w16_h40_d2:
        case xmssmt_sha2-512_w16_h40_d4:
        case xmssmt_sha2-512_w16_h40_d8:



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        case xmssmt_sha2-512_w16_h60_d3:
        case xmssmt_sha2-512_w16_h60_d6:
        case xmssmt_sha2-512_w16_h60_d12:
        case xmssmt_shake256_w16_h20_d2:
        case xmssmt_shake256_w16_h20_d4:
        case xmssmt_shake256_w16_h40_d2:
        case xmssmt_shake256_w16_h40_d4:
        case xmssmt_shake256_w16_h40_d8:
        case xmssmt_shake256_w16_h60_d3:
        case xmssmt_shake256_w16_h60_d6:
        case xmssmt_shake256_w16_h60_d12:
          bytestring64 root_n64;

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

      /* XMSS^MT public key structure */

      struct xmssmt_public_key {
        xmssmt_root root;  /* Root node */
        seed SEED;  /* Seed for bitmasks */
      };


Authors' Addresses

   Andreas Huelsing
   TU Eindhoven
   P.O. Box 513
   Eindhoven  5600 MB
   NL

   Email: ietf@huelsing.net


   Denis Butin
   TU Darmstadt
   Hochschulstrasse 10
   Darmstadt  64289
   DE

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








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   Stefan-Lukas Gazdag
   genua GmbH
   Domagkstrasse 7
   Kirchheim bei Muenchen  85551
   DE

   Email: ietf@gazdag.de


   Aziz Mohaisen
   SUNY Buffalo
   323 Davis Hall
   Buffalo, NY  14260
   US

   Phone: +1 716 645-1592
   Email: mohaisen@buffalo.edu


































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