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Versions: (draft-huelsing-cfrg-hash-sig-xmss) 00 01 02 03 04 05 06 07 08 09 10 11 12 RFC 8391

Crypto Forum Research Group                                  A. Huelsing
Internet-Draft                                              TU Eindhoven
Intended status: Informational                                  D. Butin
Expires: July 6, 2016                                       TU Darmstadt
                                                               S. Gazdag
                                                              genua GmbH
                                                             A. Mohaisen
                                                            SUNY Buffalo
                                                         January 3, 2016


                  XMSS: Extended Hash-Based Signatures
             draft-irtf-cfrg-xmss-hash-based-signatures-02

Abstract

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

Status of This Memo

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

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

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

   This Internet-Draft will expire on July 6, 2016.



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Copyright Notice

   Copyright (c) 2016 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
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   publication of this document.  Please review these documents
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   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Conventions Used In This Document . . . . . . . . . . . .   5
   2.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.1.  Data Types  . . . . . . . . . . . . . . . . . . . . . . .   5
     2.2.  Operators . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.3.  Functions . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.4.  Integer to Byte Conversion  . . . . . . . . . . . . . . .   6
     2.5.  Hash Function Address Scheme  . . . . . . . . . . . . . .   6
     2.6.  Strings of Base w Numbers . . . . . . . . . . . . . . . .   9
     2.7.  Member Functions  . . . . . . . . . . . . . . . . . . . .  10
   3.  Primitives  . . . . . . . . . . . . . . . . . . . . . . . . .  11
     3.1.  WOTS+ One-Time Signatures . . . . . . . . . . . . . . . .  11
       3.1.1.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . .  11
         3.1.1.1.  WOTS+ Functions . . . . . . . . . . . . . . . . .  12
       3.1.2.  WOTS+ Chaining Function . . . . . . . . . . . . . . .  12
       3.1.3.  WOTS+ Private Key . . . . . . . . . . . . . . . . . .  12
       3.1.4.  WOTS+ Public Key  . . . . . . . . . . . . . . . . . .  13
       3.1.5.  WOTS+ Signature Generation  . . . . . . . . . . . . .  13
       3.1.6.  WOTS+ Signature Verification  . . . . . . . . . . . .  15
       3.1.7.  Pseudorandom Key Generation . . . . . . . . . . . . .  15
   4.  Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . .  16
     4.1.  XMSS: eXtended Merkle Signature Scheme  . . . . . . . . .  16
       4.1.1.  XMSS Parameters . . . . . . . . . . . . . . . . . . .  17
       4.1.2.  XMSS Hash Functions . . . . . . . . . . . . . . . . .  17
       4.1.3.  XMSS Private Key  . . . . . . . . . . . . . . . . . .  18
       4.1.4.  Randomized Tree Hashing . . . . . . . . . . . . . . .  18
       4.1.5.  L-Trees . . . . . . . . . . . . . . . . . . . . . . .  18
       4.1.6.  TreeHash  . . . . . . . . . . . . . . . . . . . . . .  19
       4.1.7.  XMSS Public Key . . . . . . . . . . . . . . . . . . .  20
       4.1.8.  XMSS Signature  . . . . . . . . . . . . . . . . . . .  21
       4.1.9.  XMSS Signature Generation . . . . . . . . . . . . . .  22



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       4.1.10. XMSS Signature Verification . . . . . . . . . . . . .  23
       4.1.11. Pseudorandom Key Generation . . . . . . . . . . . . .  25
       4.1.12. Free Index Handling and Partial Secret Keys . . . . .  25
     4.2.  XMSS^MT: Multi-Tree XMSS  . . . . . . . . . . . . . . . .  25
       4.2.1.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . .  26
       4.2.2.  XMSS Algorithms Without Message Hash  . . . . . . . .  26
       4.2.3.  XMSS^MT Private Key . . . . . . . . . . . . . . . . .  26
       4.2.4.  XMSS^MT Public Key  . . . . . . . . . . . . . . . . .  27
       4.2.5.  XMSS^MT Signature . . . . . . . . . . . . . . . . . .  27
       4.2.6.  XMSS^MT Signature Generation  . . . . . . . . . . . .  28
       4.2.7.  XMSS^MT Signature Verification  . . . . . . . . . . .  30
       4.2.8.  Pseudorandom Key Generation . . . . . . . . . . . . .  30
       4.2.9.  Free Index Handling and Partial Secret Keys . . . . .  31
   5.  Parameter Sets  . . . . . . . . . . . . . . . . . . . . . . .  31
     5.1.  WOTS+ Parameters  . . . . . . . . . . . . . . . . . . . .  31
     5.2.  XMSS Parameters . . . . . . . . . . . . . . . . . . . . .  32
     5.3.  XMSS^MT Parameters  . . . . . . . . . . . . . . . . . . .  32
   6.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .  33
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  34
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  37
     8.1.  Security Proofs . . . . . . . . . . . . . . . . . . . . .  38
     8.2.  Security Assumptions  . . . . . . . . . . . . . . . . . .  39
     8.3.  Post-Quantum Security . . . . . . . . . . . . . . . . . .  39
   9.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  39
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  40
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  40
     10.2.  Informative References . . . . . . . . . . . . . . . . .  40
   Appendix A.  WOTS+ XDR Formats  . . . . . . . . . . . . . . . . .  41
   Appendix B.  XMSS XDR Formats . . . . . . . . . . . . . . . . . .  42
   Appendix C.  XMSS^MT XDR Formats  . . . . . . . . . . . . . . . .  47
   Appendix D.  Changed since draft-irtf-cfrg-xmss-hash-based-
                signatures-01  . . . . . . . . . . . . . . . . . . .  52
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  53

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 system can be used to sign multiple messages.

   One-Time Signature schemes, and Many-Time Signature (MTS) schemes
   composed of them, were proposed by Merkle in 1979 [Merkle79].  They
   were well-studied in the 1990s and have regained interest from 2006
   onwards because of their resistance against quantum-computer-aided



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   attacks.  These kinds of signature schemes are called hash-based
   signature schemes as they are built out of a cryptographic hash
   function.  Hash-based signature schemes generally feature small
   private and public keys as well as fast signature generation and
   verification but large signatures and relatively slow key generation.
   In addition, they are suitable for compact implementations that
   benefit various applications and are naturally resistant to most
   kinds of side-channel attacks.

   Some progress has already been made toward standardizing and
   introducing hash-based signatures.  McGrew and Curcio have published
   an Internet-Draft [DC14] specifying the "textbook" Lamport-Diffie-
   Winternitz-Merkle (LDWM) scheme based on early publications.
   Independently, Buchmann, Dahmen and Huelsing have proposed XMSS
   [BDH11], the eXtended Merkle Signature Scheme, offering better
   efficiency and a modern security proof.  Very recently, 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 would
   be 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.

   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.  Conventions Used In This Document

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

2.  Notation

2.1.  Data Types

   Bytes and byte strings are the fundamental data types.  A byte is a
   sequence of eight bits.  A single byte is denoted as a pair of
   hexadecimal digits with a leading "0x".  A byte string is an ordered
   sequence of zero or more bytes and is denoted as an ordered sequence
   of hexadecimal characters with a leading "0x".  For example, 0xe534f0
   is a byte string of length 3.  An array of byte strings is an
   ordered, indexed set starting with index 0 in which all byte strings
   have identical length.  If not stated or handled otherwise, we assume
   big-endian representation of data types.

2.2.  Operators

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

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

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

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

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

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

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

   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.




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

      A AND B denotes the bitwise logical conjunction operation.

      A XOR B denotes the bitwise logical exclusive disjunction
      operation.

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

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

2.3.  Functions

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

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

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

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

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 of 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
   generator that takes a seed S and a 16-byte address A.  The latter is
   used to select the A-th n-byte block from the PRG output where n is
   the security parameter.  Here we explain the structure of address A.
   We explain the construction of the addresses in the following
   sections where they are used.

   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 2n-byte inputs and an n-byte
   key to n-byte outputs is used.  To randomize this function, 3n bytes



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   are needed - n bytes for the key and 2n bytes for a bitmask.  For the
   one-time signature scheme constructions a hash function that maps
   n-byte inputs and n-byte keys 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 address formats for the different use
   cases.  One format for the hashes used in one-time signature schemes,
   one for hashes used within the main Merkle-tree construction, and one
   for hashes used in the L-trees.  The latter being used to compress
   one-time public keys.  All these formats share as much format as
   possible.  In the following we describe these formats in detail.

   An address is structured as follows.  It always starts with 46 zero
   bits in the most significant bits.  These are followed by a layer
   address of 8 bits, and a tree address of 24 bits.  The next bit
   decides whether it is an OTS construction or a hash tree address.
   This OTS bit is set to zero for a tree hash address and it is set to
   one for an OTS hash address.

   We first describe the OTS address case as the hash tree case again
   splits into two cases.  In this case, the OTS bit is followed by a
   24-bit OTS address that encodes the index of the OTS key pair within
   a tree.  The next 16 bits encode the chain address followed by 8 bits
   that encode the address of the hash function call within a chain.
   The key bit is used to generate two different addresses for one hash
   function call.  The bit is set to one to generate the key.  To
   generate the n-byte bitmask, the key bit is set to zero.

                       Index i for OTS hash
                     +------------------------+
                     | Padding = 0    (46 bit)|
                     +------------------------+
                     | layer address   (8 bit)|
                     +------------------------+
                     | tree address   (24 bit)|
                     +------------------------+
                     | OTS bit = 1     (1 bit)|
                     +------------------------+
                     | OTS address    (24 bit)|
                     +------------------------+
                     | chain address  (16 bit)|
                     +------------------------+
                     | hash address    (8 bit)|
                     +------------------------+
                     | key bit         (1 bit)|
                     +------------------------+



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   Now we describe the hash tree address case.  This case again splits
   into two.  The OTS bit is followed by an L-tree bit.  This bit is set
   to zero in case of an L-tree and set to one for main tree nodes.  We
   first discuss the L-tree case.  In this case the L-tree bit is
   followed by a 24 bit L-tree address, encoding the index of the leaf
   computed with this L-tree.  The next 6 bits encode the height of the
   node inside the L-tree and the following 16 bit encode the index of
   the node at that height, inside the L-tree.  The last two bits are
   used to generate three different addresses for one node.  The first
   of these bits is set to one to generate the key.  In that case the
   last bit is always zero.  To generate the 2n-byte bitmask, the key
   bit is set to zero.  The most significant n bytes are generated using
   the address with the last bit zero.  The least significant bytes are
   generated using the address with the last bit set to one.

                          An L-tree address
                     +------------------------+
                     | Padding = 0    (46 bit)|
                     +------------------------+
                     | layer address   (8 bit)|
                     +------------------------+
                     | tree address   (24 bit)|
                     +------------------------+
                     | OTS bit = 0     (1 bit)|
                     +------------------------+
                     | L-tree bit = 1  (1 bit)|
                     +------------------------+
                     | L-tree address (24 bit)|
                     +------------------------+
                     | tree height     (6 bit)|
                     +------------------------+
                     | tree index     (16 bit)|
                     +------------------------+
                     | key bit         (1 bit)|
                     +------------------------+
                     | block bit       (1 bit)|
                     +------------------------+

   We now describe the remaining format for the main tree hash
   addresses.  In this case the L-tree bit is set to zero and followed
   by 14 zero bits padding as there are less hash tree addresses
   required.  The next 8 bits encode the height of the tree node to be
   computed within the tree, followed by 24 bits that encode the index
   of this node at that height.  The last two bits are used to generate
   three different addresses for one node as described for the L-tree
   case.  The first of these bits is set to one to generate the key.  In
   that case the last bit is always zero.  To generate the 2n-byte
   bitmask, the key bit is set to zero.  The most significant n bytes



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   are generated using the address with the last bit zero.  The least
   significant bytes are generated using the address with the last bit
   set to one.

                       A hash tree address
                     +------------------------+
                     | Padding = 0    (46 bit)|
                     +------------------------+
                     | layer address   (8 bit)|
                     +------------------------+
                     | tree address   (24 bit)|
                     +------------------------+
                     | OTS bit = 0     (1 bit)|
                     +------------------------+
                     | L-tree bit = 0  (1 bit)|
                     +------------------------+
                     | Padding = 0    (14 bit)|
                     +------------------------+
                     | tree height     (8 bit)|
                     +------------------------+
                     | tree index     (24 bit)|
                     +------------------------+
                     | key bit         (1 bit)|
                     +------------------------+
                     | block bit       (1 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 also
   assume that setting the L-tree bit to zero, does also set the
   (second) padding block 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) as follows.  If X is an len_X-byte
   string, w is a member of the set {4, 16}, then base_w(X, w) outputs a
   length 8 * len_X / lg(w) array of integers between 0 and w - 1.










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   Algorithm 1: base_w(X, w)

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

       for ( consumed = 0; consumed < 8 * len_X; consumed += lg(w) ) {
           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) 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)
         +-----------+-----------+-----------+-----------+
         |     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.
   Accordingly, setX(PK, X, Y) sets value X in PK to the value hold by
   Y.




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3.  Primitives

3.1.  WOTS+ One-Time Signatures

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

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

3.1.1.  WOTS+ Parameters

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

      m : the message length in bytes

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

      w : the Winternitz parameter; it is a member of the set {4, 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+ secret key,
      public key, and signature.  It is computed as len = len_1 + len_2,
      with len_1 = ceil(8m/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 m 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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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.
   Security requirements on F are discussed in Section 8.  In addition,
   WOTS+ uses a pseudorandom generator G.  G takes as input an n-byte
   key and a 16-byte index and generates pseudorandom outputs of length
   n.  Security requirements on G 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 G.  It takes a hash function address as
   input.  This address will have the first 119 bits set to encode the
   address of this chain.  In each iteration, one output of G is used as
   key for F and a second output is XORed to the intermediate result
   before it is processed by F.  In the following, ADRS is a 16-byte
   hash function address as specified in Section 2.5 and SEED is an
   n-byte string, both used to generate the outputs of G.  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 G.

   Algorithm 2: Chaining Function

     if ( s is equal to 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.setKeyBit(0);
     BM = G(SEED, ADRS);
     ADRS.setKeyBit(1);
     KEY = G(SEED, ADRS);
     tmp = F(KEY, tmp XOR BM);
     return tmp;

3.1.3.  WOTS+ Private Key

   The private key in WOTS+, denoted by sk, is a length len array of
   n-byte strings.  This private key MUST be only used to sign exactly
   one message.  Each n-byte string MUST either be selected randomly
   from the uniform distribution or using a cryptographically secure
   pseudorandom procedure.  In the latter case, the security of the used



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   procedure MUST at least match that of the WOTS+ parameters used.  For
   a further discussion on pseudorandom key generation see the end of
   this section.  The following pseudocode (Algorithm 3) describes an
   algorithm for generating sk.

   Algorithm 3: Generating a WOTS+ Private Key

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

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 secret key
   each define the start node for one hash chain.  The public key
   consists of the end nodes of these hash chains.  Therefore, like the
   secret key, the public key is also a length len array of n-byte
   strings.  To compute the hash chain, the chaining function (Algorithm
   2) is used.  A hash function address ADRS and a seed SEED has 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 fields of ADRS than
   chain address, hash address and key bit.  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

     for ( i = 0; i < len; i = i + 1 ) {
       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 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.  Each of the base w
   integers is used to select a node from a different hash chain.  The
   signature is formed by concatenating the selected nodes.  The



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   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

     csum = 0;

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

     // compute checksum
     for ( i = 0; i < len_1; i = i + 1 ) {
           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);
     for ( i = 0; i < len; i = i + 1 ) {
          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.

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

     csum = 0;

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

     // compute checksum
     for ( i = 0; i < len_1; i = i + 1 ) {
           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);
     for ( i = 0; i < len; i = i + 1 ) {
          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.

3.1.7.  Pseudorandom Key Generation

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

   The advantage of generating the secret key elements from a random
   n-byte string is that only this n-byte string needs to be stored



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   instead of the full secret key.  The key can be regenerated when
   needed.  The suggested method from [BDH11] can be described using G.
   During key generation a uniformly random n-byte string S is sampled
   from a secure source of randomness.  This string S is stored as
   secret key.  The secret key elements are computed as sk[i] = G'(S,
   toByte(i,16)) 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
   five cryptographic components: WOTS+ as OTS method, two additional
   cryptographic hash functions H and H_m, a pseudorandom function
   PRF_m, and a pseudorandom generator G.  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 generator G.  The value
   corresponding to the root of the XMSS tree forms the XMSS public key
   together with the seed.

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




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   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 valid.
   The XMSS secret key consists of all WOTS+ secret keys and the actual
   index.  To reduce storage, a pseudorandom key generation procedure,
   as described in [BDH11], MAY be used.  The security of the used
   method MUST at least match the security of the XMSS instance.

4.1.1.  XMSS Parameters

   XMSS has the following parameters:

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

      n : the length in bytes of each node

      m : the length of the message digest

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

   There are N = 2^h leaves in the tree.

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

4.1.2.  XMSS Hash Functions

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

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

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

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

      A pseudorandom generator G.  G takes as input an n-byte key and a
      16-byte index and generates pseudorandom outputs of length n.





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4.1.3.  XMSS Private Key

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

4.1.4.  Randomized Tree Hashing

   To improve readability we introduce a function RAND_HASH(LEFT, RIGHT,
   SEED, ADRS) that does the randomized hashing.  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 for G and the address
   ADRS of this hash function call.  RAND_HASH first uses G 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)).

   Algorithm 7: RAND_HASH

       ADRS.setKeyBit(0);
       ADRS.setBlockBit(0);
       BM_0 = G(SEED, ADRS);
       ADRS.setBlockBit(1);
       BM_1 = G(SEED, ADRS);
       ADRS.setKeyBit(1);
       ADRS.setBlockBit(0);
       KEY = G(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
   address ADRS as input that encodes the address of the L-tree.  The
   algorithm uses G and the seed SEED generated during public key
   generation.



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   Algorithm 8: ltree

     unsigned int len' = len;
     unsigned int i = 0;
     ADRS.setTreeHeight(0);
     while ( len' > 1 ) {
       for ( i = 0; i < floor(len' / 2); i = i + 1 ) {
         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];

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 secret key SK,
   an unsigned integer s (the start index), an unsigned integer t (the
   target node height), a seed SEED, 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 uses a stack holding up to
   (t-1) n-byte strings, with the usual stack functions push() and
   pop().


















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   Algorithm 9: treeHash

     if( s % (1 << t) != 0 ) return -1;
     for ( i = 0; i < 2^t; i = i + 1 ) {
       ADRS.setOTSBit(1);
       ADRS.setOTSAddress(s+i);
       pk = WOTS_genPK (getWOTS_SK(SK, s+i), SEED, ADRS);
       ADRS.setOTSBit(0);
       ADRS.setLTreeBit(1);
       ADRS.setLTreeAddress(s+i);
       node = ltree(pk, SEED, ADRS);
       ADRS.setLTreeBit(0);
       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();

4.1.7.  XMSS Public Key

   The XMSS public key is computed as described in XMSS_genPK (Algorithm
   10).  The algorithm takes the XMSS secret key SK, and the tree height
   h.  The XMSS public key PK consists of the root of the binary hash
   tree and the seed SEED.  SEED is generated as a uniformly random
   n-byte string.  Although SEED is public, it is important that it is
   generated using a good entropy source.  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.

   Algorithm 10: XMSS_genPK - Generate an XMSS public key from an XMSS
   private key

     set SEED to a uniformly random n-byte string;
     ADRS = toByte(0,16);
     root = treeHash(SK, 0, h, SEED, ADRS);
     PK = root || SEED;
     return PK;

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




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   The format of an XMSS public key is given below.

   XMSS Public Key

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

4.1.8.  XMSS Signature

   An XMSS signature is a (4 + m + (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 (m 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 x^th node on level y with x = 0 being
   the leftmost node on a level.  The leaves are on level 0, the root is
   on level h.  An authentication path contains exactly one node on
   every layer 0 <= x <= h-1.  For the i^th WOTS+ key pair, counting
   from zero, the j^th authentication path node is

      Node(j, floor(i / (2^j)) XOR 1)

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





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     ADRS = toByte(0, 16);
     for (j = 0; j < h; j++) {
       k = floor(i / (2^j)) XOR 1;
       auth[j] = treeHash(SK, k * 2^j, j, SEED, ADRS);
       }

   The data format for a signature is given below.

   XMSS Signature

             +---------------------------------+
             |                                 |
             |          index idx_sig          |    4 bytes
             |                                 |
             +---------------------------------+
             |                                 |
             |          randomness r           |    m 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.  Then a
   WOTS+ signature of the message is computed using the next unused
   WOTS+ private key.  Next, the authentication path is computed.
   Finally, the secret key is updated, i.e.  idx is incremented.  An
   implementation MUST NOT output the signature before the updated
   private key.

   The node values of the authentication path MAY be computed in any
   way.  This computation is assumed to be performed by the subroutine



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   buildAuth for the function XMSS_sign, as below.  The fastest
   alternative is to store all tree nodes and set the array in the
   signature by copying them, respectively.  The least storage-intensive
   alternative is to recompute all nodes for each signature online.
   There exist several algorithms in between, with different time/
   storage trade-offs.  For an overview see [BDS09].  Note that the
   details of this procedure are not relevant to interoperability; it is
   not necessary to know any of these details in order to perform the
   signature verification operation.  As a consequence, buildAuth is not
   specified here.

   The algorithm XMSS_sign (Algorithm 11) described below calculates an
   updated secret key SK and a signature on a message M.  XMSS_sign
   takes as inputs a message M of an arbitrary length, an XMSS secret
   key SK and seed SEED.  It returns the byte string containing the
   concatenation of the updated secret key SK and the signature Sig.

   Algorithm 11: XMSS_sign - Generate an XMSS signature and update the
   XMSS secret key

     idx_sig = getIdx(SK);
     ADRS = toByte(0,16);
     auth = buildAuth(SK, idx_sig, SEED, ADRS);
     byte[m] r = PRF_m(getSK_PRF(SK), M);
     byte[m] M' = H_m(r, M);
     ADRS.setOTSBit(1);
     ADRS.setOTSAddress(idx_sig);
     sig_ots = WOTS_sign(getWOTS_SK(SK, idx_sig), M', SEED, ADRS);
     Sig = (idx_sig || r || sig_ots || auth);
     setIdx(SK, idx_sig + 1);
     return (SK || Sig);

4.1.10.  XMSS Signature Verification

   An XMSS signature is verified by first computing the message digest
   using randomness r and a message M.  Then the used WOTS+ public key
   pk_ots is computed from the WOTS+ signature using WOTS_pkFromSig.
   The WOTS+ public key in turn is used to compute the corresponding
   leaf using an L-tree.  The leaf, together with index idx_sig and
   authentication path auth is used to compute an alternative root value
   for the tree.  These first steps are done by XMSS_rootFromSig
   (Algorithm 12).  The verification succeeds if and only if the
   computed root value matches the one in the XMSS public key.  In any
   other case it MUST return fail.

   The main part of XMSS signature verification is done by the function
   XMSS_rootFromSig (Algorithm 12) described below.  XMSS_rootFromSig
   takes as inputs an XMSS signature Sig, a message M, and seed SEED.



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   XMSS_rootFromSig returns an n-byte string holding the value of the
   root of a tree defined by the input data.

   Algorithm 12: XMSS_rootFromSig - Compute a root node using an XMSS
   signature, a message, and seed SEED

     byte[m] M' = H_m(r, M);
     ADRS = toByte(0,16);
     ADRS.setOTSBit(1);
     ADRS.setOTSAddress(idx_sig);
     pk_ots = WOTS_pkFromSig(sig_ots, M', SEED, ADRS);
     ADRS.setOTSBit(0);
     ADRS.setLTreeBit(1);
     ADRS.setLTreeAddress(idx_sig);
     byte[n][2] node;
     node[0] = ltree(pk_ots, SEED, ADRS);
     ADRS.setLTreeBit(0);
     ADRS.setTreeIndex(idx_sig);
     for ( k = 0; k < h; k = k + 1 ) {
       ADRS.setTreeHeight(k);
       if ( floor(idx_sig / (2^k)) % 2 is equal to 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.  XMSS^MT uses
   only XMSS_rootFromSig and delegates the comparison to a later
   comparison of data depending on its output.

   Algorithm 13: XMSS_verify - Verify an XMSS signature using an XMSS
   signature, the corresponding XMSS public key and a message

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







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4.1.11.  Pseudorandom Key Generation

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

   For XMSS a similar method than the one used for WOTS+ can be used.
   The suggested method from [BDH11] can be described using G.  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+
   secret key using the method described in Section 3.1.7.  The seeds
   for the WOTS+ key pairs are computed as S_ots[i] = G(S,i).  The
   second parameter of G is the index i of the WOTS+ key pair,
   represented as 16-byte string in the common way.  An advantage of
   this method is that a WOTS+ key can be computed using only len + 1
   evaluations of G when S is given.

4.1.12.  Free Index Handling and Partial Secret Keys

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

4.2.  XMSS^MT: Multi-Tree XMSS

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

   Consider an XMSS^MT tree of total height h that has d layers of XMSS
   trees of height h / d.  Then layer d - 1 contains one XMSS tree,




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   layer d - 2 contains 2^(h / d) XMSS trees, and so on.  Finally, layer
   0 contains 2^(h - h / d) XMSS trees.

4.2.1.  XMSS^MT Parameters

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

   All the trees on higher layers sign root nodes of other trees which
   are n-byte strings.  Hence, no message compression is needed and
   WOTS+ is used to sign the root nodes themselves instead of their hash
   values.  Hence the WOTS+ message length for these layers is n not m.
   Accordingly, the values of len_1, len_2 and len change for these
   layers.  The parameters len_1_n, len_2_n, and len_n denote the
   respective values computed using n as message length for WOTS+.

4.2.2.  XMSS Algorithms Without Message Hash

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

4.2.3.  XMSS^MT Private Key

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

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






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4.2.4.  XMSS^MT Public Key

   The XMSS^MT public key PK_MT contains the root of the single XMSS
   tree on layer d-1 and the seed SEED.  The pseudorandom generator G is
   used with SEED to generate the bitmasks and keys for all XMSS trees.
   Algorithm 14 shows pseudocode to generate PK_MT.  First, the n-byte
   SEED is chosen uniformly at random.  The n-byte root node of the top
   layer tree is computed using treeHash.  The algorithm XMSSMT_genPK
   takes the XMSS^MT secret key SK_MT as an input and outputs an XMSS^MT
   public key PK_MT.

   Algorithm 14: XMSSMT_genPK - Generate an XMSS^MT public key from an
   XMSS^MT private key

     set SEED to a uniformly random n-byte string;
     ADRS = toByte(0,16);
     ADRS.setLayerAddress(d-1);
     root = treeHash(getXMSS_SK(SK_MT, 0, d - 1), 0, h / d, SEED, ADRS);
     PK_MT = root || SEED;
     return PK_MT;

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

   XMSS^MT Public Key

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

4.2.5.  XMSS^MT Signature

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

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

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

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



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

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

   The data format for a signature is given below.

   XMSS^MT signature

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

4.2.6.  XMSS^MT Signature Generation

   To compute the XMSS^MT signature Sig_MT of a message M using an
   XMSS^MT private key SK_MT and seed SEED, XMSSMT_sign (Algorithm 15)
   described below uses XMSS_sign and XMSS_sign_wo_hash as defined in



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   Section 4.2.2.  First, the signature index is set to idx.  Next,
   PRF_m is used to compute a pseudorandom m-byte string r.  This m-byte
   string is then used to compute a randomized message digest of length
   m.  The message digest is signed using the WOTS+ key pair on the
   bottom layer with absolute index idx.  The authentication path for
   the WOTS+ key pair is computed as well as the root of the containing
   XMSS tree.  The root is signed by the parent XMSS tree.  This is
   repeated until the top tree is reached.

   Algorithm 15: XMSSMT_sign - Generate an XMSS^MT signature and update
   the XMSS^MT secret key

     ADRS = toByte(0,16);
     SK_PRF = getSK_PRF(SK_MT);
     idx_sig = getIdx(SK_MT);
     setIdx(SK_MT, idx_sig + 1);
     Sig_MT = idx_sig;
     unsigned int idx_tree = (h - h/d) most significant bits of idx_sig;
     unsigned int idx_leaf = (h / d) least significant bits of idx_sig;
     SK = idx_leaf || SK_PRF || getXMSS_SK(SK_MT, idx_tree, 0);
     ADRS.setLayerAddress(0);
     ADRS.setTreeAddress(idx_tree);
     Sig_tmp = XMSS_sign(M, SK, SEED, ADRS);
     Sig_tmp = Sig_tmp without idx;
     Sig_MT = Sig_MT || Sig_tmp;
     for ( j = 1; j < d; j = j + 1 ) {
        root = treeHash(SK, 0, h / d, SEED, 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 || SK_PRF || getXMSS_SK(SK_MT, idx_tree, j);
        ADRS.setLayerAddress(j);
        ADRS.setTreeAddress(idx_tree);
        Sig_tmp = XMSS_sign_wo_hash(root, SK, SEED, ADRS)
             with idx removed;
        Sig_MT = Sig_MT || Sig_tmp;
     }
     return SK_MT || Sig_MT;

   Algorithm 15 is only one method to compute XMSS^MT signatures.
   Especially, there exist time-memory trade-offs that allow to reduce
   the signing time to less than the signing time of an XMSS scheme with
   tree height h / d.  These trade-offs prevent certain values from
   being recomputed several times by keeping a state and distribute all
   computations over all signature generations.  Details can be found in
   [Huelsing13a].






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4.2.7.  XMSS^MT Signature Verification

   XMSS^MT signature verification (Algorithm 16) can be summarized as d
   XMSS signature verifications with small changes.  First, only the
   message is hashed.  The remaining XMSS signatures are on the root
   nodes of trees which have a fixed length.  Second, instead of
   comparing the computed root node to a given value, a signature on the
   root is verified.  Only the root node of the top tree is compared to
   the value in the XMSS^MT public key.  XMSSMT_verify uses
   XMSS_rootFromSig and XMSS_rootFromSig_wo_hash.  XMSSMT_verify takes
   as inputs an XMSS^MT signature Sig_MT, a message M and a public key
   PK_MT.  It outputs a boolean.

   Algorithm 16: XMSSMT_verify - Verify an XMSS^MT signature Sig_MT on a
   message M using an XMSS^MT public key PK_MT

     idx = getIdx(Sig_MT);
     SEED = getSEED(PK_MT);
     ADRS = toByte(0,16);
     unsigned int idx_leaf = (h / d) least significant bits of idx;
     unsigned int idx_tree = (h - h / d) most significant bits of idx;
     Sig' = leaf || getR(Sig_MT) || getXMSSSignature(Sig, 0);
     ADRS.setLayerAddress(0);
     ADRS.setTreeAddress(idx_tree);
     byte[n] node = XMSS_rootFromSig(Sig', M, SEED, ADRS);
     for ( j = 1; j < d; j = j + 1 ) {
        idx_leaf = (h / d) least significant bytes of idx_tree;
        idx_tree = (h - j * h / d) most significant bytes of idx_tree;
        Sig' = idx_leaf || getXMSSSignature(Sig, j);
        ADRS.setLayerAddress(j);
        ADRS.setTreeAddress(idx_tree);
        node = XMSS_rootFromSig_wo_hash(Sig', node, SEED, ADRS);
     }
     if ( node is equal to getRoot(PK_MT) ) {
       return true;
     } else {
       return false;
     }

4.2.8.  Pseudorandom Key Generation

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




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   For XMSS^MT a method similar to that for XMSS and WOTS+ can be used.
   The method uses a G as pseudorandom generator.  During key generation
   a uniformly random n-byte string S_MT is sampled from a secure source
   of randomness.  This seed S_MT is used to generate one n-byte value S
   for each XMSS key pair.  This n-byte value can be used to compute the
   respective XMSS secret key using the method described in
   Section 4.1.11.  Let S[x][y] be the seed for the x^th XMSS secret key
   on layer y.  The seeds are computed as S[x][y] = G(G(S, y), x).  The
   second parameter of G is the index x (resp. level y), represented as
   16-byte string in the common way.

4.2.9.  Free Index Handling and Partial Secret Keys

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

5.  Parameter Sets

   This note provides a first basic set of parameter sets which are
   assumed to cover most relevant applicants.  Parameter sets for two
   classical security levels are defined: 256 and 512 bits.  Function
   output sizes are n = m = 32 and 64 bytes.  Considering quantum-
   computer-aided attacks, these output sizes yield post-quantum
   security of 128 and 256 bits, respectively.

   For the n = m = 32 and n = m = 64 settings, we give parameters that
   use SHA2-256 and SHA2-512 as defined in [FIPS180], respectively, and
   ChaCha20 as defined in [RFC7539].  SHA2 does not provide a keyed-mode
   itself.  To implement a keyed hash-function, SHA2-256(toByte(0,32) ||
   KEY || M) and SHA2-512(toByte(0,64) || KEY || M) are used.  This
   construction is used for the functions F, H, and H_m.  To implement
   PRF_m, HMAC-SHA2-256 and HMAC-SHA2-512 are used, respectively.  The
   pseudorandom generator G for n=32 is implemented as ChaCha20 using
   SEED as key, the most significant 12 bytes of the address input as
   nonce and the least significant 4 bytes as counter.  The output
   consists of the first 32 bytes of the key stream.  The pseudorandom
   generator G for n=64 is implemented as HMAC-SHA2-512.

5.1.  WOTS+ Parameters

   To fully describe a WOTS+ signature method, the parameters m, n, and
   w, as well as the functions F and G 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     | G        | m  | n  | w  | len |
    +------------------------+-------+----------+----+----+----+-----+
    | WOTSP_SHA2-256_M32_W16 | SHA-2 | ChaCha20 | 32 | 32 | 16 | 67  |
    |                        |       |          |    |    |    |     |
    | WOTSP_SHA2-512_M64_W16 | SHA-2 | SHA-2    | 64 | 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 m, n, w,
   and h, as well as the functions F, H, H_m, PRF_m, and G 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 ChaCha20 for n = 32 and only
   SHA2 for n=64 as described above.

          +---------------------------+----+----+----+-----+----+
          | Name                      | m  | n  | w  | len | h  |
          +---------------------------+----+----+----+-----+----+
          | XMSS_SHA2-256_M32_W16_H10 | 32 | 32 | 16 | 67  | 10 |
          |                           |    |    |    |     |    |
          | XMSS_SHA2-256_M32_W16_H16 | 32 | 32 | 16 | 67  | 16 |
          |                           |    |    |    |     |    |
          | XMSS_SHA2-256_M32_W16_H20 | 32 | 32 | 16 | 67  | 20 |
          |                           |    |    |    |     |    |
          | XMSS_SHA2-512_M64_W16_H10 | 64 | 64 | 16 | 131 | 10 |
          |                           |    |    |    |     |    |
          | XMSS_SHA2-512_M64_W16_H16 | 64 | 64 | 16 | 131 | 16 |
          |                           |    |    |    |     |    |
          | XMSS_SHA2-512_M64_W16_H20 | 64 | 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 m, n,
   w, h, and d, as well as the functions F, H, H_m, PRF_m, and G MUST be
   specified.  This section defines several XMSS^MT signature systems,
   each of which is identified by a name.  We define parameter sets that



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   implement the functions using SHA2 and ChaCha20 for n = 32 and only
   SHA2 for n=64 as described above.

    +---------------------------------+----+----+----+-----+----+----+
    | Name                            | m  | n  | w  | len | h  | d  |
    +---------------------------------+----+----+----+-----+----+----+
    | XMSSMT_SHA2-256_M32_W16_H20_D2  | 32 | 32 | 16 | 67  | 20 | 2  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H20_D4  | 32 | 32 | 16 | 67  | 20 | 4  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H40_D2  | 32 | 32 | 16 | 67  | 40 | 2  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H40_D4  | 32 | 32 | 16 | 67  | 40 | 4  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H40_D8  | 32 | 32 | 16 | 67  | 40 | 8  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H60_D3  | 32 | 32 | 16 | 67  | 60 | 3  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H60_D6  | 32 | 32 | 16 | 67  | 60 | 6  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-256_M32_W16_H60_D12 | 32 | 32 | 16 | 67  | 60 | 12 |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H20_D2  | 64 | 64 | 16 | 131 | 20 | 2  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H20_D4  | 64 | 64 | 16 | 131 | 20 | 4  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H40_D2  | 64 | 64 | 16 | 131 | 40 | 2  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H40_D4  | 64 | 64 | 16 | 131 | 40 | 4  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H40_D8  | 64 | 64 | 16 | 131 | 40 | 8  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H60_D3  | 64 | 64 | 16 | 131 | 60 | 3  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H60_D6  | 64 | 64 | 16 | 131 | 60 | 6  |
    |                                 |    |    |    |     |    |    |
    | XMSSMT_SHA2-512_M64_W16_H60_D12 | 64 | 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.  Other signature
   methods are out of scope and may be an interesting follow-on work.



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   The description is done in a modular way that allows to base a
   description of stateless hash-based signature algorithms like SPHINCS
   [BHH15] on it.

   The draft slightly deviates from the scientific literature using a
   tweak that prevents multi-target attacks against the underlying hash-
   function.  The security assumptions for this tweak are discussed in
   Section 8.  The main difference to literature is that security now
   relies either on the random oracle model or some other seemingly
   natural heuristic assumptions.

   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 secret keys must be properly
   converted to and from network byte order; this is the only field that
   requires such conversion.  There are 2^32 XDR enumeration values,
   2^16 of which are palindromes, which is adequate for the foreseeable
   future.  If there is a need for more assignments, non-palindromes can
   be assigned.

7.  IANA Considerations

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




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      a short name, such as "XMSS_SHA2-512_M64_W16_H20",

      a positive number, and

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

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

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

   The WOTS+ registry is as follows.

      +-------------------------+-------------+--------------------+
      | Name                    |  Reference  | Numeric Identifier |
      +-------------------------+-------------+--------------------+
      | WOTSP_SHA2-256_M32_W16  | Section 5.1 |     0x01000001     |
      |                         |             |                    |
      | WOTSP_SHA2-512_M64_W16  | Section 5.1 |     0x02000002     |
      +-------------------------+-------------+--------------------+

                                  Table 4

   The XMSS registry is as follows.
















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     +----------------------------+-------------+--------------------+
     | Name                       |  Reference  | Numeric Identifier |
     +----------------------------+-------------+--------------------+
     | XMSS_SHA2-256_M32_W16_H10  | Section 5.2 |     0x01000001     |
     |                            |             |                    |
     | XMSS_SHA2-256_M32_W16_H16  | Section 5.2 |     0x02000002     |
     |                            |             |                    |
     | XMSS_SHA2-256_M32_W16_H20  | Section 5.2 |     0x03000003     |
     |                            |             |                    |
     | XMSS_SHA2-512_M64_W16_H10  | Section 5.2 |     0x04000004     |
     |                            |             |                    |
     | XMSS_SHA2-512_M64_W16_H16  | Section 5.2 |     0x05000005     |
     |                            |             |                    |
     | XMSS_SHA2-512_M64_W16_H20  | Section 5.2 |     0x06000006     |
     +----------------------------+-------------+--------------------+

                                  Table 5

   The XMSS^MT registry is as follows.
































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   +---------------------------------+-------------+-------------------+
   | Name                            |  Reference  |      Numeric      |
   |                                 |             |     Identifier    |
   +---------------------------------+-------------+-------------------+
   | XMSSMT_SHA2-256_M32_W16_H20_D2  | Section 5.3 |     0x01000001    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H20_D4  | Section 5.3 |     0x02000002    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H40_D2  | Section 5.3 |     0x03000003    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H40_D4  | Section 5.3 |     0x04000004    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H40_D8  | Section 5.3 |     0x05000005    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H60_D3  | Section 5.3 |     0x06000006    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H60_D6  | Section 5.3 |     0x07000007    |
   |                                 |             |                   |
   | XMSSMT_SHA2-256_M32_W16_H60_D12 | Section 5.3 |     0x08000008    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H20_D2  | Section 5.3 |     0x09000009    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H20_D4  | Section 5.3 |     0x0a00000a    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H40_D2  | Section 5.3 |     0x0b00000b    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H40_D4  | Section 5.3 |     0x0c00000c    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H40_D8  | Section 5.3 |     0x0d00000d    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H60_D3  | Section 5.3 |     0x0e00000e    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H60_D6  | Section 5.3 |     0x0f00000f    |
   |                                 |             |                   |
   | XMSSMT_SHA2-512_M64_W16_H60_D12 | Section 5.3 |     0x01010101    |
   +---------------------------------+-------------+-------------------+

                                  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



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   is secure if, even in this setting, the attacker can not produce a
   message signature pair of his choosing such that the verification
   algorithm accepts.

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

8.1.  Security Proofs

   There exist formal security proofs for schemes very similar to those
   described here in the literature [Huelsing13a].  These proofs show
   that an attacker has to break at least one out of certain security
   properties of the used hash functions and PRFs to forge a signature.
   The proofs in [Huelsing13a] do not consider the initial message
   compression and the extended randomized hashing used here.  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.

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

   While the basic randomized hashing used in the original descriptions
   of the schemes allows to prove that it is not enough for an adversary
   to break the collision resistance of the underlying hash function.
   However, it turns out that an attacker could launch a multi-target
   second-preimage attack.  The (simplified) reason is that the
   adversary learns in the order of 2^h hash function input-output pairs



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   and it suffices to find a second-preimage for one out of those.
   Hence, an attacker can do a brute force search until he finds an
   input that matches one of the given outputs.

   The extended randomized hashing used here makes the hash function
   calls position dependent.  Hence, the above attack does not work
   anymore because each hash function evaluation during an attack can
   only target one output value.  This can also be shown formally.

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

8.2.  Security Assumptions

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

8.3.  Post-Quantum Security

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

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

9.  Acknowledgements

   We would like to thank Scott Fluhrer, Burt Kaliski, Adam Langley,
   David McGrew, and Sean Parkinson for their help and comments.







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10.  References

10.1.  Normative References

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

   [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>.

   [RFC2434]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", RFC 2434,
              DOI 10.17487/RFC2434, October 1998,
              <http://www.rfc-editor.org/info/rfc2434>.

   [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>.

   [RFC7539]  Nir, Y. and A. Langley, "ChaCha20 and Poly1305 for IETF
              Protocols", RFC 7539, DOI 10.17487/RFC7539, May 2015,
              <http://www.rfc-editor.org/info/rfc7539>.

10.2.  Informative References

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

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

   [BHH15]    Bernstein, D., Hopwood, D., Huelsing, A., Lange, T.,
              Niederhagen, R., Papachristodoulou, L., Schneider, M.,
              Schwabe, P., and Z. Wilcox-O'Hearn, "SPHINCS: Practical
              Stateless Hash-Based Signatures", Lecture Notes in
              Computer Science volume 9056. Advances in Cryptology -
              EUROCRYPT, 2015.

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






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

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

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

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

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

Appendix A.  WOTS+ XDR Formats

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

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


      /* ots_algorithm_type identifies a particular
         signature algorithm */

      enum ots_algorithm_type {
        wotsp_reserved         = 0x00000000,
        wotsp_sha2-256_m32_w16 = 0x01000001,
        wotsp_sha2-512_m64_w16 = 0x02000002,
      };







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   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_m32_w16:
          bytestring32 ots_sig_m32_len67[67];

        case wotsp_sha2-512_m64_w16:
          bytestring64 ots_sig_m64_len18[131];

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


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


      union ots_pubkey switch (ots_algorithm_type type) {
        case wotsp_sha2-256_m32_w16:
          bytestring32 ots_pubk_m32_len67[67];

        case wotsp_sha2-512_m64_w16:
          bytestring64 ots_pubk_m64_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_m32_w16_h10 = 0x01000001,
        xmss_sha2-256_m32_w16_h16 = 0x02000002,
        xmss_sha2-256_m32_w16_h20 = 0x03000003,

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

        xmss_sha2-512_m64_w16_h10 = 0x04000004,
        xmss_sha2-512_m64_w16_h16 = 0x05000005,
        xmss_sha2-512_m64_w16_h20 = 0x06000006,
      };


   XMSS signatures are defined using XDR syntax as follows:


      /* Authentication path types */

      union xmss_path switch (xmss_algorithm_type type) {
        case xmss_sha2-256_m32_w16_h10:
          bytestring32 path_n32_t10[10];

        case xmss_sha2-256_m32_w16_h16:
          bytestring32 path_n32_t16[16];

        case xmss_sha2-256_m32_w16_h20:
          bytestring32 path_n32_t20[20];

        case xmss_sha2-512_m64_w16_h10:
          bytestring64 path_n64_t10[10];

        case xmss_sha2-512_m64_w16_h16:
          bytestring64 path_n64_t16[16];

        case xmss_sha2-512_m64_w16_h20:



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          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_m32_w16_h10:
        case xmss_sha2-256_m32_w16_h16:
        case xmss_sha2-256_m32_w16_h20:
          bytestring32 rand_m32;

        case xmss_sha2-512_m64_w16_h10:
        case xmss_sha2-512_m64_w16_h16:
        case xmss_sha2-512_m64_w16_h20:
          bytestring64 rand_m64;

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

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

      union xmss_ots_signature switch (xmss_algorithm_type type) {
        case xmss_sha2-256_m32_w16_h10:
        case xmss_sha2-256_m32_w16_h16:
        case xmss_sha2-256_m32_w16_h20:
          wotsp_sha2-256_m32_w16;

        case xmss_sha2-512_m64_w16_h10:
        case xmss_sha2-512_m64_w16_h16:
        case xmss_sha2-512_m64_w16_h20:
          wotsp_sha2-512_m64_w16;

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

      /* XMSS signature structure */

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



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        xmss_ots_signature sig_ots;
        /* authentication path */
        xmss_path nodes;
      };















































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   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_m32_w16_h10:
        case xmss_sha2-256_m32_w16_h16:
        case xmss_sha2-256_m32_w16_h20:
          bytestring32 seed_n32;

        case xmss_sha2-512_m64_w16_h10:
        case xmss_sha2-512_m64_w16_h16:
        case xmss_sha2-512_m64_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_m32_w16_h10:
        case xmss_sha2-256_m32_w16_h16:
        case xmss_sha2-256_m32_w16_h20:
          bytestring32 root_n32;

        case xmss_sha2-512_m64_w16_h10:
        case xmss_sha2-512_m64_w16_h16:
        case xmss_sha2-512_m64_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_m32_w16_h20_d2  = 0x01000001,
        xmssmt_sha2-256_m32_w16_h20_d4  = 0x02000002,
        xmssmt_sha2-256_m32_w16_h40_d2  = 0x03000003,
        xmssmt_sha2-256_m32_w16_h40_d4  = 0x04000004,
        xmssmt_sha2-256_m32_w16_h40_d8  = 0x05000005,
        xmssmt_sha2-256_m32_w16_h60_d3  = 0x06000006,
        xmssmt_sha2-256_m32_w16_h60_d6  = 0x07000007,
        xmssmt_sha2-256_m32_w16_h60_d12 = 0x08000008,

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

        xmssmt_sha2-512_m64_w16_h20_d2  = 0x09000009,
        xmssmt_sha2-512_m64_w16_h20_d4  = 0x0a00000a,
        xmssmt_sha2-512_m64_w16_h40_d2  = 0x0b00000b,
        xmssmt_sha2-512_m64_w16_h40_d4  = 0x0c00000c,
        xmssmt_sha2-512_m64_w16_h40_d8  = 0x0d00000d,
        xmssmt_sha2-512_m64_w16_h60_d3  = 0x0e00000e,
        xmssmt_sha2-512_m64_w16_h60_d6  = 0x0f00000f,
        xmssmt_sha2-512_m64_w16_h60_d12 = 0x01010101,
      };


   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_m32_w16_h20_d2:



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        case xmssmt_sha2-256_m32_w16_h20_d4:
        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h20_d4:
          bytestring3 idx3;

        case xmssmt_sha2-256_m32_w16_h40_d2:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-256_m32_w16_h40_d8:
        case xmssmt_sha2-512_m64_w16_h40_d2:
        case xmssmt_sha2-512_m64_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h40_d8:
          bytestring5 idx5;

        case xmssmt_sha2-256_m32_w16_h60_d3:
        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-256_m32_w16_h60_d12:
        case xmssmt_sha2-512_m64_w16_h60_d3:
        case xmssmt_sha2-512_m64_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h60_d12:
          bytestring8 idx8;

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

      union random_string_xmssmt switch (xmssmt_algorithm_type type) {
        case xmssmt_sha2-256_m32_w16_h20_d2:
        case xmssmt_sha2-256_m32_w16_h20_d4:
        case xmssmt_sha2-256_m32_w16_h40_d2:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-256_m32_w16_h40_d8:
        case xmssmt_sha2-256_m32_w16_h60_d3:
        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-256_m32_w16_h60_d12:
          bytestring32 rand_m32;

        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h20_d4:
        case xmssmt_sha2-512_m64_w16_h40_d2:
        case xmssmt_sha2-512_m64_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h40_d8:
        case xmssmt_sha2-512_m64_w16_h60_d3:
        case xmssmt_sha2-512_m64_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h60_d12:
          bytestring64 rand_m64;

        default:
          void;     /* error condition */



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      };

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

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

      union xmss_reduced_others (xmss_algorithm_type type) {
        case xmssmt_sha2-256_m32_w16_h20_d2:
        case xmssmt_sha2-256_m32_w16_h20_d4:
          bytestring32 xmss_reduced_n32_t87[87];

        case xmssmt_sha2-256_m32_w16_h40_d2:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-256_m32_w16_h40_d8:
          bytestring32 xmss_reduced_n32_t107[107];

        case xmssmt_sha2-256_m32_w16_h60_d3:
        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-256_m32_w16_h60_d12:
          bytestring32 xmss_reduced_n32_t127[127];

        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h20_d4:
          bytestring64 xmss_reduced_n64_t151[151];

        case xmssmt_sha2-512_m64_w16_h40_d2:
        case xmssmt_sha2-512_m64_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h40_d8:
          bytestring64 xmss_reduced_n64_t171[171];

        case xmssmt_sha2-512_m64_w16_h60_d3:
        case xmssmt_sha2-512_m64_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h60_d12:
          bytestring64 xmss_reduced_n64_t191[191];

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

      /* xmss_reduced_array depends on d */

      union xmss_reduced_array (xmss_algorithm_type type) {
        case xmssmt_sha2-256_m32_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-256_m32_w16_h40_d2:



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        case xmssmt_sha2-512_m64_w16_h40_d2:
          xmss_reduced_others xmss_red_arr_d2[1];

        case xmssmt_sha2-256_m32_w16_h60_d3:
        case xmssmt_sha2-512_m64_w16_h60_d3:
          xmss_reduced_others xmss_red_arr_d3[2];

        case xmssmt_sha2-256_m32_w16_h20_d4:
        case xmssmt_sha2-512_m64_w16_h20_d4:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h40_d4:
          xmss_reduced_others xmss_red_arr_d4[3];

        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h60_d6:
          xmss_reduced_others xmss_red_arr_d6[5];

        case xmssmt_sha2-256_m32_w16_h40_d8:
        case xmssmt_sha2-512_m64_w16_h40_d8:
          xmss_reduced_others xmss_red_arr_d8[7];

        case xmssmt_sha2-256_m32_w16_h60_d12:
        case xmssmt_sha2-512_m64_w16_h60_d12:
          xmss_reduced_others xmss_red_arr_d12[11];

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

      /* XMSS^MT signature structure */

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


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


      /* Types for bitmask seed */




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      union seed switch (xmssmt_algorithm_type type) {
        case xmssmt_sha2-256_m32_w16_h20_d2:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-256_m32_w16_h20_d4:
        case xmssmt_sha2-256_m32_w16_h40_d8:
        case xmssmt_sha2-256_m32_w16_h60_d12:
        case xmssmt_sha2-256_m32_w16_h40_d2:
        case xmssmt_sha2-256_m32_w16_h60_d3:
          bytestring32 seed_n32;

        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h20_d4:
        case xmssmt_sha2-512_m64_w16_h40_d8:
        case xmssmt_sha2-512_m64_w16_h60_d12:
        case xmssmt_sha2-512_m64_w16_h40_d2:
        case xmssmt_sha2-512_m64_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_m32_w16_h20_d2:
        case xmssmt_sha2-256_m32_w16_h20_d4:
        case xmssmt_sha2-256_m32_w16_h40_d2:
        case xmssmt_sha2-256_m32_w16_h40_d4:
        case xmssmt_sha2-256_m32_w16_h40_d8:
        case xmssmt_sha2-256_m32_w16_h60_d3:
        case xmssmt_sha2-256_m32_w16_h60_d6:
        case xmssmt_sha2-256_m32_w16_h60_d12:
          bytestring32 root_n32;

        case xmssmt_sha2-512_m64_w16_h20_d2:
        case xmssmt_sha2-512_m64_w16_h20_d4:
        case xmssmt_sha2-512_m64_w16_h40_d2:
        case xmssmt_sha2-512_m64_w16_h40_d4:
        case xmssmt_sha2-512_m64_w16_h40_d8:
        case xmssmt_sha2-512_m64_w16_h60_d3:
        case xmssmt_sha2-512_m64_w16_h60_d6:
        case xmssmt_sha2-512_m64_w16_h60_d12:
          bytestring64 root_n64;




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

      /* XMSS^MT public key structure */

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


Appendix D.  Changed since draft-irtf-cfrg-xmss-hash-based-signatures-01

   1: Renamed m to len_X in description of base_w.

   2: base_w now uses big-endian byte order.

   3: Changed base_w example to use w=16 instead of w=4.

   4: toByte now uses big-endian byte order.

   5: The OID for the WOTS+ signatures is now omitted.  For the correct
   use of WOTS+ in combination with XMSS or XMSS^MT please have a look
   at Section 5.

   6: Corrected WOTS_sign and WOTS_pkFromSig.  In the two lines

         csum = csum << ( 8 - ( len_2 % 8 ));
         len_2_bytes = ceil( len_2 / 8 );

   len_2 had to be multiplied by lg(w):

         csum = csum << ( 8 - ( ( len_2 * lg(w) ) % 8 ));
         len_2_bytes = ceil( ( len_2 * lg(w) ) / 8 );

   7: For WOTS+ signature verification the sentence "Otherwise, the
   signature is rejected" was changed to "Otherwise, the signature MUST
   be rejected.".

   8: Removed "padded with zeros in the most significant bit positions"
   in Section 2.4.

   9: Changed "pk[floor(l / 2) + 1] = pk[l];" to "pk[floor(l / 2)] =
   pk[l-1];" and "unsigned int j = 0;" to "unsigned int i = 0;" in
   algorithm 8.





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   10: Changed "ADRS.setOTSBit(0)" to "ADRS.setOTSBit(1)" in algorithm
   11.

   11: Changed "setR" to "getR" in algorithm 16.

Authors' Addresses

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

   Email: ietf@huelsing.net


   Denis Butin
   TU Darmstadt
   Hochschulstrasse 10
   Darmstadt  64289
   Germany

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


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

   Email: stefan-lukas_gazdag@genua.eu


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

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









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