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INFORMATIONAL

Internet Research Task Force (IRTF)                          A. Huelsing
Request for Comments: 8391                                  TU Eindhoven
Category: Informational                                         D. Butin
ISSN: 2070-1721                                             TU Darmstadt
                                                               S. Gazdag
                                                              genua GmbH
                                                            J. Rijneveld
                                                      Radboud University
                                                             A. Mohaisen
                                           University of Central Florida
                                                                May 2018


                 XMSS: eXtended Merkle Signature Scheme

Abstract

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



















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Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Research Task Force
   (IRTF).  The IRTF publishes the results of Internet-related research
   and development activities.  These results might not be suitable for
   deployment.  This RFC represents the consensus of the Crypto Forum
   Research Group of the Internet Research Task Force (IRTF).  Documents
   approved for publication by the IRSG are not candidates for any level
   of Internet Standard; see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   https://www.rfc-editor.org/info/rfc8391.

Copyright Notice

   Copyright (c) 2018 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
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.























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Table of Contents

   1. Introduction ....................................................5
      1.1. CFRG Note on Post-Quantum Cryptography .....................6
      1.2. Conventions Used in This Document ..........................7
   2. Notation ........................................................7
      2.1. Data Types .................................................7
      2.2. Functions ..................................................7
      2.3. Operators ..................................................8
      2.4. Integer-to-Byte Conversion .................................9
      2.5. Hash Function Address Scheme ...............................9
      2.6. Strings of Base w Numbers .................................12
      2.7. Member Functions ..........................................13
   3. Primitives .....................................................14
      3.1. WOTS+: One-Time Signatures ................................14
           3.1.1. WOTS+ Parameters ...................................14
                  3.1.1.1. WOTS+ Functions ...........................15
           3.1.2. WOTS+ Chaining Function ............................15
           3.1.3. WOTS+ Private Key ..................................16
           3.1.4. WOTS+ Public Key ...................................17
           3.1.5. WOTS+ Signature Generation .........................17
           3.1.6. WOTS+ Signature Verification .......................19
           3.1.7. Pseudorandom Key Generation ........................20
   4. Schemes ........................................................20
      4.1. XMSS: eXtended Merkle Signature Scheme ....................20
           4.1.1. XMSS Parameters ....................................21
           4.1.2. XMSS Hash Functions ................................22
           4.1.3. XMSS Private Key ...................................22
           4.1.4. Randomized Tree Hashing ............................23
           4.1.5. L-Trees ............................................23
           4.1.6. TreeHash ...........................................24
           4.1.7. XMSS Key Generation ................................25
           4.1.8. XMSS Signature .....................................27
           4.1.9. XMSS Signature Generation ..........................28
           4.1.10. XMSS Signature Verification .......................30
           4.1.11. Pseudorandom Key Generation .......................32
           4.1.12. Free Index Handling and Partial Private Keys ......33
      4.2. XMSS^MT: Multi-Tree XMSS ..................................33
           4.2.1. XMSS^MT Parameters .................................33
           4.2.2. XMSS^MT Key Generation .............................33
           4.2.3. XMSS^MT Signature ..................................36
           4.2.4. XMSS^MT Signature Generation .......................37
           4.2.5. XMSS^MT Signature Verification .....................39
           4.2.6. Pseudorandom Key Generation ........................40
           4.2.7. Free Index Handling and Partial Private Keys .......40






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   5. Parameter Sets .................................................40
      5.1. Implementing the Functions ................................41
      5.2. WOTS+ Parameters ..........................................43
      5.3. XMSS Parameters ...........................................43
           5.3.1. Parameter Guide ....................................44
      5.4. XMSS^MT Parameters ........................................45
           5.4.1. Parameter Guide ....................................47
   6. Rationale ......................................................49
   7. Reference Code .................................................50
   8. IANA Considerations ............................................50
   9. Security Considerations ........................................54
      9.1. Security Proofs ...........................................55
      9.2. Minimal Security Assumptions ..............................56
      9.3. Post-Quantum Security .....................................56
   10. References ....................................................57
      10.1. Normative References .....................................57
      10.2. Informative References ...................................58
   Appendix A.  WOTS+ XDR Formats ....................................60
     A.1.  WOTS+ Parameter Sets ......................................60
     A.2.  WOTS+ Signatures ..........................................60
     A.3.  WOTS+ Public Keys .........................................61
   Appendix B.  XMSS XDR Formats .....................................61
     B.1.  XMSS Parameter Sets .......................................61
     B.2.  XMSS Signatures ...........................................62
     B.3.  XMSS Public Keys ..........................................64
   Appendix C.  XMSS^MT XDR Formats ..................................65
     C.1.  XMSS^MT Parameter Sets ....................................65
     C.2.  XMSS^MT Signatures ........................................67
     C.3.  XMSS^MT Public Keys .......................................71
   Acknowledgements ..................................................73
   Authors' Addresses ................................................74




















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1.  Introduction

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

   OTS schemes, and MTS schemes composed from them, were proposed by
   Merkle in 1979 [Merkle83].  They were well-studied in the 1990s and
   have regained interest from the mid 2000s onwards because of their
   resistance against quantum-computer-aided attacks.  These kinds of
   signature schemes are called hash-based signature schemes as they are
   built out of a cryptographic hash function.  Hash-based signature
   schemes generally feature small private and public keys as well as
   fast signature generation and verification; however, they also
   feature large signatures and relatively slow key generation.  In
   addition, they are suitable for compact implementations that benefit
   various applications and are naturally resistant to most kinds of
   side-channel attacks.

   Some progress has already been made toward introducing and
   standardizing hash-based signatures.  Buchmann, Dahmen, and Huelsing
   proposed the eXtended Merkle Signature Scheme (XMSS) [BDH11], which
   offers better efficiency than Merkle's original scheme and a modern
   security proof in the standard model.  McGrew, Curcio, and Fluhrer
   authored an Internet-Draft [MCF18] specifying the Leighton-Micali
   Signature (LMS) scheme, which builds on the seminal works by Lamport,
   Diffie, Winternitz, and Merkle, taking a different approach than XMSS
   and relying entirely on security arguments in the random oracle
   model.  Very recently, the stateless hash-based signature scheme
   SPHINCS was introduced [BHH15], with the intent of being easier to
   deploy in current applications.  A reasonable next step toward
   introducing hash-based signatures is to complete the specifications
   of the basic algorithms -- LMS, XMSS, SPHINCS, and/or variants.

   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.  In particular, it is not required that the cryptographic
   hash function is collision-resistant for the security of XMSS.
   Improvements upon XMSS, as described in [HRS16], are part of this
   note.



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   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.  Section 7 gives
   information about the reference code.  The IANA registry for these
   signature systems is described in Section 8.  Finally, security
   considerations are presented in Section 9.

1.1.  CFRG Note on Post-Quantum Cryptography

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

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

   Additionally, for XMSS:

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

   In contrast to traditional signature schemes, the signature schemes
   described in this document are stateful, meaning the secret key
   changes over time.  If a secret key state is used twice, no
   cryptographic security guarantees remain.  In consequence, it becomes



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   feasible to forge a signature on a new message.  This is a new
   property that most developers will not be familiar with and requires
   careful handling of secret keys.  Developers should not use the
   schemes described here except in systems that prevent the reuse of
   secret key states.

   Note that the fact that the schemes described in this document are
   stateful also implies that classical APIs for digital signatures
   cannot be used without modification.  The API MUST be able to handle
   a secret key state; in particular, this means that the API MUST allow
   to return an updated secret key state.

1.2.  Conventions Used in This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in
   BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

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

2.2.  Functions

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

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

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

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







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2.3.  Operators

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

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

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

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

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

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

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

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

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

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

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

   Arrays are used in the common way, where the i^th element of an array
   A is denoted A[i].  Byte strings are treated as arrays of bytes where
   necessary: if X is a byte string, then X[i] denotes its i^th byte,
   where X[0] is the leftmost byte.

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

   o  A AND B denotes the bitwise logical conjunction operation.

   o  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.






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

2.4.  Integer-to-Byte Conversion

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

2.5.  Hash Function Address Scheme

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

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

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

   The structure of an address complies with word borders, with a word
   being 32 bits long in this context.  Only the tree address is too
   long to fit a single word, but it can fit a double word.  An address
   is structured as follows.  It always starts with a layer address of
   one word in the most significant bits, followed by a tree address of
   two words.  Both addresses are needed for the multi-tree variant (see
   Section 4.2) and describe the position of a tree within a multi-tree.
   They are therefore set to zero in single-tree applications.  For



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   multi-tree hash-based signatures, the layer address describes the
   height of a tree within the multi-tree, starting from height zero for
   trees at the bottom layer.  The tree address describes the position
   of a tree within a layer of a multi-tree starting with index zero for
   the leftmost tree.  The next word defines the type of the address.
   It is set to 0 for an OTS address, to 1 for an L-tree address, and to
   2 for a hash tree address.  Whenever the type word of an address is
   changed, all following words should be initialized with 0 to prevent
   non-zero values in unused padding words.

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

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

                            An OTS Hash Address

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



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

                             An L-tree Address

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

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

                            A Hash Tree Address





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

2.6.  Strings of Base w Numbers

   A byte string can be considered as a string of base w numbers, i.e.,
   integers in the set {0, ... , w - 1}.  The correspondence is defined
   by the function base_w(X, w, out_len) (Algorithm 1) as follows.  If X
   is a len_X-byte string, and w is a member of the set {4, 16}, then
   base_w(X, w, out_len) outputs an array of out_len integers between 0
   and w - 1.  The length out_len is REQUIRED to be less than or equal
   to 8 * len_X / lg(w).

   Algorithm 1: base_w

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

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

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

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









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

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

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

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

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

                                  Example

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












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3.  Primitives

3.1.  WOTS+: One-Time Signatures

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

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

3.1.1.  WOTS+ Parameters

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

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

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

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

      len: the number of n-byte string elements in a WOTS+ private key,
      public key, and signature.  It is computed as len = len_1 + len_2,
      with len_1 = ceil(8n / lg(w)) and len_2 = floor(lg(len_1 *
      (w - 1)) / lg(w)) + 1.

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

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






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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.
   More detail on specific instantiations can be found in Section 5.
   Security requirements on F are discussed in Section 9.  In addition,
   WOTS+ uses a pseudorandom function PRF.  PRF takes as input an n-byte
   key and a 32-byte index and generates pseudorandom outputs of length
   n.  More detail on specific instantiations can be found in Section 5.
   Security requirements on PRF are discussed in Section 9.

3.1.2.  WOTS+ Chaining Function

   The chaining function (Algorithm 2) computes an iteration of F on an
   n-byte input using outputs of PRF.  It takes an OTS hash address as
   input.  This address will have the first six 32-bit words set to
   encode the address of this chain.  In each iteration, PRF is used to
   generate a key for F and a bitmask that is XORed to the intermediate
   result before it is processed by F.  In the following, ADRS is a
   32-byte OTS hash address as specified in Section 2.5 and SEED is an
   n-byte string.  To generate the keys and bitmasks, PRF is called with
   SEED as key and ADRS as input.  The chaining function takes as input
   an n-byte string X, a start index i, a number of steps s, as well as
   ADRS and SEED.  The chaining function returns as output the value
   obtained by iterating F for s times on input X, using the outputs of
   PRF.

























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   Algorithm 2: chain - Chaining Function

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

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

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

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

3.1.3.  WOTS+ Private Key

   The private key in WOTS+, denoted by sk (s for secret), is a length
   len array of n-byte strings.  This private key MUST be only used to
   sign at most one message.  Each n-byte string MUST either be selected
   randomly from the uniform distribution or be selected using a
   cryptographically secure pseudorandom procedure.  In the latter case,
   the security of the used procedure MUST at least match that of the
   WOTS+ parameters used.  For a further discussion on pseudorandom key
   generation, see Section 3.1.7.  The following pseudocode (Algorithm
   3) describes an algorithm for generating sk.

   Algorithm 3: WOTS_genSK - Generating a WOTS+ Private Key

     Input: No input
     Output: WOTS+ private key sk

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







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3.1.4.  WOTS+ Public Key

   A WOTS+ key pair defines a virtual structure that consists of len
   hash chains of length w.  The len n-byte strings in the private key
   each define the start node for one hash chain.  The public key
   consists of the end nodes of these hash chains.  Therefore, like the
   private key, the public key is also a length len array of n-byte
   strings.  To compute the hash chain, the chaining function (Algorithm
   2) is used.  An OTS hash address ADRS and a seed SEED have to be
   provided by the calling algorithm.  This address will encode the
   address of the WOTS+ key pair within a greater structure.  Hence, a
   WOTS+ algorithm MUST NOT manipulate any parts of ADRS except for the
   last three 32-bit words.  Please note that the SEED used here is
   public information also available to a verifier.  The following
   pseudocode (Algorithm 4) describes an algorithm for generating the
   public key pk, where sk is the private key.

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

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

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

3.1.5.  WOTS+ Signature Generation

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



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   ADRS except for the last three 32-bit words.  Please note that the
   SEED used here is public information also available to a verifier.
   The pseudocode for signature generation is shown below (Algorithm 5),
   where M is the message and sig is the resulting signature.

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

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

     csum = 0;

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

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

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

   The data format for a signature is given below.

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

                              WOTS+ Signature



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3.1.6.  WOTS+ Signature Verification

   In order to verify a signature sig on a message M, the verifier
   computes a WOTS+ public key value from the signature.  This can be
   done by "completing" the chain computations starting from the
   signature values, using the base w values of the message hash and its
   checksum.  This step, called WOTS_pkFromSig, is described below in
   Algorithm 6.  The result of WOTS_pkFromSig is then compared to the
   given public key.  If the values are equal, the signature is
   accepted.  Otherwise, the signature MUST be rejected.  An OTS hash
   address ADRS and a seed SEED have to be provided by the calling
   algorithm.  This address will encode the address of the WOTS+ key
   pair within a greater structure.  Hence, a WOTS+ algorithm MUST NOT
   manipulate any parts of ADRS except for the last three 32-bit words.
   Please note that the SEED used here is public information also
   available to a verifier.

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

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

     csum = 0;

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

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

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

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






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3.1.7.  Pseudorandom Key Generation

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

   The advantage of generating the private key elements from a random
   n-byte string is that only this n-byte string needs to be stored
   instead of the full private key.  The key can be regenerated when
   needed.  The suggested method from [BDH11] can be described using
   PRF.  During key generation, a uniformly random n-byte string S is
   sampled from a secure source of randomness.  This string S is stored
   as private key.  The private key elements are computed as sk[i] =
   PRF(S, toByte(i, 32)) whenever needed.  Please note that this seed S
   MUST be different from the seed SEED used to randomize the hash
   function calls.  Also, this seed S MUST be kept secret.  The seed S
   MUST NOT be a low entropy, human-memorable value since private key
   elements are derived from S deterministically and their
   confidentiality is security-critical.

4.  Schemes

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

4.1.  XMSS: eXtended Merkle Signature Scheme

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



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   (public) seed that is part of the public key using the pseudorandom
   function PRF.  The value corresponding to the root of the XMSS tree
   forms the XMSS public key together with the seed.

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

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

4.1.1.  XMSS Parameters

   XMSS has the following parameters:

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

      n: the length in bytes of the message digest as well as each node

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

   There are 2^h leaves in the tree.

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

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






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4.1.2.  XMSS Hash Functions

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

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

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

   More detail on specific instantiations can be found in Section 5.
   Security requirements on H and H_msg are discussed in Section 9.

4.1.3.  XMSS Private Key

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

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

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









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4.1.4.  Randomized Tree Hashing

   To improve readability, we introduce a function RAND_HASH(LEFT,
   RIGHT, SEED, ADRS) (Algorithm 7) that does the randomized hashing in
   the tree.  It takes as input two n-byte values LEFT and RIGHT that
   represent the left and the right halves of the hash function input,
   the seed SEED used as key for PRF, and the address ADRS of this hash
   function call.  RAND_HASH first uses PRF with SEED and ADRS to
   generate a key KEY and n-byte bitmasks BM_0, BM_1.  Then, it returns
   the randomized hash H(KEY, (LEFT XOR BM_0) || (RIGHT XOR BM_1)).

   Algorithm 7: RAND_HASH

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

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

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

4.1.5.  L-Trees

   To compute the leaves of the binary hash tree, a so-called L-tree is
   used.  An L-tree is an unbalanced binary hash tree, distinct but
   similar to the main XMSS binary hash tree.  The algorithm ltree
   (Algorithm 8) takes as input a WOTS+ public key pk and compresses it
   to a single n-byte value pk[0].  It also takes as input an L-tree
   address ADRS that encodes the address of the L-tree and the seed
   SEED.
















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   Algorithm 8: ltree

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

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

4.1.6.  TreeHash

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














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   Algorithm 9: treeHash

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

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

4.1.7.  XMSS Key Generation

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













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   Algorithm 10: XMSS_keyGen - Generate an XMSS key pair

     Input: No input
     Output: XMSS private key SK, XMSS public key PK

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

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

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

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

   The format of an XMSS public key is given below.

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

                              XMSS Public Key





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4.1.8.  XMSS Signature

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

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

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

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

   o  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.  A verifier needs
   these nodes to compute a root node for the tree from the WOTS+ public
   key.  A node Node is addressed by its position in the tree.  Node(x,
   y) denotes the y^th node on level x with y = 0 being the leftmost
   node on a level.  The leaves are on level 0; the root is on level h.
   An authentication path contains exactly one node on every layer 0 <=
   x <= (h - 1).  For the i^th WOTS+ key pair, counting from zero, the
   j^th authentication path node is:

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

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






















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   The data format for a signature is given below.

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

                              XMSS Signature

4.1.9.  XMSS Signature Generation

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

   The node values of the authentication path MAY be computed in any
   way.  This computation is assumed to be performed by the subroutine
   buildAuth for the function XMSS_sign (Algorithm 12).  The fastest
   alternative is to store all tree nodes and set the array in the
   signature by copying the respective nodes.  The least storage-
   intensive alternative is to recompute all nodes for each signature



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   online using the treeHash algorithm (Algorithm 9).  Several
   algorithms exist in between, with different time/storage trade-offs.
   For an overview, see [BDS09].  A further approach can be found in
   [KMN14].  Note that the details of this procedure are not relevant to
   interoperability; it is not necessary to know any of these details in
   order to perform the signature verification operation.  The following
   version of buildAuth is given for completeness.  It is a simple
   example for understanding, but extremely inefficient.  The use of one
   of the alternative algorithms is strongly RECOMMENDED.

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

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

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

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

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

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














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

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

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

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

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

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

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

4.1.10.  XMSS Signature Verification

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




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

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

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

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

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

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





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   message M, and an XMSS public key PK.  XMSS_verify returns true if
   and only if Sig is a valid signature on M under public key PK.
   Otherwise, it returns false.

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

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

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

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

4.1.11.  Pseudorandom Key Generation

   An implementation MAY use a cryptographically secure pseudorandom
   method to generate the XMSS private key from a single n-byte value.
   For example, the method suggested in [BDH11] and explained below MAY
   be used.  Other methods, such as the one in [HRS16], MAY be used.
   The choice of a pseudorandom method does not affect interoperability,
   but the cryptographic strength MUST match that of the used XMSS
   parameters.

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







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4.1.12.  Free Index Handling and Partial Private Keys

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

4.2.  XMSS^MT: Multi-Tree XMSS

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

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

4.2.1.  XMSS^MT Parameters

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

   All the trees on higher layers sign root nodes of other trees, with
   the root nodes being n-byte strings.  Hence, no message compression
   is needed, and WOTS+ is used to sign the root nodes themselves
   instead of their hash values.

4.2.2.  XMSS^MT Key Generation

   An XMSS^MT private key SK_MT (S for secret) consists of one reduced
   XMSS private key for each XMSS tree.  These reduced XMSS private keys
   just contain the WOTS+ private keys corresponding to that XMSS key
   pair; they do not contain a pseudorandom function key, index, public
   seed, or root node.  Instead, SK_MT contains a single n-byte
   pseudorandom function key SK_PRF, a single (ceil(h / 8))-byte index
   idx_MT, a single n-byte seed SEED, and a single root value root



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   (which is the root of the single tree on the top layer).  The index
   is a global index over all WOTS+ key pairs of all XMSS trees on layer
   0.  It is initialized with 0.  It stores the index of the last used
   WOTS+ key pair on the bottom layer, i.e., a number between 0 and 2^h
   - 1.

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

   This document does not define any specific format for the XMSS^MT
   private key SK_MT as it is not required for interoperability.
   Algorithms 15 and 16 use a function getXMSS_SK(SK, x, y) that outputs
   the reduced private key of the x^th XMSS tree on the y^th layer.

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















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

     Input: No input
     Output: XMSS^MT private key SK_MT, XMSS^MT public key PK_MT

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

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

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

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

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









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

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

                            XMSS^MT Public Key

4.2.3.  XMSS^MT Signature

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

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

   o  a byte string r used for randomized message hashing (n bytes), and

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

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




















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   The data format for a signature is given below.

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

                             XMSS^MT Signature

4.2.4.  XMSS^MT Signature Generation

   To compute the XMSS^MT signature Sig_MT of a message M using an
   XMSS^MT private key SK_MT, XMSSMT_sign (Algorithm 16) described below
   uses treeSig as defined in Section 4.1.9.  First, the signature index
   is set to idx_sig.  Next, PRF is used to compute a pseudorandom
   n-byte string r.  This n-byte string, idx_sig, and the root node from
   PK_MT are then used to compute a randomized message digest of length
   n.  The message digest is signed using the WOTS+ key pair on the
   bottom layer with absolute index idx.  The authentication path for
   the WOTS+ key pair and the root of the containing XMSS tree are
   computed.  The root is signed by the parent XMSS tree.  This is
   repeated until the top tree is reached.





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   Algorithm 16: XMSSMT_sign - Generate an XMSS^MT signature and update
   the XMSS^MT private key

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

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

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

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

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









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   Algorithm 16 is only one method to compute XMSS^MT signatures.  Time-
   memory trade-offs exist that allow reduction of the signing time to
   less than the signing time of an XMSS scheme with tree height h / d.
   These trade-offs 1) prevent certain values from being recomputed
   several times by keeping a state and 2) distribute all computations
   over all signature generations.  Details can be found in
   [Huelsing13a].

4.2.5.  XMSS^MT Signature Verification

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

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

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

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

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

     unsigned int idx_leaf
                   = (h / d) least significant bits of idx_sig;
     unsigned int idx_tree
                   = (h - h / d) most significant bits of idx_sig;
     Sig' = getXMSSSignature(Sig_MT, 0);
     ADRS.setLayerAddress(0);
     ADRS.setTreeAddress(idx_tree);
     byte[n] node = XMSS_rootFromSig(idx_leaf, getSig_ots(Sig'),
                                      getAuth(Sig'), M', SEED, ADRS);





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

4.2.6.  Pseudorandom Key Generation

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

   For XMSS^MT, a method similar to that for XMSS and WOTS+ can be used.
   The method uses PRF.  During key generation, a uniformly random
   n-byte string S_MT is sampled from a secure source of randomness.
   This seed S_MT is used to generate one n-byte value S for each XMSS
   key pair.  This n-byte value can be used to compute the respective
   XMSS private key using the method described in Section 4.1.11.  Let
   S[x][y] be the seed for the x^th XMSS private key on layer y.  The
   seeds are computed as S[x][y] = PRF(PRF(S, toByte(y, 32)), toByte(x,
   32)).

4.2.7.  Free Index Handling and Partial Private Keys

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

5.  Parameter Sets

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



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   While this document specifies several parameter sets, an
   implementation is only REQUIRED to provide support for verification
   of all REQUIRED parameter sets.  The REQUIRED parameter sets all use
   SHA2-256 to instantiate all functions.  The REQUIRED parameter sets
   are only distinguished by the tree height parameter h (which
   determines the number of signatures that can be done with a single
   key pair) and the number of layers d (which defines a trade-off
   between speed and signature size).  An implementation MAY provide
   support for signature generation using any of the proposed parameter
   sets.  For convenience, this document defines a default option for
   XMSS (XMSS_SHA2_20_256) and XMSS^MT (XMSSMT-SHA2_60/3_256).  These
   are supposed to match the most generic requirements.

5.1.  Implementing the Functions

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

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

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

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

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

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

   Accordingly, for SHA2 with n = 64 we use:

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

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

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

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






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

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

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

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

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

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

   Accordingly, for SHA3 with n = 64, we use:

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

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

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

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

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

















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5.2.  WOTS+ Parameters

   To fully describe a WOTS+ signature method, the parameters n and w,
   as well as the functions F and PRF, MUST be specified.  The following
   table defines several WOTS+ signature systems, each of which is
   identified by a name.  Naming follows this convention:
   WOTSP-[Hashfamily]_[n in bits].  Naming does not include w as all
   parameter sets in this document use w=16.  Values for len are
   provided for convenience.

              +-----------------+----------+----+----+-----+
              | Name            | F / PRF  |  n |  w | len |
              +-----------------+----------+----+----+-----+
              | REQUIRED:       |          |    |    |     |
              |                 |          |    |    |     |
              | WOTSP-SHA2_256  | SHA2-256 | 32 | 16 |  67 |
              |                 |          |    |    |     |
              | OPTIONAL:       |          |    |    |     |
              |                 |          |    |    |     |
              | WOTSP-SHA2_512  | SHA2-512 | 64 | 16 | 131 |
              |                 |          |    |    |     |
              | WOTSP-SHAKE_256 | SHAKE128 | 32 | 16 |  67 |
              |                 |          |    |    |     |
              | WOTSP-SHAKE_512 | SHAKE256 | 64 | 16 | 131 |
              +-----------------+----------+----+----+-----+

                                  Table 1

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

5.3.  XMSS Parameters

   To fully describe an XMSS signature method, the parameters n, w, and
   h, as well as the functions F, H, H_msg, and PRF, MUST be specified.
   The following table defines different XMSS signature systems, each of
   which is identified by a name.  Naming follows this convention:
   XMSS-[Hashfamily]_[h]_[n in bits].  Naming does not include w as all
   parameter sets in this document use w=16.











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          +-------------------+-----------+----+----+-----+----+
          | Name              | Functions |  n |  w | len |  h |
          +-------------------+-----------+----+----+-----+----+
          | REQUIRED:         |           |    |    |     |    |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_10_256  | SHA2-256  | 32 | 16 |  67 | 10 |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_16_256  | SHA2-256  | 32 | 16 |  67 | 16 |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_20_256  | SHA2-256  | 32 | 16 |  67 | 20 |
          |                   |           |    |    |     |    |
          | OPTIONAL:         |           |    |    |     |    |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_10_512  | SHA2-512  | 64 | 16 | 131 | 10 |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_16_512  | SHA2-512  | 64 | 16 | 131 | 16 |
          |                   |           |    |    |     |    |
          | XMSS-SHA2_20_512  | SHA2-512  | 64 | 16 | 131 | 20 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_10_256 | SHAKE128  | 32 | 16 |  67 | 10 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_16_256 | SHAKE128  | 32 | 16 |  67 | 16 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_20_256 | SHAKE128  | 32 | 16 |  67 | 20 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_10_512 | SHAKE256  | 64 | 16 | 131 | 10 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_16_512 | SHAKE256  | 64 | 16 | 131 | 16 |
          |                   |           |    |    |     |    |
          | XMSS-SHAKE_20_512 | SHAKE256  | 64 | 16 | 131 | 20 |
          +-------------------+-----------+----+----+-----+----+

                                  Table 2

   The XDR formats for XMSS are listed in Appendix B.

5.3.1.  Parameter Guide

   In contrast to traditional signature schemes like RSA or Digital
   Signature Algorithm (DSA), XMSS has a tree height parameter h that
   determines the number of messages that can be signed with one key
   pair.  Increasing the height allows using a key pair for more
   signatures, but it also increases the signature size and slows down
   key generation, signing, and verification.  To demonstrate the impact
   of different values of h, the following table shows signature size
   and runtimes.  Runtimes are given as the number of calls to F and H
   when the BDS algorithm is used to compute authentication paths for




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RFC 8391                          XMSS                          May 2018


   the worst case.  The last column shows the number of messages that
   can be signed with one key pair.  The numbers are the same for the
   XMSS-SHAKE instances with same parameters h and n.

    +------------------+-------+------------+--------+--------+-------+
    | Name             | |Sig| |     KeyGen |   Sign | Verify | #Sigs |
    +------------------+-------+------------+--------+--------+-------+
    | REQUIRED:        |       |            |        |        |       |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_10_256 | 2,500 |  1,238,016 |  5,725 |  1,149 |  2^10 |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_16_256 | 2,692 |    79*10^6 |  9,163 |  1,155 |  2^16 |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_20_256 | 2,820 | 1,268*10^6 | 11,455 |  1,159 |  2^20 |
    |                  |       |            |        |        |       |
    | OPTIONAL:        |       |            |        |        |       |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_10_512 | 9,092 |  2,417,664 | 11,165 |  2,237 |  2^10 |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_16_512 | 9,476 |   155*10^6 | 17,867 |  2,243 |  2^16 |
    |                  |       |            |        |        |       |
    | XMSS-SHA2_20_512 | 9,732 | 2,476*10^6 | 22,335 |  2,247 |  2^20 |
    +------------------+-------+------------+--------+--------+-------+

                                  Table 3

   As a default, users without special requirements should use option
   XMSS-SHA2_20_256, which allows signing of 2^20 messages with one key
   pair and provides reasonable speed and signature size.  Users that
   require more signatures per key pair or faster key generation should
   consider XMSS^MT.

5.4.  XMSS^MT Parameters

   To fully describe an XMSS^MT signature method, the parameters n, w,
   h, and d, as well as the functions F, H, H_msg, and PRF, MUST be
   specified.  The following table defines different XMSS^MT signature
   systems, each of which is identified by a name.  Naming follows this
   convention: XMSSMT-[Hashfamily]_[h]/[d]_[n in bits].  Naming does not
   include w as all parameter sets in this document use w=16.











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     +------------------------+-----------+----+----+-----+----+----+
     | Name                   | Functions |  n |  w | len |  h |  d |
     +------------------------+-----------+----+----+-----+----+----+
     | REQUIRED:              |           |    |    |     |    |    |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_20/2_256   | SHA2-256  | 32 | 16 |  67 | 20 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_20/4_256   | SHA2-256  | 32 | 16 |  67 | 20 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/2_256   | SHA2-256  | 32 | 16 |  67 | 40 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/4_256   | SHA2-256  | 32 | 16 |  67 | 40 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/8_256   | SHA2-256  | 32 | 16 |  67 | 40 |  8 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/3_256   | SHA2-256  | 32 | 16 |  67 | 60 |  3 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/6_256   | SHA2-256  | 32 | 16 |  67 | 60 |  6 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/12_256  | SHA2-256  | 32 | 16 |  67 | 60 | 12 |
     |                        |           |    |    |     |    |    |
     | OPTIONAL:              |           |    |    |     |    |    |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_20/2_512   | SHA2-512  | 64 | 16 | 131 | 20 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_20/4_512   | SHA2-512  | 64 | 16 | 131 | 20 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/2_512   | SHA2-512  | 64 | 16 | 131 | 40 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/4_512   | SHA2-512  | 64 | 16 | 131 | 40 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_40/8_512   | SHA2-512  | 64 | 16 | 131 | 40 |  8 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/3_512   | SHA2-512  | 64 | 16 | 131 | 60 |  3 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/6_512   | SHA2-512  | 64 | 16 | 131 | 60 |  6 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHA2_60/12_512  | SHA2-512  | 64 | 16 | 131 | 60 | 12 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_20/2_256  | SHAKE128  | 32 | 16 |  67 | 20 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_20/4_256  | SHAKE128  | 32 | 16 |  67 | 20 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/2_256  | SHAKE128  | 32 | 16 |  67 | 40 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/4_256  | SHAKE128  | 32 | 16 |  67 | 40 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/8_256  | SHAKE128  | 32 | 16 |  67 | 40 |  8 |



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     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/3_256  | SHAKE128  | 32 | 16 |  67 | 60 |  3 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/6_256  | SHAKE128  | 32 | 16 |  67 | 60 |  6 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/12_256 | SHAKE128  | 32 | 16 |  67 | 60 | 12 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_20/2_512  | SHAKE256  | 64 | 16 | 131 | 20 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_20/4_512  | SHAKE256  | 64 | 16 | 131 | 20 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/2_512  | SHAKE256  | 64 | 16 | 131 | 40 |  2 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/4_512  | SHAKE256  | 64 | 16 | 131 | 40 |  4 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_40/8_512  | SHAKE256  | 64 | 16 | 131 | 40 |  8 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/3_512  | SHAKE256  | 64 | 16 | 131 | 60 |  3 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/6_512  | SHAKE256  | 64 | 16 | 131 | 60 |  6 |
     |                        |           |    |    |     |    |    |
     | XMSSMT-SHAKE_60/12_512 | SHAKE256  | 64 | 16 | 131 | 60 | 12 |
     +------------------------+-----------+----+----+-----+----+----+

                                  Table 4

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

5.4.1.  Parameter Guide

   In addition to the tree height parameter already used for XMSS,
   XMSS^MT has the parameter d that determines the number of tree
   layers.  XMSS can be understood as XMSS^MT with a single layer, i.e.,
   d=1.  Hence, the choice of h has the same effect as for XMSS.  The
   number of tree layers provides a trade-off between signature size on
   the one side and key generation and signing speed on the other side.
   Increasing the number of layers reduces key generation time
   exponentially and signing time linearly at the cost of increasing the
   signature size linearly.  Essentially, an XMSS^MT signature contains
   one WOTSP signature per layer.  Speed roughly corresponds to d-times
   the speed for XMSS with trees of height h/d.

   To demonstrate the impact of different values of h and d, the
   following table shows signature size and runtimes.  Runtimes are
   given as the number of calls to F and H when the BDS algorithm and
   distributed signature generation are used.  Timings are worst-case
   times.  The last column shows the number of messages that can be
   signed with one key pair.  The numbers are the same for the XMSS-



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   SHAKE instances with same parameters h and n.  Due to formatting
   limitations, only the parameter part of the parameter set names are
   given, omitting the name "XMSSMT".

    +----------------+---------+------------+--------+--------+-------+
    | Name           |   |Sig| |     KeyGen |   Sign | Verify | #Sigs |
    +----------------+---------+------------+--------+--------+-------+
    | REQUIRED:      |         |            |        |        |       |
    |                |         |            |        |        |       |
    | SHA2_20/2_256  |   4,963 |  2,476,032 |  7,227 |  2,298 |  2^20 |
    |                |         |            |        |        |       |
    | SHA2_20/4_256  |   9,251 |    154,752 |  4,170 |  4,576 |  2^20 |
    |                |         |            |        |        |       |
    | SHA2_40/2_256  |   5,605 | 2,535*10^6 | 13,417 |  2,318 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_40/4_256  |   9,893 |  4,952,064 |  7,227 |  4,596 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_40/8_256  |  18,469 |    309,504 |  4,170 |  9,152 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_60/3_256  |   8,392 | 3,803*10^6 | 13,417 |  3,477 |  2^60 |
    |                |         |            |        |        |       |
    | SHA2_60/6_256  |  14,824 |  7,428,096 |  7,227 |  6,894 |  2^60 |
    |                |         |            |        |        |       |
    | SHA2_60/12_256 |  27,688 |    464,256 |  4,170 | 13,728 |  2^60 |
    |                |         |            |        |        |       |
    | OPTIONAL:      |         |            |        |        |       |
    |                |         |            |        |        |       |
    | SHA2_20/2_512  |  18,115 |  4,835,328 | 14,075 |  4,474 |  2^20 |
    |                |         |            |        |        |       |
    | SHA2_20/4_512  |  34,883 |    302,208 |  8,138 |  8,928 |  2^20 |
    |                |         |            |        |        |       |
    | SHA2_40/2_512  |  19,397 | 4,951*10^6 | 26,025 |  4,494 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_40/4_512  |  36,165 |  9,670,656 | 14,075 |  8,948 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_40/8_512  |  69,701 |    604,416 |  8,138 | 17,856 |  2^40 |
    |                |         |            |        |        |       |
    | SHA2_60/3_512  |  29,064 | 7,427*10^6 | 26,025 |  6,741 |  2^60 |
    |                |         |            |        |        |       |
    | SHA2_60/6_512  |  54,216 | 14,505,984 | 14,075 | 13,422 |  2^60 |
    |                |         |            |        |        |       |
    | SHA2_60/12_512 | 104,520 |    906,624 |  8,138 | 26,784 |  2^60 |
    +----------------+---------+------------+--------+--------+-------+

                                  Table 5






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   As a default, users without special requirements should use option
   XMSSMT-SHA2_60/3_256, which allows signing of 2^60 messages with one
   key pair (this is a virtually unbounded number of signatures).  At
   the same time, signature size and speed are well balanced.

6.  Rationale

   The goal of this note is to describe the WOTS+, XMSS, and XMSS^MT
   algorithms based on the scientific literature.  The description is
   done in a modular way that allows basing a description of stateless
   hash-based signature algorithms like SPHINCS [BHH15] on it.

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

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

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

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








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   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.

7.  Reference Code

   For testing purposes, a reference implementation in C is available.
   The code contains a basic implementation that closely follows the
   pseudocode in this document and an optimized implementation that uses
   the BDS algorithm [BDS08] to compute authentication paths and
   distributed signature generation as described in [HRB13] for XMSS^MT.

   The code is permanently available at
   <https://github.com/joostrijneveld/xmss-reference>.

8.  IANA Considerations

   The Internet Assigned Numbers Authority (IANA) has created three
   registries: one for WOTS+ signatures (as defined in Section 3), one
   for XMSS signatures (as defined in Section 4), and one for XMSS^MT
   signatures (as defined in Section 4).  For the sake of clarity and
   convenience, the first collection of WOTS+, XMSS, and XMSS^MT
   parameter sets is defined in Section 5.  Additions to these
   registries require that a specification be documented in an RFC or
   another permanent and readily available reference in sufficient
   detail as defined by the "Specification Required" policy described in
   [RFC8126] to make interoperability between independent
   implementations possible.  Each entry in these registries contains
   the following elements:

   o  a short name, such as "XMSS_SHA2_20_256",

   o  a positive number, and

   o  a reference to a specification that completely defines the
      signature method test cases or provides a reference implementation
      that can be used to verify the correctness of an implementation
      (or a reference to such an implementation).

   Requests to add an entry to these registries MUST include the name
   and the reference.  The number is assigned by IANA.  These number
   assignments SHOULD use the smallest available positive number.
   Submitters MUST have their requests reviewed and approved.
   Designated Experts for this task as requested by the "Specification
   Required" policy defined by [RFC8126] will be assigned by the




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   Internet Engineering Steering Group (IESG).  The IESG can be
   contacted at iesg@ietf.org.  Interested applicants that are
   unfamiliar with IANA processes should visit <http://www.iana.org>.

   The number 0x00000000 (decimal 0) is Reserved.  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
   [RFC8126].

   The "WOTS+ Signatures" registry is as follows.

          +--------------------+-----------------+-------------+
          | Numeric Identifier | Name            |  Reference  |
          +--------------------+-----------------+-------------+
          |     0x00000000     | Reserved        |   this RFC  |
          |                    |                 |             |
          |     0x00000001     | WOTSP-SHA2_256  | Section 5.2 |
          |                    |                 |             |
          |     0x00000002     | WOTSP-SHA2_512  | Section 5.2 |
          |                    |                 |             |
          |     0x00000003     | WOTSP-SHAKE_256 | Section 5.2 |
          |                    |                 |             |
          |     0x00000004     | WOTSP-SHAKE_512 | Section 5.2 |
          +--------------------+-----------------+-------------+

                                  Table 6























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   The "XMSS Signatures" registry is as follows.

         +--------------------+-------------------+-------------+
         | Numeric Identifier | Name              |  Reference  |
         +--------------------+-------------------+-------------+
         |     0x00000000     | Reserved          |   this RFC  |
         |                    |                   |             |
         |     0x00000001     | XMSS-SHA2_10_256  | Section 5.3 |
         |                    |                   |             |
         |     0x00000002     | XMSS-SHA2_16_256  | Section 5.3 |
         |                    |                   |             |
         |     0x00000003     | XMSS-SHA2_20_256  | Section 5.3 |
         |                    |                   |             |
         |     0x00000004     | XMSS-SHA2_10_512  | Section 5.3 |
         |                    |                   |             |
         |     0x00000005     | XMSS-SHA2_16_512  | Section 5.3 |
         |                    |                   |             |
         |     0x00000006     | XMSS-SHA2_20_512  | Section 5.3 |
         |                    |                   |             |
         |     0x00000007     | XMSS-SHAKE_10_256 | Section 5.3 |
         |                    |                   |             |
         |     0x00000008     | XMSS-SHAKE_16_256 | Section 5.3 |
         |                    |                   |             |
         |     0x00000009     | XMSS-SHAKE_20_256 | Section 5.3 |
         |                    |                   |             |
         |     0x0000000A     | XMSS-SHAKE_10_512 | Section 5.3 |
         |                    |                   |             |
         |     0x0000000B     | XMSS-SHAKE_16_512 | Section 5.3 |
         |                    |                   |             |
         |     0x0000000C     | XMSS-SHAKE_20_512 | Section 5.3 |
         +--------------------+-------------------+-------------+

                                  Table 7


















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   The "XMSS^MT Signatures" registry is as follows.

       +--------------------+------------------------+-------------+
       | Numeric Identifier | Name                   |  Reference  |
       +--------------------+------------------------+-------------+
       |     0x00000000     | Reserved               |   this RFC  |
       |                    |                        |             |
       |     0x00000001     | XMSSMT-SHA2_20/2_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000002     | XMSSMT-SHA2_20/4_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000003     | XMSSMT-SHA2_40/2_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000004     | XMSSMT-SHA2_40/4_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000005     | XMSSMT-SHA2_40/8_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000006     | XMSSMT-SHA2_60/3_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000007     | XMSSMT-SHA2_60/6_256   | Section 5.4 |
       |                    |                        |             |
       |     0x00000008     | XMSSMT-SHA2_60/12_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000009     | XMSSMT-SHA2_20/2_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000A     | XMSSMT-SHA2_20/4_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000B     | XMSSMT-SHA2_40/2_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000C     | XMSSMT-SHA2_40/4_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000D     | XMSSMT-SHA2_40/8_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000E     | XMSSMT-SHA2_60/3_512   | Section 5.4 |
       |                    |                        |             |
       |     0x0000000F     | XMSSMT-SHA2_60/6_512   | Section 5.4 |
       |                    |                        |             |
       |     0x00000010     | XMSSMT-SHA2_60/12_512  | Section 5.4 |
       |                    |                        |             |
       |     0x00000011     | XMSSMT-SHAKE_20/2_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000012     | XMSSMT-SHAKE_20/4_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000013     | XMSSMT-SHAKE_40/2_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000014     | XMSSMT-SHAKE_40/4_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000015     | XMSSMT-SHAKE_40/8_256  | Section 5.4 |



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       |                    |                        |             |
       |     0x00000016     | XMSSMT-SHAKE_60/3_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000017     | XMSSMT-SHAKE_60/6_256  | Section 5.4 |
       |                    |                        |             |
       |     0x00000018     | XMSSMT-SHAKE_60/12_256 | Section 5.4 |
       |                    |                        |             |
       |     0x00000019     | XMSSMT-SHAKE_20/2_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001A     | XMSSMT-SHAKE_20/4_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001B     | XMSSMT-SHAKE_40/2_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001C     | XMSSMT-SHAKE_40/4_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001D     | XMSSMT-SHAKE_40/8_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001E     | XMSSMT-SHAKE_60/3_512  | Section 5.4 |
       |                    |                        |             |
       |     0x0000001F     | XMSSMT-SHAKE_60/6_512  | Section 5.4 |
       |                    |                        |             |
       |     0x00000020     | XMSSMT-SHAKE_60/12_512 | Section 5.4 |
       +--------------------+------------------------+-------------+

                                  Table 8

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

9.  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 its choice.  A signature system
   is secure if, even in this setting, the attacker cannot produce a new
   message/signature pair of his choosing such that the verification
   algorithm accepts.

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




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   1/2.  Hence, to mount a successful attack, an attacker needs to
   perform 2^b bit operations on average.  The given values for bit
   security were estimated according to [HRS16].

9.1.  Security Proofs

   A full security proof for all schemes described in this document can
   be found in [HRS16].  This proof shows that an attacker has to break
   at least one out of certain security properties of the used hash
   functions and PRFs to forge a signature in any of the described
   schemes.  The proof in [HRS16] considers an initial message
   compression different from the randomized hashing used here.  We
   comment on this below.  For the original schemes, these proofs show
   that an attacker has to break certain minimal security properties.
   In particular, it is not sufficient to break the collision resistance
   of the hash functions to generate a forgery.

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

   The proof in [HRS16] considers classical randomized hashing for the
   initial message compression, i.e., H(r, M) instead of H(r ||
   getRoot(PK) || index, M).  This classical randomized hashing allows
   getting a security reduction from extended target collision
   resistance [HRS16], a property that is conjectured to be strictly
   weaker than collision resistance.  However, it turns out that in this
   case, an attacker could still launch a multi-target attack even
   against multiple users at the same time.  The reason is that the
   adversary attacking u users at the same time learns u * 2^h



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   randomized hashes H(r_i_j || M_i_j) with signature index i in [0, 2^h
   - 1] and user index j in [0, u].  It suffices to find a single pair
   (r*, M*) such that H(r* || M*) = H(r_i_u || M_i_u) for one out of the
   u * 2^h learned hashes.  Hence, an attacker can do a brute-force
   search in time 2^n / u * 2^h instead of 2^n.

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

   The given bit security values were estimated based on the complexity
   of the best-known generic attacks against the required security
   properties of the used hash and pseudorandom functions, assuming
   conventional and quantum adversaries.  At the time of writing,
   generic attacks are the best-known attacks for the parameters
   suggested in the classical setting.  Also, in the quantum setting,
   there are no dedicated attacks known that perform better than generic
   attacks.  Nevertheless, the topic of quantum cryptanalysis of hash
   functions is not as well understood as in the classical setting.

9.2.  Minimal Security Assumptions

   The assumptions one has to make to prove security of the described
   schemes are minimal in the following sense.  Any signature algorithm
   that allows arbitrary size messages relies on the security of a
   cryptographic hash function, either on collision resistance or on
   extended target collision resistance if randomized hashing is used
   for message compression.  For the schemes described here, this is
   already sufficient to be secure.  In contrast, common signature
   schemes like RSA, DSA, and Elliptic Curve Digital Signature Algorithm
   (ECDSA) additionally rely on the conjectured hardness of certain
   mathematical problems.

9.3.  Post-Quantum Security

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



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   In contrast to RSA, DSA, and ECDSA, the described signature systems
   are post-quantum-secure if they are used with an appropriate
   cryptographic hash function.  In particular, for post-quantum
   security, the size of n must be twice the size required for classical
   security.  This is in order to protect against quantum square-root
   attacks due to Grover's algorithm.  [HRS16] shows that variants of
   Grover's algorithm are the optimal generic attacks against the
   security properties of hash functions required for the described
   schemes.

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

10.  References

10.1.  Normative References

   [FIPS180]  National Institute of Standards and Technology, "Secure
              Hash Standard (SHS)", FIPS PUB 180-4,
              DOI 10.6028/NIST.FIPS.180-4, August 2015.

   [FIPS202]  National Institute of Standards and Technology, "SHA-3
              Standard: Permutation-Based Hash and Extendable-Output
              Functions", FIPS PUB 202, DOI 10.6028/NIST.FIPS.202,
              August 2015.

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

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

   [RFC8126]  Cotton, M., Leiba, B., and T. Narten, "Guidelines for
              Writing an IANA Considerations Section in RFCs", BCP 26,
              RFC 8126, DOI 10.17487/RFC8126, June 2017,
              <https://www.rfc-editor.org/info/rfc8126>.






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RFC 8391                          XMSS                          May 2018


   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

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,
              DOI 10.1007/978-3-642-25405-5_8, 2011.

   [BDS08]    Buchmann, J., Dahmen, E., and M. Schneider, "Merkle Tree
              Traversal Revisited", Lecture Notes in Computer Science,
              Volume 5299, Post-Quantum Cryptography,
              DOI 10.1007/978-3-540-88403-3_5, 2008.

   [BDS09]    Buchmann, J., Dahmen, E., and M. Szydlo, "Hash-based
              Digital Signature Schemes", Book chapter, Post-Quantum
              Cryptography, DOI 10.1007/978-3-540-88702-7_3, 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, DOI 10.1007/978-3-662-46800-5_15, 2015.

   [HRB13]    Huelsing, A., Rausch, L., and J. Buchmann, "Optimal
              Parameters for XMSS^MT", Lecture Notes in Computer
              Science, Volume 8128, CD-ARES,
              DOI 10.1007/978-3-642-40588-4_14, 2013.

   [HRS16]    Huelsing, A., Rijneveld, J., and F. Song, "Mitigating
              Multi-Target Attacks in Hash-based Signatures", Lecture
              Notes in Computer Science, Volume 9614, Public-Key
              Cryptography - PKC, DOI 10.1007/978-3-662-49384-7_15,
              2016.

   [Huelsing13]
              Huelsing, A., "W-OTS+ - Shorter Signatures for Hash-Based
              Signature Schemes", Lecture Notes in Computer Science,
              Volume 7918, Progress in Cryptology - AFRICACRYPT,
              DOI 10.1007/978-3-642-38553-7_10, 2013.







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   [Huelsing13a]
              Huelsing, A., "Practical Forward Secure Signatures using
              Minimal Security Assumptions", PhD thesis TU Darmstadt,
              2013,
              <http://tuprints.ulb.tu-darmstadt.de/3651/1/Thesis.pdf>.

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

   [MCF18]    McGrew, D., Curcio, M., and S. Fluhrer, "Hash-Based
              Signatures", Work in Progress, draft-mcgrew-hash-sigs-11,
              April 2018.

   [Merkle83] Merkle, R., "Secrecy, Authentication, and Public Key
              Systems", Computer Science Series, UMI Research Press,
              ISBN: 9780835713849, 1983.

































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Appendix A.  WOTS+ XDR Formats

   The WOTS+ signature and public key formats are formally defined using
   XDR [RFC4506] in order to provide an unambiguous, machine readable
   definition.  Though XDR is used, these formats are simple and easy to
   parse without any special tools.  Note that this representation
   includes all optional parameter sets.  The same applies for the XMSS
   and XMSS^MT formats below.

A.1.  WOTS+ Parameter Sets

   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  = 0x00000001,
        wotsp-sha2_512  = 0x00000002,
        wotsp-shake_256 = 0x00000003,
        wotsp-shake_512 = 0x00000004,
      };

A.2.  WOTS+ Signatures

   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:
        case wotsp-shake_256:
          bytestring32 ots_sig_n32_len67[67];

        case wotsp-sha2_512:
        case wotsp-shake_512:
          bytestring64 ots_sig_n64_len18[131];

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






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A.3.  WOTS+ Public Keys

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

      union ots_pubkey switch (ots_algorithm_type type) {
        case wotsp-sha2_256:
        case wotsp-shake_256:
          bytestring32 ots_pubk_n32_len67[67];

        case wotsp-sha2_512:
        case wotsp-shake_512:
          bytestring64 ots_pubk_n64_len18[131];

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

Appendix B.  XMSS XDR Formats

B.1.  XMSS Parameter Sets

   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_10_256  = 0x00000001,
        xmss-sha2_16_256  = 0x00000002,
        xmss-sha2_20_256  = 0x00000003,

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

        xmss-sha2_10_512  = 0x00000004,
        xmss-sha2_16_512  = 0x00000005,
        xmss-sha2_20_512  = 0x00000006,








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

        xmss-shake_10_256 = 0x00000007,
        xmss-shake_16_256 = 0x00000008,
        xmss-shake_20_256 = 0x00000009,

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

        xmss-shake_10_512 = 0x0000000A,
        xmss-shake_16_512 = 0x0000000B,
        xmss-shake_20_512 = 0x0000000C,
      };

B.2.  XMSS Signatures

   XMSS signatures are defined using XDR syntax as follows:

      /* Authentication path types */

      union xmss_path switch (xmss_algorithm_type type) {
        case xmss-sha2_10_256:
        case xmss-shake_10_256:
          bytestring32 path_n32_t10[10];

        case xmss-sha2_16_256:
        case xmss-shake_16_256:
          bytestring32 path_n32_t16[16];

        case xmss-sha2_20_256:
        case xmss-shake_20_256:
          bytestring32 path_n32_t20[20];

        case xmss-sha2_10_512:
        case xmss-shake_10_512:
          bytestring64 path_n64_t10[10];

        case xmss-sha2_16_512:
        case xmss-shake_16_512:
          bytestring64 path_n64_t16[16];

        case xmss-sha2_20_512:
        case xmss-shake_20_512:
          bytestring64 path_n64_t20[20];

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




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      /* Types for XMSS random strings */

      union random_string_xmss switch (xmss_algorithm_type type) {
        case xmss-sha2_10_256:
        case xmss-sha2_16_256:
        case xmss-sha2_20_256:
        case xmss-shake_10_256:
        case xmss-shake_16_256:
        case xmss-shake_20_256:
          bytestring32 rand_n32;

        case xmss-sha2_10_512:
        case xmss-sha2_16_512:
        case xmss-sha2_20_512:
        case xmss-shake_10_512:
        case xmss-shake_16_512:
        case xmss-shake_20_512:
          bytestring64 rand_n64;

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

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

      union xmss_ots_signature switch (xmss_algorithm_type type) {
        case xmss-sha2_10_256:
        case xmss-sha2_16_256:
        case xmss-sha2_20_256:
          wotsp-sha2_256;

        case xmss-sha2_10_512:
        case xmss-sha2_16_512:
        case xmss-sha2_20_512:
          wotsp-sha2_512;

        case xmss-shake_10_256:
        case xmss-shake_16_256:
        case xmss-shake_20_256:
          wotsp-shake_256;

        case xmss-shake_10_512:
        case xmss-shake_16_512:
        case xmss-shake_20_512:
          wotsp-shake_512;






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

      /* XMSS signature structure */

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

B.3.  XMSS Public Keys

   XMSS public keys are defined using XDR syntax as follows:

      /* Types for bitmask seed */

      union seed switch (xmss_algorithm_type type) {
        case xmss-sha2_10_256:
        case xmss-sha2_16_256:
        case xmss-sha2_20_256:
        case xmss-shake_10_256:
        case xmss-shake_16_256:
        case xmss-shake_20_256:
          bytestring32 seed_n32;

        case xmss-sha2_10_512:
        case xmss-sha2_16_512:
        case xmss-sha2_20_512:
        case xmss-shake_10_512:
        case xmss-shake_16_512:
        case xmss-shake_20_512:
          bytestring64 seed_n64;

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








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      /* Types for XMSS root node */

      union xmss_root switch (xmss_algorithm_type type) {
        case xmss-sha2_10_256:
        case xmss-sha2_16_256:
        case xmss-sha2_20_256:
        case xmss-shake_10_256:
        case xmss-shake_16_256:
        case xmss-shake_20_256:
          bytestring32 root_n32;

        case xmss-sha2_10_512:
        case xmss-sha2_16_512:
        case xmss-sha2_20_512:
        case xmss-shake_10_512:
        case xmss-shake_16_512:
        case xmss-shake_20_512:
          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 */
      };

Appendix C.  XMSS^MT XDR Formats

C.1.  XMSS^MT Parameter Sets

   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,





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RFC 8391                          XMSS                          May 2018


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

        xmssmt-sha2_20/2_256   = 0x00000001,
        xmssmt-sha2_20/4_256   = 0x00000002,
        xmssmt-sha2_40/2_256   = 0x00000003,
        xmssmt-sha2_40/4_256   = 0x00000004,
        xmssmt-sha2_40/8_256   = 0x00000005,
        xmssmt-sha2_60/3_256   = 0x00000006,
        xmssmt-sha2_60/6_256   = 0x00000007,
        xmssmt-sha2_60/12_256  = 0x00000008,

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

        xmssmt-sha2_20/2_512   = 0x00000009,
        xmssmt-sha2_20/4_512   = 0x0000000A,
        xmssmt-sha2_40/2_512   = 0x0000000B,
        xmssmt-sha2_40/4_512   = 0x0000000C,
        xmssmt-sha2_40/8_512   = 0x0000000D,
        xmssmt-sha2_60/3_512   = 0x0000000E,
        xmssmt-sha2_60/6_512   = 0x0000000F,
        xmssmt-sha2_60/12_512  = 0x00000010,

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

        xmssmt-shake_20/2_256  = 0x00000011,
        xmssmt-shake_20/4_256  = 0x00000012,
        xmssmt-shake_40/2_256  = 0x00000013,
        xmssmt-shake_40/4_256  = 0x00000014,
        xmssmt-shake_40/8_256  = 0x00000015,
        xmssmt-shake_60/3_256  = 0x00000016,
        xmssmt-shake_60/6_256  = 0x00000017,
        xmssmt-shake_60/12_256 = 0x00000018,

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

        xmssmt-shake_20/2_512  = 0x00000019,
        xmssmt-shake_20/4_512  = 0x0000001A,
        xmssmt-shake_40/2_512  = 0x0000001B,
        xmssmt-shake_40/4_512  = 0x0000001C,
        xmssmt-shake_40/8_512  = 0x0000001D,
        xmssmt-shake_60/3_512  = 0x0000001E,
        xmssmt-shake_60/6_512  = 0x0000001F,
        xmssmt-shake_60/12_512 = 0x00000020,
      };







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RFC 8391                          XMSS                          May 2018


C.2.  XMSS^MT Signatures

   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_20/2_256:
        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_20/4_512:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_20/4_512:
          bytestring3 idx3;

        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_40/2_512:
        case xmssmt-sha2_40/4_512:
        case xmssmt-sha2_40/8_512:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_40/2_512:
        case xmssmt-shake_40/4_512:
        case xmssmt-shake_40/8_512:
          bytestring5 idx5;

        case xmssmt-sha2_60/3_256:
        case xmssmt-sha2_60/6_256:
        case xmssmt-sha2_60/12_256:
        case xmssmt-sha2_60/3_512:
        case xmssmt-sha2_60/6_512:
        case xmssmt-sha2_60/12_512:
        case xmssmt-shake_60/3_256:
        case xmssmt-shake_60/6_256:
        case xmssmt-shake_60/12_256:
        case xmssmt-shake_60/3_512:
        case xmssmt-shake_60/6_512:
        case xmssmt-shake_60/12_512:
          bytestring8 idx8;






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RFC 8391                          XMSS                          May 2018


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

      union random_string_xmssmt switch (xmssmt_algorithm_type type) {
        case xmssmt-sha2_20/2_256:
        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_60/3_256:
        case xmssmt-sha2_60/6_256:
        case xmssmt-sha2_60/12_256:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_60/3_256:
        case xmssmt-shake_60/6_256:
        case xmssmt-shake_60/12_256:
          bytestring32 rand_n32;

        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_20/4_512:
        case xmssmt-sha2_40/2_512:
        case xmssmt-sha2_40/4_512:
        case xmssmt-sha2_40/8_512:
        case xmssmt-sha2_60/3_512:
        case xmssmt-sha2_60/6_512:
        case xmssmt-sha2_60/12_512:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_20/4_512:
        case xmssmt-shake_40/2_512:
        case xmssmt-shake_40/4_512:
        case xmssmt-shake_40/8_512:
        case xmssmt-shake_60/3_512:
        case xmssmt-shake_60/6_512:
        case xmssmt-shake_60/12_512:
          bytestring64 rand_n64;

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

      /* Type for reduced XMSS signatures */





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RFC 8391                          XMSS                          May 2018


      union xmss_reduced (xmss_algorithm_type type) {
        case xmssmt-sha2_20/2_256:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_60/6_256:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_60/6_256:
          bytestring32 xmss_reduced_n32_t77[77];

        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_60/12_256:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_60/12_256:
          bytestring32 xmss_reduced_n32_t72[72];

        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_60/3_256:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_60/3_256:
          bytestring32 xmss_reduced_n32_t87[87];

        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_40/4_512:
        case xmssmt-sha2_60/6_512:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_40/4_512:
        case xmssmt-shake_60/6_512:
          bytestring64 xmss_reduced_n32_t141[141];

        case xmssmt-sha2_20/4_512:
        case xmssmt-sha2_40/8_512:
        case xmssmt-sha2_60/12_512:
        case xmssmt-shake_20/4_512:
        case xmssmt-shake_40/8_512:
        case xmssmt-shake_60/12_512:
          bytestring64 xmss_reduced_n32_t136[136];

        case xmssmt-sha2_40/2_512:
        case xmssmt-sha2_60/3_512:
        case xmssmt-shake_40/2_512:
        case xmssmt-shake_60/3_512:
          bytestring64 xmss_reduced_n32_t151[151];







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RFC 8391                          XMSS                          May 2018


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

      /* xmss_reduced_array depends on d */

      union xmss_reduced_array (xmss_algorithm_type type) {
        case xmssmt-sha2_20/2_256:
        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_40/2_512:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_40/2_512:
          xmss_reduced xmss_red_arr_d2[2];

        case xmssmt-sha2_60/3_256:
        case xmssmt-sha2_60/3_512:
        case xmssmt-shake_60/3_256:
        case xmssmt-shake_60/3_512:
          xmss_reduced xmss_red_arr_d3[3];

        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_20/4_512:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_40/4_512:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_20/4_512:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_40/4_512:
          xmss_reduced xmss_red_arr_d4[4];

        case xmssmt-sha2_60/6_256:
        case xmssmt-sha2_60/6_512:
        case xmssmt-shake_60/6_256:
        case xmssmt-shake_60/6_512:
          xmss_reduced xmss_red_arr_d6[6];

        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_40/8_512:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_40/8_512:
          xmss_reduced xmss_red_arr_d8[8];







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RFC 8391                          XMSS                          May 2018


        case xmssmt-sha2_60/12_256:
        case xmssmt-sha2_60/12_512:
        case xmssmt-shake_60/12_256:
        case xmssmt-shake_60/12_512:
          xmss_reduced xmss_red_arr_d12[12];

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

      /* XMSS^MT signature structure */

      struct xmssmt_signature {
        /* WOTS+ key pair index */
        idx_sig_xmssmt idx_sig;
        /* Random string for randomized hashing */
        random_string_xmssmt randomness;
        /* Array of d reduced XMSS signatures */
        xmss_reduced_array;
      };

C.3.  XMSS^MT Public Keys

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

      /* Types for bitmask seed */

      union seed switch (xmssmt_algorithm_type type) {
        case xmssmt-sha2_20/2_256:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_60/6_256:
        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_60/12_256:
        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_60/3_256:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_60/6_256:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_60/12_256:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_60/3_256:
          bytestring32 seed_n32;






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RFC 8391                          XMSS                          May 2018


        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_40/4_512:
        case xmssmt-sha2_60/6_512:
        case xmssmt-sha2_20/4_512:
        case xmssmt-sha2_40/8_512:
        case xmssmt-sha2_60/12_512:
        case xmssmt-sha2_40/2_512:
        case xmssmt-sha2_60/3_512:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_40/4_512:
        case xmssmt-shake_60/6_512:
        case xmssmt-shake_20/4_512:
        case xmssmt-shake_40/8_512:
        case xmssmt-shake_60/12_512:
        case xmssmt-shake_40/2_512:
        case xmssmt-shake_60/3_512:
          bytestring64 seed_n64;

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

      /* Types for XMSS^MT root node */

      union xmssmt_root switch (xmssmt_algorithm_type type) {
        case xmssmt-sha2_20/2_256:
        case xmssmt-sha2_20/4_256:
        case xmssmt-sha2_40/2_256:
        case xmssmt-sha2_40/4_256:
        case xmssmt-sha2_40/8_256:
        case xmssmt-sha2_60/3_256:
        case xmssmt-sha2_60/6_256:
        case xmssmt-sha2_60/12_256:
        case xmssmt-shake_20/2_256:
        case xmssmt-shake_20/4_256:
        case xmssmt-shake_40/2_256:
        case xmssmt-shake_40/4_256:
        case xmssmt-shake_40/8_256:
        case xmssmt-shake_60/3_256:
        case xmssmt-shake_60/6_256:
        case xmssmt-shake_60/12_256:
          bytestring32 root_n32;

        case xmssmt-sha2_20/2_512:
        case xmssmt-sha2_20/4_512:
        case xmssmt-sha2_40/2_512:
        case xmssmt-sha2_40/4_512:
        case xmssmt-sha2_40/8_512:



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RFC 8391                          XMSS                          May 2018


        case xmssmt-sha2_60/3_512:
        case xmssmt-sha2_60/6_512:
        case xmssmt-sha2_60/12_512:
        case xmssmt-shake_20/2_512:
        case xmssmt-shake_20/4_512:
        case xmssmt-shake_40/2_512:
        case xmssmt-shake_40/4_512:
        case xmssmt-shake_40/8_512:
        case xmssmt-shake_60/3_512:
        case xmssmt-shake_60/6_512:
        case xmssmt-shake_60/12_512:
          bytestring64 root_n64;

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

      /* XMSS^MT public key structure */

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

Acknowledgements

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





















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RFC 8391                          XMSS                          May 2018


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: ietf@gazdag.de


   Joost Rijneveld
   Radboud University
   Toernooiveld 212
   Nijmegen  6525 EC
   The Netherlands

   Email: ietf@joostrijneveld.nl


   Aziz Mohaisen
   University of Central Florida
   4000 Central Florida Blvd
   Orlando, FL  32816
   United States of America

   Phone: +1 407 823-1294
   Email: mohaisen@ieee.org





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