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Crypto Forum Research Group                                    D. McGrew
Internet-Draft                                                 M. Curcio
Intended status: Informational                             Cisco Systems
Expires: April 21, 2016                                 October 19, 2015


                         Hash-Based Signatures
                       draft-mcgrew-hash-sigs-03

Abstract

   This note describes a digital signature system based on cryptographic
   hash functions, following the seminal work in this area of Lamport,
   Diffie, Winternitz, and Merkle, as adapted by Leighton and Micali in
   1995.  It specifies a one-time signature scheme and a general
   signature scheme.  These systems provide asymmetric authentication
   without using large integer mathematics and can achieve a high
   security level.  They are suitable for compact implementations, are
   relatively simple to implement, and naturally resist side-channel
   attacks.  Unlike most other signature systems, hash-based signatures
   would still be secure even if it proves feasible for an attacker to
   build a quantum computer.

Status of This Memo

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   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on April 21, 2016.

Copyright Notice

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

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (http://trustee.ietf.org/license-info) in effect on the date of



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   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Conventions Used In This Document . . . . . . . . . . . .   4
   2.  Interface . . . . . . . . . . . . . . . . . . . . . . . . . .   4
   3.  Notation  . . . . . . . . . . . . . . . . . . . . . . . . . .   4
     3.1.  Data Types  . . . . . . . . . . . . . . . . . . . . . . .   4
       3.1.1.  Operators . . . . . . . . . . . . . . . . . . . . . .   5
       3.1.2.  Strings of w-bit elements . . . . . . . . . . . . . .   5
     3.2.  Security string . . . . . . . . . . . . . . . . . . . . .   6
     3.3.  Functions . . . . . . . . . . . . . . . . . . . . . . . .   8
   4.  LM-OTS One-Time Signatures  . . . . . . . . . . . . . . . . .   8
     4.1.  Parameters  . . . . . . . . . . . . . . . . . . . . . . .   8
     4.2.  Hashing Functions . . . . . . . . . . . . . . . . . . . .   9
     4.3.  Signature Methods . . . . . . . . . . . . . . . . . . . .   9
     4.4.  Private Key . . . . . . . . . . . . . . . . . . . . . . .  10
     4.5.  Public Key  . . . . . . . . . . . . . . . . . . . . . . .  10
     4.6.  Checksum  . . . . . . . . . . . . . . . . . . . . . . . .  11
     4.7.  Signature Generation  . . . . . . . . . . . . . . . . . .  11
     4.8.  Signature Verification  . . . . . . . . . . . . . . . . .  12
     4.9.  Notes . . . . . . . . . . . . . . . . . . . . . . . . . .  13
     4.10. Formats . . . . . . . . . . . . . . . . . . . . . . . . .  13
   5.  Leighton Micali Signatures  . . . . . . . . . . . . . . . . .  16
     5.1.  LMS Private Key . . . . . . . . . . . . . . . . . . . . .  16
     5.2.  LMS Public Key  . . . . . . . . . . . . . . . . . . . . .  17
     5.3.  LMS Signature . . . . . . . . . . . . . . . . . . . . . .  17
       5.3.1.  LMS Signature Generation  . . . . . . . . . . . . . .  18
     5.4.  LMS Signature Verification  . . . . . . . . . . . . . . .  18
     5.5.  LMS Formats . . . . . . . . . . . . . . . . . . . . . . .  19
   6.  Hierarchical signatures . . . . . . . . . . . . . . . . . . .  21
     6.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  21
     6.2.  Signature Generation  . . . . . . . . . . . . . . . . . .  21
     6.3.  Signature Verification  . . . . . . . . . . . . . . . . .  22
   7.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .  22
   8.  History . . . . . . . . . . . . . . . . . . . . . . . . . . .  23
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  23
   10. Security Considerations . . . . . . . . . . . . . . . . . . .  26
     10.1.  Stateful signature algorithm . . . . . . . . . . . . . .  26
     10.2.  Security of LM-OTS Checksum  . . . . . . . . . . . . . .  27
   11. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  28
   12. References  . . . . . . . . . . . . . . . . . . . . . . . . .  28



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     12.1.  Normative References . . . . . . . . . . . . . . . . . .  28
     12.2.  Informative References . . . . . . . . . . . . . . . . .  28
   Appendix A.  LM-OTS Parameter Options . . . . . . . . . . . . . .  29
   Appendix B.  An iterative algorithm for computing an LMS public
                key  . . . . . . . . . . . . . . . . . . . . . . . .  30
   Appendix C.  Example implementation . . . . . . . . . . . . . . .  31
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  42

1.  Introduction

   One-time signature systems, and general purpose signature systems
   built out of one-time signature systems, have been known since 1979
   [Merkle79], were well studied in the 1990s [USPTO5432852], and have
   benefited from renewed attention in the last decade.  The
   characteristics of these signature systems are small private and
   public keys and fast signature generation and verification, but large
   signatures and relatively slow key generation.  In recent years there
   has been interest in these systems because of their post-quantum
   security and their suitability for compact implementations.

   This note describes the Leighton and Micali adaptation [USPTO5432852]
   of the original Lamport-Diffie-Winternitz-Merkle one-time signature
   system [Merkle79] [C:Merkle87][C:Merkle89a][C:Merkle89b] and general
   signature system [Merkle79] with enough specificity to ensure
   interoperability between implementations.  An example implementation
   is given in an appendix.

   A signature system provides asymmetric message authentication.  The
   key generation algorithm produces a public/private key pair.  A
   message is signed by a private key, producing a signature, and a
   message/signature pair can be verified by a public key.  A One-Time
   Signature (OTS) system can be used to sign exactly one message
   securely, but cannot securely sign more than one.  An N-time
   signature system can be used to sign N or fewer messages securely.  A
   Merkle tree signature scheme is an N-time signature system that uses
   an OTS system as a component.  In this note we describe the Leighton-
   Micali Signature (LMS) system, which is a variant of the Merkle
   scheme.  We denote the one-time signature scheme that it incorporates
   as LM-OTS.

   This note is structured as follows.  Notation is introduced in
   Section 3.  The LM-OTS signature system is described in Section 4,
   and the LMS N-time signature system is described in Section 5.
   Sufficient detail is provided to ensure interoperability.  The IANA
   registry for these signature systems is described in Section 9.
   Security considerations are presented in Section 10.





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1.1.  Conventions Used In This Document

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

2.  Interface

   The LMS signing algorithm is stateful; once a particular value of the
   private key is used to sign one message, it MUST NOT be used to sign
   another.  To make this fact explicit in the interface, we use a
   functional programming approach, in which the key generation,
   signing, and verification algorithms do not have any side effects.
   The signing algorithm returns both a signature and a different
   private key value, which can be used to sign additional messages.

      The key generation algorithm takes as input an indication of the
      parameters for the signature system.  If it is successful, it
      returns both a private key and a public key.  Otherwise, it
      returns an indication of failure.

      The signing algorithm takes as input the message to be signed and
      the current value of the private key.  If successful, it returns a
      signature and the next value of the private key, if there is such
      a value.  After the private key of an N-time signature system has
      signed N messages, the signing algorithm returns the signature and
      an indication that there is no next value of the private key that
      can be used for signing.  If unsuccessful, it returns an
      indication of failure.

      The verification algorithm takes as input the public key, a
      message, and a signature, and returns an indication of whether or
      not the signature and message pair are valid.

   A message/signature pair are valid if the signature was returned by
   the signing algorithm upon input of the message and the private key
   corresponding to the public key; otherwise, the signature and message
   pair are not valid with probability very close to one.

3.  Notation

3.1.  Data Types

   Bytes and byte strings are the fundamental data types.  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 with a length



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   of three.  An array of byte strings is an ordered set, indexed
   starting at zero, in which all strings have the same length.

   Unsigned integers are converted into byte strings by representing
   them in network byte order.  To make the number of bytes in the
   representation explicit, we define the functions uint8str(X),
   uint16str(X), and uint32str(X), which return one, two, and four byte
   values, respectively.

3.1.1.  Operators

   When a and b are real numbers, 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 multiplied by b

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

      % : a % b denotes the remainder of the integer division of a by b

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

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

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

   If A and B are bytes, then A AND B denotes the bitwise logical and
   operation.

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

   If S and T are byte strings, then S || T denotes the concatenation of
   S and T.

   The i^th byte string in an array A is denoted as A[i].

3.1.2.  Strings of w-bit elements

   If S is a byte string, then byte(S, i) denotes its i^th byte, where
   byte(S, 0) is the leftmost byte.  In addition, bytes(S, i, j) denotes
   the range of bytes from the i^th to the j^th byte, inclusive.  For
   example, if S = 0x02040608, then byte(S, 0) is 0x02 and bytes(S, 1,
   2) is 0x0406.



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   A byte string can be considered to be a string of w-bit unsigned
   integers; the correspondence is defined by the function coef(S, i, w)
   as follows:

   If S is a string, i is a positive integer, and w is a member of the
   set { 1, 2, 4, 8 }, then coef(S, i, w) is the i^th, w-bit value, if S
   is interpreted as a sequence of w-bit values.  That is,

       coef(S, i, w) = (2^w - 1) AND
                       ( byte(S, floor(i * w / 8)) >>
                         (8 - (w * (i % (8 / w)) + w)) )

   For example, if S is the string 0x1234, then coef(S, 7, 1) is 0 and
   coef(S, 0, 4) is 1.


                      S (represented as bits)
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
         | 0| 0| 0| 1| 0| 0| 1| 0| 0| 0| 1| 1| 0| 1| 0| 0|
         +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
                                ^
                                |
                          coef(S, 7, 1)


                 S (represented as four-bit values)
         +-----------+-----------+-----------+-----------+
         |     1     |     2     |     3     |     4     |
         +-----------+-----------+-----------+-----------+
               ^
               |
         coef(S, 0, 4)

   The return value of coef is an unsigned integer.  If i is larger than
   the number of w-bit values in S, then coef(S, i, w) is undefined, and
   an attempt to compute that value should raise an error.

3.2.  Security string

   To improve security against attacks that amortize their effort
   against multiple invocations of the hash function H, Leighton and
   Micali introduce a "security string" that is distinct for each
   invocation of H.  The following fields can appear in a security
   string:

      I - an identifier for the private key.  This value is 31 bytes
      long, and it MUST be distinct from all other such identifiers.  It
      SHOULD be chosen uniformly at random, or via a pseudorandom



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      process, in order to ensure that it will be distinct with
      probability close to one, but it MAY be a structured identifier.

      D - a domain separation parameter, which is a single byte that
      takes on different values in the different algorithms in which H
      is invoked.  D takes on the following values:

         D_ITER = 0x00 in the iterations of the LM-OTS algorithms

         D_PBLC = 0x01 when computing the hash of all of the iterates in
         the LM-OTS algorithm

         D_MESG = 0x02 when computing the hash of the message in the LM-
         OTS algorithms

         D_LEAF = 0x03 when computing the hash of the leaf of an LMS
         tree

         D_INTR = 0x04 when computing the hash of an interior node of an
         LMS tree

      C - an n-byte randomizer that is included with the message
      whenever it is being hashed to improve security.  C MUST be chosen
      uniformly at random, or via a pseudorandom process.

      i - in the LM-OTS one-time signature scheme, i is the index of the
      private key element upon which H is being applied.  It is
      represented as a 16-bit (two byte) unsigned integer in network
      byte order.

      j - in the LM-OTS one-time signature scheme, j is the iteration
      number used when the private key element is being iteratively
      hashed.  It is represented as an 8-bit (one byte) unsigned
      integer.

      q - in the LM-OTS one-time signature scheme, q is a
      diversification string provided as input.  In the LMS N-time
      signature scheme, a distinct value of q is provided for each
      distinct LM-OTS public/private keypair.  It is represented as a
      four byte string.

      r - in the LMS N-time signature scheme, the node number r
      associated with a particular node of the hash tree is used as an
      input to the hash used to compute that node.  This value is
      represented as a 32-bit (four byte) unsigned integer in network
      byte order.





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3.3.  Functions

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

      ceil(r) : returns the smallest integer larger than r

      floor(r) : returns the largest integer smaller than r

      lg(r) : returns the base-2 logarithm of r

4.  LM-OTS One-Time Signatures

   This section defines LM-OTS signatures.  The signature is used to
   validate the authenticity of a message by associating a secret
   private key with a shared public key.  These are one-time signatures;
   each private key MUST be used only one time to sign any given
   message.

   As part of the signing process, a digest of the original message is
   computed using the cryptographic hash function H (see Section 4.2),
   and the resulting digest is signed.

   In order to facilitate its use in an N-time signature system, the LM-
   OTS key generation, signing, and verification algorithms all take as
   input a diversification parameter q.  When the LM-OTS signature
   system is used outside of an N-time signature system, this value
   SHOULD be set to the all-zero value.

4.1.  Parameters

   The signature system uses the parameters n and w, which are both
   positive integers.  The algorithm description also makes use of the
   internal parameters p and ls, which are dependent on n and w.  These
   parameters are summarized as follows:

      n : the number of bytes of the output of the hash function

      w : the Winternitz parameter; it is a member of the set
      { 1, 2, 4, 8 }

      p : the number of n-byte string elements that make up the LM-OTS
      signature

      ls : the number of left-shift bits used in the checksum function
      Cksm (defined in Section 4.6).





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   The value of n is determined by the functions selected for use as
   part of the LM-OTS algorithm; the choice of this value has a strong
   effect on the security of the system.  The parameter w can be chosen
   to set the number of bytes in the signature; it has little effect on
   security.  Note however, that there is a larger computational cost to
   generate and verify a shorter signature.  The values of p and ls are
   dependent on the choices of the parameters n and w, as described in
   Appendix A.  A table illustrating various combinations of n, w, p,
   and ls is provided in Table 1.

4.2.  Hashing Functions

   The LM-OTS algorithm uses a hash function H that accepts byte strings
   of any length, and returns an n-byte string.

4.3.  Signature Methods

   To fully describe a LM-OTS signature method, the parameters n and w,
   as well as the function H, MUST be specified.  This section defines
   several LM-OTS signature systems, each of which is identified by a
   name.  Values for p and ls are provided as a convenience.

          +---------------------+-----------+----+---+-----+----+
          | Name                | H         | n  | w | p   | ls |
          +---------------------+-----------+----+---+-----+----+
          | LMOTS_SHA256_N32_W1 | SHA256    | 32 | 1 | 265 | 7  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N32_W2 | SHA256    | 32 | 2 | 133 | 6  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N32_W4 | SHA256    | 32 | 4 | 67  | 4  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N32_W8 | SHA256    | 32 | 8 | 34  | 0  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N16_W1 | SHA256-16 | 16 | 1 | 68  | 8  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N16_W2 | SHA256-16 | 16 | 2 | 68  | 8  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N16_W4 | SHA256-16 | 16 | 4 | 35  | 4  |
          |                     |           |    |   |     |    |
          | LMOTS_SHA256_N16_W8 | SHA256-16 | 16 | 8 | 18  | 0  |
          +---------------------+-----------+----+---+-----+----+

                                  Table 1

   Here SHA256 denotes the NIST standard hash function [FIPS180].
   SHA256-16 denotes the SHA256 hash function with its final output
   truncated to return the leftmost 16 bytes.




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4.4.  Private Key

   The LM-OTS private key consists of an array of size p containing
   n-byte strings.  Let x denote the private key.  This private key must
   be used to sign one and only one message.  It must therefore be
   unique from all other private keys.  The following algorithm shows
   pseudocode for generating x.

   Algorithm 0: Generating a Private Key

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

   An implementation MAY use a pseudorandom method to compute x[i], as
   suggested in [Merkle79], page 46.  The details of the pseudorandom
   method do not affect interoperability, but the cryptographic strength
   MUST match that of the LM-OTS algorithm.

4.5.  Public Key

   The LM-OTS public key is generated from the private key by
   iteratively applying the function H to each individual element of x,
   for 2^w - 1 iterations, then hashing all of the resulting values.

   Each public/private key pair is associated with a single identifier
   I.  This string MUST be 31 bytes long, and be generated as described
   in Section 3.2.

   The diversification parameter q is an input to the algorithm, as
   described in Section 3.2.

   The following algorithm shows pseudocode for generating the public
   key, where the array x is the private key.

   Algorithm 1: Generating a Public Key From a Private Key

    for ( i = 0; i < p; i = i + 1 ) {
      tmp = x[i]
      for ( j = 0; j < 2^w - 1; j = j + 1 ) {
         tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER)
      }
      y[i] = tmp
    }
    return H(I || q || y[0] || y[1] || ... || y[p-1] || D_PBLC)





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   The public key is the string consisting of a four-byte enumeration
   that identifies the parameters in use, followed by the value returned
   by Algorithm 1.  Section 4.10 specifies the enumeration and more
   formally defines the format.

4.6.  Checksum

   A checksum is used to ensure that any forgery attempt that
   manipulates the elements of an existing signature will be detected.
   The security property that it provides is detailed in Section 10.
   The checksum function Cksm is defined as follows, where S denotes the
   byte string that is input to that function, and the value sum is a
   16-bit unsigned integer:

   Algorithm 2: Checksum Calculation

     sum = 0
     for ( i = 0; i < u; i = i + 1 ) {
       sum = sum + (2^w - 1) - coef(S, i, w)
     }
     return (sum << ls)

   Because of the left-shift operation, the rightmost bits of the result
   of Cksm will often be zeros.  Due to the value of p, these bits will
   not be used during signature generation or verification.

4.7.  Signature Generation

   The LM-OTS signature of a message is generated by first appending the
   randomizer C, the identifier string I, and the diversification string
   q to the message, then using H to compute the hash of the resulting
   string, concatenating the checksum of the hash to the hash itself,
   then considering the resulting value as a sequence of w-bit values,
   and using each of the the w-bit values to determine the number of
   times to apply the function H to the corresponding element of the
   private key.  The outputs of the function H are concatenated together
   and returned as the signature.  The pseudocode for this procedure is
   shown below.

   The identifier string I and diversification string q are the same as
   in Section 4.5.










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   Algorithm 3: Generating a Signature From a Private Key and a Message

   set C to a uniformly random n-byte string
   set type to the appropriate ots_algorithm_type
   Q = H(message || C || I || q || D_MESG)
   for ( i = 0; i < p; i = i + 1 ) {
     a = coef(Q || Cksm(Q), i, w)
     tmp = x[i]
     for ( j = 0; j < a; j = j + 1 ) {
        tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER)
     }
     y[i] = tmp
   }
   return type || C || I || 0x00 || q || (y[0] || y[1] || ... || y[p-1])

   Note that this algorithm results in a signature whose elements are
   intermediate values of the elements computed by the public key
   algorithm in Section 4.5.

   The signature is the string consisting of a four-byte enumeration
   that identifies the parameters in use, followed by the value returned
   by Algorithm 3.  Section 4.10 specifies the enumeration and more
   formally defines the format.

4.8.  Signature Verification

   In order to verify a message with its signature (an array of n-byte
   strings, denoted as y), the receiver must "complete" the series of
   applications of H using the w-bit values of the message hash and its
   checksum.  This computation should result in a value that matches the
   provided public key.




















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   Algorithm 4: Verifying a Signature and Message Using a Public Key

    parse C, I, q, and y from the signature as follows:
       type = first 4 bytes
       C = next n bytes
       I = next 31 bytes
       NULL = next byte; this padding value is discarded
       q = next four bytes
       y[0] = next n bytes
       y[1] = next n bytes
       ...
       y[p-1] = next n bytes
    Q = H(message || C || I || q || D_MESG)
    for ( i = 0; i < p; i = i + 1 ) {
      a = (2^w - 1) - coef(Q || Cksm(Q), i, w)
      tmp = y[i]
      for ( j = a+1; j < 2^w - 1; j = j + 1 ) {
         tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER)
      }
      z[i] = tmp
    }
    candidate = H(z[0] || z[1] || ... || z[p-1] || I || q || D_PBLC)
    if (candidate = public_key)
      return 1  // message/signature pair is valid
    else
      return 0  // message/signature pair is invalid

4.9.  Notes

   A future version of this specification may define a method for
   computing the signature of a very short message in which the hash is
   not applied to the message during the signature computation.  That
   would allow the signatures to have reduced size.

4.10.  Formats

   The signature and public key formats are formally defined using the
   External Data Representation (XDR) [RFC4506] in order to provide an
   unambiguous, machine readable definition.  For clarity, we also
   include a private key format as well, though consistency is not
   needed for interoperability and an implementation MAY use any private
   key format.  Though XDR is used, these formats are simple and easy to
   parse without any special tools.  The definitions are as follows:


   /*
    * ots_algorithm_type identifies a particular signature algorithm
    */



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   enum ots_algorithm_type {
     ots_reserved        = 0,
     lmots_sha256_m16_w1  = 0x00000001,
     lmots_sha256_m16_w2  = 0x00000002,
     lmots_sha256_m16_w4  = 0x00000003,
     lmots_sha256_m16_w8  = 0x00000004,
     lmots_sha256_n32_w1  = 0x00000005,
     lmots_sha256_n32_w2  = 0x00000006,
     lmots_sha256_n32_w4  = 0x00000007,
     lmots_sha256_n32_w8  = 0x00000008
   };

   /*
    * byte strings (for n=16 and n=32)
    */
   typedef opaque bytestring16[16];
   typedef opaque bytestring32[32];

   union ots_signature switch (ots_algorithm_type type) {
    case lmots_sha256_n16_w1:
         bytestring16 y_n16_p265[265];
    case lmots_sha256_n16_w2:
         bytestring16 y_n16_p133[133];
    case lmots_sha256_n16_w4:
         bytestring16 y_n16_p67[67];
    case lmots_sha256_n16_w8:
         bytestring16 y_n16_p34[34];
    case lmots_sha256_n32_w1:
         bytestring32 y_n32_p265[265];
    case lmots_sha256_n32_w2:
         bytestring32 y_m3_p133[133];
    case lmots_sha256_n32_w4:
         bytestring32 y_n32_y_p67[67];
    case lmots_sha256_n32_w8:
         bytestring32 y_n32_p34[34];
    default:
      void;   /* error condition */
   };

   union ots_public_key switch (ots_algorithm_type type) {
    case lmots_sha256_n16_w1:
    case lmots_sha256_n16_w2:
    case lmots_sha256_n16_w4:
    case lmots_sha256_n16_w8:
    case lmots_sha256_n32_w1:
    case lmots_sha256_n32_w2:
    case lmots_sha256_n32_w4:
    case lmots_sha256_n32_w8:



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         bytestring32 y32;
    default:
      void;   /* error condition */
    };

   union ots_private_key switch (ots_algorithm_type type) {
    case lmots_sha256_m16_w1:
    case lmots_sha256_m16_w2:
    case lmots_sha256_m16_w4:
    case lmots_sha256_m16_w8:
         bytestring20 x20;
    case lmots_sha256_n32_w1:
    case lmots_sha256_n32_w2:
    case lmots_sha256_n32_w4:
    case lmots_sha256_n32_w8:
         bytestring32 x32;
    default:
      void;   /* error condition */
    };

   Though the data formats are formally defined by XDR, we include
   diagrams as well as a convenience to the reader.  An example of the
   format of an lmots_signature is illustrated below, for
   lmots_sha256_n32_w1.  An ots_signature consists of a 32-bit unsigned
   integer that indicates the ots_algorithm_type, followed by other
   data, whose format depends only on the ots_algorithm_type.  For LM-
   OTS, that data is an array of equal-length byte strings.  The number
   of bytes in each byte string, and the number of elements in the
   array, are determined by the ots_algorithm_type field.  In the case
   of lmots_sha256_n32_w1, the array has 265 elements, each of which is
   a 32-byte string.  The XDR array y_n32_p265 denotes the array y as
   used in the algorithm descriptions above, using the parameters of
   n=32 and p=265 for lmots_sha256_n32_w1.

   A verifier MUST check the ots_algorithm_type field, and a
   verification operation on a signature with an unknown
   lmots_algorithm_type MUST return FAIL.














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            +---------------------------------+
            |       ots_algorithm_type        |
            +---------------------------------+
            |                                 |
            |         y_n32_p265[0]           |
            |                                 |
            +---------------------------------+
            |                                 |
            |         y_n32_p265[1]           |
            |                                 |
            +---------------------------------+
            |                                 |
            ~             ....                ~
            |                                 |
            +---------------------------------+
            |                                 |
            |        y_n32_p265[264]          |
            |                                 |
            +---------------------------------+

5.  Leighton Micali Signatures

   The Leighton Micali Signature (LMS) method can sign a potentially
   large but fixed number of messages.  An LMS system uses two
   cryptographic components: a one-time signature method and a hash
   function.  Each LMS public/private key pair is associated with a
   perfect binary tree, each node of which contains an n-byte value.
   Each leaf of the tree contains the value of the public key of an LM-
   OTS public/private key pair.  The value contained by the root of the
   tree is the LMS public key.  Each interior node is computed by
   applying the hash function to the concatenation of the values of its
   children nodes.

   An LMS system has the following parameters:

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

      n : the number of bytes associated with each node.

   There are 2^h leaves in the tree.

5.1.  LMS Private Key

   An LMS private key consists of 2^h one-time signature private keys
   and the leaf number of the next LM-OTS private key that has not yet
   been used.  The leaf number is initialized to zero when the LMS
   private key is created.




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   An LMS private key MAY be generated pseudorandomly from a secret
   value, in which case the secret value MUST be at least n bytes long,
   be uniformly random, and MUST NOT be used for any other purpose than
   the generation of the LMS private key.  The details of how this
   process is done do not affect interoperability; that is, the public
   key verification operation is independent of these details.

5.2.  LMS Public Key

   An LMS public key is defined as follows, where we denote the public
   key associated with the i^th LM-OTS private key as OTS_PUBKEY[i],
   with i ranging from 0 to (2^h)-1.  Each instance of an LMS public/
   private key pair is associated with a perfect binary tree, and the
   nodes of that tree are indexed from 1 to 2^(h+1)-1.  Each node is
   associated with an n-byte string, and the string for the rth node is
   denoted as T[r] and is defined as

 T[r] = / H(OTS_PUBKEY[r-2^h]  || I || uint32str(r) || D_LEAF) if r >= 2^h
        \ H(T[2*r] || T[2*r+1] || I || uint32str(r) || D_INTR) otherwise.

   The LMS public key is the string consisting of a four-byte
   enumeration that identifies the parameters in use, followed by the
   string T[1].  Section 5.5 specifies the enumeration and more formally
   defines the format.  The value T[1] can be computed via recursive
   application of the above equation, or by any equivalent method.  An
   iterative procedure is outlined in Appendix B.

5.3.  LMS Signature

   An LMS signature consists of

      a typecode indicating the particular LMS algorithm,

      an LM-OTS signature, and

      an array of values that is associated with the path through the
      tree from the leaf associated with the LM-OTS signature to the
      root.

   The array of values contains the siblings of the nodes on the path
   from the leaf to the root but does not contain the nodes on the path
   itself.  The array for a tree with height h will have h values.  The
   first value is the sibling of the leaf, the next value is the sibling
   of the parent of the leaf, and so on up the path to the root.







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5.3.1.  LMS Signature Generation

   To compute the LMS signature of a message with an LMS private key,
   the signer first computes the LM-OTS signature of the message using
   the leaf number of the next unused LM-OTS private key.  Before
   releasing the signature, the leaf number in the LMS private key MUST
   be incremented to prevent the LM-OTS private key from being used
   again.  The node number in the signature is set to the leaf number of
   the LMS private key that was used in the signature.  Then the
   signature and the LMS private key are returned.

   The array of node values in the signature MAY be computed in any way.
   There are many potential time/storage tradeoffs that can be applied.
   The fastest alternative is to store all of the nodes of the tree and
   set the array in the signature by copying them.  The least storage
   intensive alternative is to recompute all of the nodes for each
   signature.  Note that the details of this procedure are not important
   for interoperability; it is not necessary to know any of these
   details in order to perform the signature verification operation.
   The internal nodes of the tree need not be kept secret, and thus a
   node-caching scheme that stores only internal nodes can sidestep the
   need for strong protections.

   One useful time/storage tradeoff is described in Column 19 of
   [USPTO5432852].

5.4.  LMS Signature Verification

   An LMS signature is verified by first using the LM-OTS signature
   verification algorithm to compute the LM-OTS public key from the LM-
   OTS signature and the message.  The value of that public key is then
   assigned to the associated leaf of the LMS tree, then the root of the
   tree is computed from the leaf value and the node array (path[]) as
   described below.  If the root value matches the public key, then the
   signature is valid; otherwise, the signature fails.
















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   Algorithm 6: LMS Signature Verification

  identify the height h of the tree from the algorithm type
  determine the leaf number the LM-OTS q value to an integer
  n = node number = 2^h + leaf_number
  tmp = candidate public key computed from LM-OTS signature and message
  tmp = H(tmp || I || uint32str(node_num) || D_LEAF)
  i = 0
  while (node_num > 1) {
      if (node_num is odd):
          tmp = H(path[i] || tmp || I || uint32str(node_num/2) || D_INTR)
      else:
          tmp = H(tmp || path[i] || I || uint32str(node_num/2) || D_INTR)
      node_num = node_num/2
      i = i + 1
  if (tmp == lms_public_key)
    return 1  // message/signature pair is valid
  else
    return 0  // message/signature pair is invalid

   Upon completion, v contains the value of the root of the LMS tree for
   comparison.

   The verifier MAY cache interior node values that have been computed
   during a successful signature verification for use in subsequent
   signature verifications.  However, any implementation that does so
   MUST make sure any nodes that are cached during a signature
   verification process are deleted if that process does not result in a
   successful match between the root of the tree and the LMS public key.

5.5.  LMS Formats

   LMS signatures and public keys are defined using XDR syntax as
   follows:

   enum lms_algorithm_type {
     lms_reserved       = 0x00000000,
     lms_sha256_n32_h20 = 0x00000001,
     lms_sha256_n32_h10 = 0x00000002,
     lms_sha256_n32_h5  = 0x00000003
     lms_sha256_n16_h20 = 0x00000004,
     lms_sha256_n16_h10 = 0x00000005,
     lms_sha256_n16_h5  = 0x00000006
   };

   union lms_path switch (lms_algorithm_type type) {
    case lms_sha256_n32_h20:
      bytestring32 path_n32_h20[20];



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    case lms_sha256_n32_h10:
      bytestring32 path_n32_h10[10];
    case lms_sha256_n32_h5:
      bytestring32 path_n32_h5[5];
    case lms_sha256_n16_h20:
      bytestring32 path_n16_h20[20];
    case lms_sha256_n16_h10:
      bytestring32 path_n16_h10[10];
    case lms_sha256_n16_h5:
      bytestring32 path_n16_h5[5];
    default:
      void;     /* error condition */
   };

   struct lms_signature {
     ots_signature ots_sig;
     lms_path nodes;
   };

   struct lms_public_key_n16 {
     ots_algorithm_type ots_alg_type;
     opaque value[16];                    /* public key */
   };

   struct lms_public_key_n64 {
     ots_algorithm_type ots_alg_type;
     opaque value[64];                    /* public key */
     opaque I[31];                        /* identity   */
   };

   union lms_public_key switch (lms_algorithm_type type) {
    case lms_sha256_n32_h20:
    case lms_sha256_n32_h10:
    case lms_sha256_n32_h5:
         lms_public_key_n32 z_n32;
    case lms_sha256_n16_h20:
    case lms_sha256_n16_h10:
    case lms_sha256_n16_h5:
         lms_public_key_n16 z_n16;
     default:
      void;     /* error condition */
   };









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6.  Hierarchical signatures

   In scenarios where it is necessary to minimize the time taken by the
   public key generation process, a hierarchical N-time signature scheme
   can be used.  Leighton and Micali describe a scheme in which an LMS
   public key is used to sign a second LMS public key, which is then
   distributed along with the signatures generated with the second
   public key [USPTO5432852].  This hierarchical scheme, which we
   describe in this section, uses an LMS scheme as a component, and it
   has two levels.  Each level is associated with an LMS public key,
   private key, and signature.  The following notation is used, where i
   is an integer between 1 and 2 inclusive:

      prv[i] is the private key of the ith level,

      pub[i] is the public key of the ith level, and

      sig[i] is the signature of the ith level.

   In this section, we say that an N-time private key is exhausted when
   it has signed all N messages, and thus it can no longer be used for
   signing.

6.1.  Key Generation

   To generate an HLMS private and public key pair, new LMS private and
   public keys are generated for prv[i] and pub[i] for i=1,2.  These key
   pairs MUST be generated independently.

   The public key of the HLMS scheme is pub[1], the public key of the
   first level.  The HLMS private key consists of prv[1] and prv[2].
   The values pub[1] and prv[1] do not change, though the values of
   pub[2] and prv[2] are dynamic, and are changed by the signature
   generation algorithm.

6.2.  Signature Generation

   To sign a message using the private key prv, the following steps are
   performed:

      The message is signed with prv[2], and the value sig[2] is set to
      that result.

      The value of the HLMS signature is set to type || pub[2] ||
      sig[1] || sig[2], where type is the typecode for the particular
      HLMS algorithm.





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      If prv[2] is exhausted, then a new LMS public and private key pair
      is generated, and pub[2] and prv[2] are set to those values.
      pub[2] is signed with prv[1], and sig[1] is set to the resulting
      value.

6.3.  Signature Verification

   To verify a signature sig and message using the public key pub, the
   following steps are performed:

      The signature sig is parsed into its components type, pub[2],
      sig[1] and sig[2].

      The signature sig[2] and message is verified using the public key
      pub[2].  If verification fails, then an indication of failure is
      returned.  Otherwise, processing continues as follows.

      The signature sig[1] of the "message" pub[2] is verified using the
      public key pub.  If verification fails, then an indication of
      failure is returned.  Otherwise, an indication of success is
      returned.

7.  Rationale

   The goal of this note is to describe the LM-OTS and LMS algorithms
   following the original references and present the modern security
   analysis of those algorithms.  Other signature methods are out of
   scope and may be interesting follow-on work.

   We adopt the techniques described by Leighton and Micali to mitigate
   attacks that amortize their work over multiple invocations of the
   hash function.

   The values taken by the identifier I across different LMS public/
   private key pairs are required to be distinct in order to improve
   security.  That distinctness ensures the uniqueness of the inputs to
   H across all of those public/private key pair instances, which is
   important for provable security in the random oracle model.  The
   length of I is set at 31 bytes so that randomly chosen values of I
   will be distinct with probability at least 1 - 1/2^128 as long as
   there are 2^60 or fewer instances of LMS public/private key pairs.

   The sizes of the parameters in the security string are such that, for
   n=16, the LM-OTS iterates a 55-byte value (that is, the string that
   is input to H() during the iteration over j during signature
   generation and verification is 55 bytes long).  Thus, when SHA-256 is
   used as the function H, only a single invocation of its compression
   function is needed.



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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.

   The Checksum Section 4.6 is calculated using a non-negative integer
   "sum", whose width was chosen to be an integer number of w-bit fields
   such that it is capable of holding the difference of the total
   possible number of applications of the function H as defined in the
   signing algorithm of Section 4.7 and the total actual number.  In the
   worst case (i.e. the actual number of times H is iteratively applied
   is 0), the sum is (2^w - 1) * ceil(8*n/w).  Thus for the purposes of
   this document, which describes signature methods based on H = SHA256
   (n = 32 bytes) and w = { 1, 2, 4, 8 }, the sum variable is a 16-bit
   non-negative integer for all combinations of n and w.  The
   calculation uses the parameter ls defined in Section 4.1 and
   calculated in Appendix A, which indicates the number of bits used in
   the left-shift operation.

8.  History

   This is the third version version of this draft.  It has the
   following changes:

      It adopts the "security string" approach of Leighton and Micali
      [USPTO5432852] in order to improve security.

      It adopts Leighton and Micali's idea of hashing a randomizer
      string (C, as defined in Section 3.2) with the message, so that
      finding an arbitrary collision in H will not lead to a forgery.

      It defines a multi-level signature scheme, again following that
      described by Leighton and Micali.

      It eliminates the function F and its iterates; the function H is
      used in its stead.  The adoption of the security string makes this
      simplification possible.

      It fixes the branching number at two for simplicity.

   This section is to be removed by the RFC editor upon publication.

9.  IANA Considerations

   The Internet Assigned Numbers Authority (IANA) is requested to create
   two registries: one for OTS signatures, which includes all of the LM-
   OTS signatures as defined in Section 3, and one for Leighton-Micali



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   Signatures, as defined in Section 4.  Additions to these registries
   require that a specification be documented in an RFC or another
   permanent and readily available reference in sufficient detail that
   interoperability between independent implementations is possible.
   Each entry in the registry contains the following elements:

      a short name, such as "LMS_SHA256_n32_h10",

      a positive number, and

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

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

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

   The LM-OTS registry is as follows.






















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         +----------------------+-----------+--------------------+
         | Name                 | Reference | Numeric Identifier |
         +----------------------+-----------+--------------------+
         | LMOTS_SHA256_N16_W1  | Section 4 |     0x00000001     |
         |                      |           |                    |
         | LMOTS_SHA256_N16_W2  | Section 4 |     0x00000002     |
         |                      |           |                    |
         | LMOTS_SHA256_N16_W4  | Section 4 |     0x00000003     |
         |                      |           |                    |
         | LMOTS_SHA256_N16_W8  | Section 4 |     0x00000004     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W1  | Section 4 |     0x00000005     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W2  | Section 4 |     0x00000006     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W4  | Section 4 |     0x00000007     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W8  | Section 4 |     0x00000008     |
         +----------------------+-----------+--------------------+

                                  Table 2

   The LMS registry is as follows.

          +--------------------+-----------+--------------------+
          | Name               | Reference | Numeric Identifier |
          +--------------------+-----------+--------------------+
          | LMS_SHA256_N32_H20 | Section 5 |     0x00000001     |
          |                    |           |                    |
          | LMS_SHA256_N32_H10 | Section 5 |     0x00000002     |
          |                    |           |                    |
          | LMS_SHA256_N32_H5  | Section 5 |     0x00000003     |
          |                    |           |                    |
          | LMS_SHA256_N16_H20 | Section 5 |     0x00000004     |
          |                    |           |                    |
          | LMS_SHA256_N16_H10 | Section 5 |     0x00000005     |
          |                    |           |                    |
          | LMS_SHA256_N16_H5  | Section 5 |     0x00000006     |
          +--------------------+-----------+--------------------+

                                  Table 3

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







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10.  Security Considerations

   The security goal of a signature system is to prevent forgeries.  A
   successful forgery occurs when an attacker who does not know the
   private key associated with a public key can find a message and
   signature that are valid with that public key (that is, the Signature
   Verification algorithm applied to that signature and message and
   public key will return "valid").  Such an attacker, in the strongest
   case, may have the ability to forge valid signatures for an arbitrary
   number of other messages.

   LM-OTS and LMS are provably secure in the random oracle model, as
   shown by Katz [Katz15].  From Theorem 8 of that reference:

      For any adversary attacking arbitrarily many instances of the one-
      time signature scheme, and making at most q hash queries, the
      probability with which the adversary is able to forge a signature
      with respect to any of the instances is at most q2^(1-8n).

   Here n is the number of bytes in the output of the hash function (as
   defined in Section 4.1).  Thus, the security of the algorithms
   defined in this note can be roughly described as follows.  For a
   security level of roughly 128 bits, assuming that there are no
   quantum computers, use n=16 by selecting an algorithm identifier with
   N16 in its name.  For a security level of roughly 128 bits, assuming
   that there are quantum computers that can compute the input to an
   arbitrary function with computational cost equivalent to the square
   root of the size of the domain of that function [Grover96], use n=32
   by selecting an algorithm identifier with N32 in its name.

10.1.  Stateful signature algorithm

   The LMS signature system, like all N-time signature systems, requires
   that the signer maintain state across different invocations of the
   signing algorithm, to ensure that none of the component one-time
   signature systems are used more than once.  This section calls out
   some important practical considerations around this statefulness.

   In a typical computing environment, a private key will be stored in
   non-volatile media such as on a hard drive.  Before it is used to
   sign a message, it will be read into an application's Random Access
   Memory (RAM).  After a signature is generated, the value of the
   private key will need to be updated by writing the new value of the
   private key into non-volatile storage.  It is essential for security
   that the application ensure that this value is actually written into
   that storage, yet there may be one or more memory caches between it
   and the application.  Memory caching is commonly done in the file
   system, and in a physical memory unit on the hard disk that is



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   dedicated to that purpose.  To ensure that the updated value is
   written to physical media, the application may need to take several
   special steps.  In a POSIX environment, for instance,the O_SYNC flag
   (for the open() system call) will cause invocations of the write()
   system call to block the calling process until the data has been to
   the underlying hardware.  However, if that hardware has its own
   memory cache, it must be separately dealt with using an operating
   system or device specific tool such as hdparm to flush the on-drive
   cache, or turn off write caching for that drive.  Because these
   details vary across different operating systems and devices, this
   note does not attempt to provide complete guidance; instead, we call
   the implementer's attention to these issues.

   When hierarchical signatures are used, an easy way to minimize the
   private key synchronization issues is to have the private key for the
   second level resident in RAM only, and never write that value into
   non-volatile memory.  A new second level public/private key pair will
   be generated whenever the application (re)starts; thus, failures such
   as a power outage or application crash are automatically
   accommodated.  Implementations SHOULD use this approach wherever
   possible.

10.2.  Security of LM-OTS Checksum

   To show the security of LM-OTS checksum, we consider the signature y
   of a message with a private key x and let h = H(message) and
   c = Cksm(H(message)) (see Section 4.7).  To attempt a forgery, an
   attacker may try to change the values of h and c.  Let h' and c'
   denote the values used in the forgery attempt.  If for some integer j
   in the range 0 to (u-1), inclusive,

      a' = coef(h', j, w),

      a = coef(h, j, w), and

      a' > a

   then the attacker can compute F^a'(x[j]) from F^a(x[j]) = y[j] by
   iteratively applying function F to the j^th term of the signature an
   additional (a' - a) times.  However, as a result of the increased
   number of hashing iterations, the checksum value c' will decrease
   from its original value of c.  Thus a valid signature's checksum will
   have, for some number k in the range u to (p-1), inclusive,

      b' = coef(c', k, w),

      b = coef(c, k, w), and




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      b' < b

   Due to the one-way property of F, the attacker cannot easily compute
   F^b'(x[k]) from F^b(x[k]) = y[k].

11.  Acknowledgements

   Thanks are due to Chirag Shroff, Andreas Hulsing, Burt Kaliski, Eric
   Osterweil, Ahmed Kosba, Russ Housley, and Scott Fluhrer for
   constructive suggestions and valuable detailed review.  We esepcially
   acknowledge Jerry Solinas, Laurie Law, and Kevin Igoe, who pointed
   out the security benefits of the approach of Leighton and Micali
   [USPTO5432852] and Jonathan Katz, who gave us security guidance.

12.  References

12.1.  Normative References

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

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

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

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

   [USPTO5432852]
              Leighton, T. and S. Micali, "Large provably fast and
              secure digital signature schemes from secure hash
              functions", U.S. Patent 5,432,852, July 1995.

12.2.  Informative References

   [C:Merkle87]
              Merkle, R., "A Digital Signature Based on a Conventional
              Encryption Function", Lecture Notes in Computer
              Science crypto87vol, 1988.





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   [C:Merkle89a]
              Merkle, R., "A Certified Digital Signature", Lecture Notes
              in Computer Science crypto89vol, 1990.

   [C:Merkle89b]
              Merkle, R., "One Way Hash Functions and DES", Lecture
              Notes in Computer Science crypto89vol, 1990.

   [Grover96]
              Grover, L., "A fast quantum mechanical algorithm for
              database search", 28th ACM Symposium on the Theory of
              Computing p. 212, 1996.

   [Katz15]   Katz, J., "Analysis of a proposed hash-based signature
              standard", Contribution to IRTF
              http://www.cs.umd.edu/~jkatz/papers/HashBasedSigs.pdf,
              2015.

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

Appendix A.  LM-OTS Parameter Options

   A table illustrating various combinations of n and w with the
   associated values of u, v, ls, and p is provided in Table 4.

   The parameters u, v, ls, and p are computed as follows:

     u = ceil(8*n/w)
     v = ceil((floor(lg((2^w - 1) * u)) + 1) / w)
     ls = (number of bits in sum) - (v * w)
     p = u + v

   Here u and v represent the number of w-bit fields required to contain
   the hash of the message and the checksum byte strings, respectively.
   The "number of bits in sum" is defined according to Section 4.6.  And
   as the value of p is the number of w-bit elements of
   ( H(message) || Cksm(H(message)) ), it is also equivalently the
   number of byte strings that form the private key and the number of
   byte strings in the signature.









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   +---------+------------+-----------+-----------+-------+------------+
   |   Hash  | Winternitz |   w-bit   |   w-bit   |  Left |   Total    |
   |  Length | Parameter  |  Elements |  Elements | Shift | Number of  |
   |    in   |    (w)     |  in Hash  |     in    |  (ls) |   w-bit    |
   |  Bytes  |            |    (u)    |  Checksum |       |  Elements  |
   |   (n)   |            |           |    (v)    |       |    (p)     |
   +---------+------------+-----------+-----------+-------+------------+
   |    16   |     1      |    128    |     8     |   8   |    137     |
   |         |            |           |           |       |            |
   |    16   |     2      |     64    |     4     |   8   |     68     |
   |         |            |           |           |       |            |
   |    16   |     4      |     32    |     3     |   4   |     35     |
   |         |            |           |           |       |            |
   |    16   |     8      |     16    |     2     |   0   |     18     |
   |         |            |           |           |       |            |
   |    32   |     1      |    256    |     9     |   7   |    265     |
   |         |            |           |           |       |            |
   |    32   |     2      |    128    |     5     |   6   |    133     |
   |         |            |           |           |       |            |
   |    32   |     4      |     64    |     3     |   4   |     67     |
   |         |            |           |           |       |            |
   |    32   |     8      |     32    |     2     |   0   |     34     |
   +---------+------------+-----------+-----------+-------+------------+

                                  Table 4

Appendix B.  An iterative algorithm for computing an LMS public key

   The LMS public key can be computed using the following algorithm or
   any equivalent method.  The algorithm uses a stack of hashes for data
   and a separate stack of integers to keep track of the level of the
   tree.  It also makes use of a hash function with the typical
   init/update/final interface to hash functions; the result of the
   invocations hash_init(), hash_update(N[1]), hash_update(N[2]), ... ,
   hash_update(N[n]), v = hash_final(), in that order, is identical to
   that of the invocation of H(N[1] || N[2] || ... || N[n]).















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   Generating an LMS Public Key From an LMS Private Key

  for ( i = 0; i < num_lmots_keys; i = i + 2 ) {
    level = 0;
    for ( j = 0; j < 2; j = j + 1 ) {
      r = node number
      push H(OTS_PUBKEY[i+j] || I || uint32str(r) || D_LEAF) onto data stack
      push level onto the integer stack
    }
    while ( height of the integer stack >= 2 ) {
      if level of the top 2 elements on the integer stack are equal {
        hash_init()
        siblings = ""
        repeat ( 2 ) {
          siblings = (pop(data stack) || siblings)
          level = pop(integer stack)
        }
        hash_update(siblings)
        r = node number
        hash_update(I || uint32str(r) || D_INTR)
        push hash_final() onto the data stack
        push (level + 1) onto the integer stack
      }
    }
  }
  public_key = pop(data stack)

   Note that this pseudocode expects that all 2^h leaves of the tree
   have equal depth.  Neither stack ever contains more than h+1
   elements.  For typical parameters, these stacks will hold around 512
   bytes of data.

Appendix C.  Example implementation

# example implementation for Leighton-Micali hash based signatures
# Internet draft
#
# Notes:
#
#     * only a limted set of parameters are supported; in particular,
#     * w=8 and n=32
#
#     * HLMS, LMS, and LM-OTS are all implemented
#
#     * uncommenting print statements may be useful for debugging, or
#       for understanding the mechanics of
#
#



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# LMOTS constants
#
D_ITER = chr(0x00) # in the iterations of the LM-OTS algorithms
D_PBLC = chr(0x01) # when computing the hash of all of the iterates in the LM-OTS algorithm
D_MESG = chr(0x02) # when computing the hash of the message in the LMOTS algorithms
D_LEAF = chr(0x03) # when computing the hash of the leaf of an LMS tree
D_INTR = chr(0x04) # when computing the hash of an interior node of an LMS tree

NULL   = chr(0)    # used as padding for encoding

lmots_sha256_n32_w8 = 0x08000008 # typecode for LM-OTS with n=32, w=8
lms_sha256_n32_h10  = 0x02000002 # typecode for LMS with n=32, h=10
hlms_sha256_n32_l2  = 0x01000001 # typecode for two-level HLMS with n=32


# LMOTS parameters
#
n = 32; p = 34; w = 8; ls = 0

def bytes_in_lmots_sig():
    return n*(p+1)+40 # 4 + n + 31 + 1 + 4 + n*p

from Crypto.Hash import SHA256
from Crypto import Random

# SHA256 hash function
#
def H(x):
#    print "hash input: " + stringToHex(x)
    h = SHA256.new()
    h.update(x)
    return h.digest()[0:n]

def sha256_iter(x, num):
    tmp = x
    for j in range(0, num):
        tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER)

# entropy source
#
entropySource = Random.new()

# integer to string conversion
#
def uint32ToString(x):
    c4 = chr(x & 0xff)
    x = x >> 8
    c3 = chr(x & 0xff)



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    x = x >> 8
    c2 = chr(x & 0xff)
    x = x >> 8
    c1 = chr(x & 0xff)
    return c1 + c2 + c3 + c4

def uint16ToString(x):
    c2 = chr(x & 0xff)
    x = x >> 8
    c1 = chr(x & 0xff)
    return c1 + c2

def uint8ToString(x):
    return chr(x)

def stringToUint(x):
    sum = 0
    for c in x:
        sum = sum * 256 + ord(c)
    return sum

# string-to-hex function needed for debugging
#
def stringToHex(x):
    return "".join("{:02x}".format(ord(c)) for c in x)

# LM-OTS functions
#
def encode_lmots_sig(C, I, q, y):
    result = uint32ToString(lmots_sha256_n32_w8) + C + I + NULL + q
    for i, e in enumerate(y):
        result = result + y[i]
    return result

def decode_lmots_sig(sig):
    if (len(sig) != bytes_in_lmots_sig()):
        print "error decoding signature; incorrect length (" + str(len(sig)) + " bytes)"
    typecode = sig[0:4]
    if (typecode != uint32ToString(lmots_sha256_n32_w8)):
        print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(lmots_sha256_n32_w8))
        return ""
    C = sig[4:n+4]
    I = sig[n+4:n+35]
    q = sig[n+36:n+40] # note: skip over NULL
    y = list()
    pos = n+40
    for i in range(0, p):
        y.append(sig[pos:pos+n])



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        pos = pos + n
    return C, I, q, y

def print_lmots_sig(sig):
    C, I, q, y = decode_lmots_sig(sig)
    print "C:\t" + stringToHex(C)
    print "I:\t" + stringToHex(I)
    print "q:\t" + stringToHex(q)
    for i, e in enumerate(y):
        print "y[" + str(i) + "]:\t" + stringToHex(e)

# Algorithm 0: Generating a Private Key
#
def lmots_gen_priv():
    priv = list()
    for i in range(0, p):
        priv.append(entropySource.read(n))
    return priv

# Algorithm 1: Generating a Public Key From a Private Key
#
def lmots_gen_pub(private_key, I, q):
    hash = SHA256.new()
    hash.update(I + q)
    for i, x in enumerate(private_key):
        tmp = x
        # print "i:" + str(i) + " range: " + str(range(0, 256))
        for j in range(0, 256):
            tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER)
        hash.update(tmp)
    hash.update(D_PBLC)
    return hash.digest()

# Algorithm 2: Merkle Checksum Calculation
#
def checksum(x):
    sum = 0
    for c in x:
        sum = sum + ord(c)
    # print format(sum, '04x')
    c1 = chr(sum >> 8)
    c2 = chr(sum & 0xff)
    return c1 + c2

# Algorithm 3: Generating a Signature From a Private Key and a Message
#
def lmots_gen_sig(private_key, I, q, message):
    C = entropySource.read(n)



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    hashQ = H(message + C + I + q + D_MESG)
    V = hashQ + checksum(hashQ)
    # print "V: " + stringToHex(V)
    y = list()
    for i, x in enumerate(private_key):
        tmp = x
        # print "i:" + str(i) + " range: " + str(range(0, ord(V[i])))
        for j in range(0, ord(V[i])):
            tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER)
        y.append(tmp)
    return encode_lmots_sig(C, I, q, y)

def lmots_sig_to_pub(sig, message):
    C, I, q, y = decode_lmots_sig(sig)
    hashQ = H(message + C + I + q + D_MESG)
    V = hashQ + checksum(hashQ)
    # print "V: " + stringToHex(V)
    hash = SHA256.new()
    hash.update(I + q)
    for i, y in enumerate(y):
        tmp = y
        # print "i:" + str(i) + " range: " + str(range(ord(V[i]), 256))
        for j in range(ord(V[i]), 256):
            tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER)
        hash.update(tmp)
    hash.update(D_PBLC)
    return hash.digest()

# Algorithm 4: Verifying a Signature and Message Using a Public Key
#
def lmots_verify_sig(public_key, sig, message):
    z = lmots_sig_to_pub(sig, message)
    # print "z: " + stringToHex(z)
    if z == public_key:
        return 1
    else:
        return 0

# LM-OTS test functions
#
I = entropySource.read(31)
q = uint32ToString(0)
private_key = lmots_gen_priv()

print "LMOTS private key: "
for i, x in enumerate(private_key):
    print "x[" + str(i) + "]:\t" + stringToHex(x)




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public_key = lmots_gen_pub(private_key, I, q)

print "LMOTS public key: "
print stringToHex(public_key)

message = "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized."

print "message: " + message

sig = lmots_gen_sig(private_key, I, q, message)

print "LMOTS signature byte length: " + str(len(sig))

print "LMOTS signature: "
print_lmots_sig(sig)

print "verification: "
print "true positive test: "
if (lmots_verify_sig(public_key, sig, message) == 1):
    print "passed: message/signature pair is valid as expected"
else:
    print "failed: message/signature pair is invalid"

print "false positive test: "
if (lmots_verify_sig(public_key, sig, "some other message") == 1):
    print "failed: message/signature pair is valid (expected failure)"
else:
    print "passed: message/signature pair is invalid as expected"



# LMS N-time signatures functions
#
h = 10 # height (number of levels -1) of tree

def encode_lms_sig(lmots_sig, path):
    result = uint32ToString(lms_sha256_n32_h10) + lmots_sig
    for i, e in enumerate(path):
        result = result + path[i]
    return result

def decode_lms_sig(sig):
    typecode = sig[0:4]
    if (typecode != uint32ToString(lms_sha256_n32_h10)):
        print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(lms_sha256_h10))
        return ""
    pos = 4 + bytes_in_lmots_sig()
    lmots_sig = sig[4:pos]



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    path = list()
    for i in range(0,h):
        # print "sig[" + str(i) + "]:\t" + stringToHex(sig[pos:pos+n])
        path.append(sig[pos:pos+n])
        pos = pos + n
    return lmots_sig, path

def print_lms_sig(sig):
    lmots_sig, path = decode_lms_sig(sig)
    print_lmots_sig(lmots_sig)
    for i, e in enumerate(path):
        print "path[" + str(i) + "]:\t" + str(stringToHex(e))

def bytes_in_lms_sig():
    return bytes_in_lmots_sig() + h*n + 4

class lms_private_key(object):

    # Algorithm for computing root and other nodes (alternative to Algorithm 6)
    #
    def T(self, j):
        # print "T(" + str(j) + ")"
        if (j >= 2**h):
            self.nodes[j] = H(self.pub[j - 2**h] + self.I + uint32ToString(j) + D_LEAF)
            return self.nodes[j]
        else:
            self.nodes[j] = H(self.T(2*j) + self.T(2*j+1) + self.I + uint32ToString(j) + D_INTR)
            return self.nodes[j]

    def __init__(self):
        self.I = entropySource.read(31)
        self.priv = list()
        self.pub = list()
        for q in range(0, 2**h):
            # print "generating " + str(q) + "th OTS key"
            ots_priv = lmots_gen_priv()
            ots_pub = lmots_gen_pub(ots_priv, self.I, uint32ToString(q))
            self.priv.append(ots_priv)
            self.pub.append(ots_pub)
        self.leaf_num = 0
        self.nodes = {}
        self.lms_public_key = self.T(1)

    def num_sigs_remaining():
        return 2**h - self.leaf_num

    def printHex(self):
        for i, p in enumerate(self.priv):



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            print "priv[" + str(i) + "]:"
            for j, x in enumerate(p):
                print "x[" + str(j) + "]:\t" + stringToHex(x)
            print "pub[" + str(i) + "]:\t" + stringToHex(self.pub[i])
        for t, T in self.nodes.items():
            print "T(" + str(t) + "):\t" + stringToHex(T)
        print "pub: \t" + stringToHex(self.lms_public_key)

    def get_public_key(self):
        return self.lms_public_key

    def get_path(self, leaf_num):
        node_num = leaf_num + 2**h
        # print "signing node " + str(node_num)
        path = list()
        while node_num > 1:
            if (node_num % 2):
                # print "path" + str(node_num - 1) + ": " + stringToHex(self.nodes[node_num - 1])
                path.append(self.nodes[node_num - 1])
            else:
                # print "path " + str(node_num + 1) + ": " + stringToHex(self.nodes[node_num + 1])
                path.append(self.nodes[node_num + 1])
            node_num = node_num/2
        return path

    def sign(self, message):
        if (self.leaf_num >= 2**h):
            return ""
        sig = lmots_gen_sig(self.priv[self.leaf_num], self.I, uint32ToString(self.leaf_num), message)
        # C, I, q, y = decode_lmots_sig(sig)
        path = self.get_path(self.leaf_num)
        leaf_num = self.leaf_num
        self.leaf_num = self.leaf_num + 1
        return encode_lms_sig(sig, path)


class lms_public_key(object):

    def __init__(self, value):
        self.value = value

    def verify(self, message, sig):
        lmots_sig, path = decode_lms_sig(sig)
        C, I, q, y = decode_lmots_sig(lmots_sig)       # note: only q is actually needed here
        node_num = stringToUint(q) + 2**h
        # print "verifying node " + str(node_num)
        pathvalue = iter(path)
        tmp = lmots_sig_to_pub(lmots_sig, message)



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        tmp = H(tmp + I + uint32ToString(node_num) + D_LEAF)
        while node_num > 1:
            # print "S(" + str(node_num) + "):\t" + stringToHex(tmp)
            if (node_num % 2):
                # print "adding node " + str(node_num - 1)
                tmp = H(pathvalue.next() + tmp + I + uint32ToString(node_num/2) + D_INTR)
            else:
                # print "adding node " + str(node_num + 1)
                tmp = H(tmp + pathvalue.next() + I + uint32ToString(node_num/2) + D_INTR)
            node_num = node_num/2
        # print "pubkey: " + stringToHex(tmp)
        if (tmp == self.value):
            return 1
        else:
            return 0




# test LMS signatures
#

print "LMS test"

lms_priv = lms_private_key()
lms_pub = lms_public_key(lms_priv.get_public_key())

# lms_priv.printHex()

for i in range(0, 2**h):
    sig = lms_priv.sign(message)

    print "LMS signature byte length: " + str(len(sig))

    # print_lms_sig(sig)

    print "true positive test"
    if (lms_pub.verify(message, sig) == 1):
        print "passed: LMS message/signature pair is valid"
    else:
        print "failed: LMS message/signature pair is invalid"

    print "false positive test"
    if (lms_pub.verify("other message", sig) == 1):
        print "failed: LMS message/signature pair is valid (expected failure)"
    else:
        print "passed: LMS message/signature pair is invalid as expected"




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Internet-Draft            Hash-Based Signatures             October 2015


# Hierarchical LMS signatures (HLMS)

def encode_hlms_sig(pub2, sig1, lms_sig):
    result = uint32ToString(hlms_sha256_n32_l2)
    result = result + pub2
    result = result + sig1
    result = result + lms_sig
    return result

def decode_hlms_sig(sig):
    typecode = sig[0:4]
    if (typecode != uint32ToString(hlms_sha256_n32_l2)):
        print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(hlms_sha256_n32_l2))
        return ""
    pub2 = sig[4:36]
    lms_sig_len = bytes_in_lms_sig()
    sig1 = sig[36:36+lms_sig_len]
    lms_sig = sig[36+lms_sig_len:36+2*lms_sig_len]
    return pub2, sig1, lms_sig

def print_hlms_sig(sig):
    pub2, sig1, lms_sig = decode_hlms_sig(sig)
    print "pub2:\t" + stringToHex(pub2)
    print "sig1: "
    print_lms_sig(sig1)
    print "sig2: "
    print_lms_sig(lms_sig)

class hlms_private_key(object):
    def __init__(self):
        self.prv1 = lms_private_key()
        self.init_level_2()

    def init_level_2(self):
        self.prv2 = lms_private_key()
        self.sig1 = self.prv1.sign(self.prv2.get_public_key())

    def get_public_key(self):
        return self.prv1.get_public_key()

    def sign(self, message):
        lms_sig = self.prv2.sign(message)
        if (lms_sig == ""):
            print "refreshing level 2 public/private key pair"
            self.init_level_2()
            lms_sig = self.prv2.sign(message)
        return encode_hlms_sig(self.prv2.get_public_key(), self.sig1, lms_sig)




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Internet-Draft            Hash-Based Signatures             October 2015


class hlms_public_key(object):
    def __init__(self, value):
        self.pub1 = lms_public_key(value)

    def verify(self, message, sig):
        pub2, sig1, lms_sig = decode_hlms_sig(sig)
        if (self.pub1.verify(pub2, sig1) == 1):
            if (lms_public_key(pub2).verify(message, lms_sig) == 1):
                return 1
            else:
                print "pub2 verification of lms_sig did not pass"
        else:
            print "pub1 verification of sig1 did not pass"
        return 0



print "HLMS testing"

hlms_prv = hlms_private_key()

hlms_pub = hlms_public_key(hlms_prv.get_public_key())

for i in range(0, 4096):

    sig = hlms_prv.sign(message)

    # print_hlms_sig(sig)

    print "HLMS signature byte length: " + str(len(sig))

    print "testing verification (" + str(i) + "th iteration)"
    print "true positive test"
    if (hlms_pub.verify(message, sig) == 1):
        print "passed; HLMS message/signature pair is valid"
    else:
        print "failed; HLMS message/signature pair is invalid"

        print "false positive test"
        if (hlms_pub.verify("other message", sig) == 1):
            print "failed; HLMS message/signature pair is valid (expected failure)"
        else:
            print "passed; HLMS message/signature pair is invalid as expected"








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Internet-Draft            Hash-Based Signatures             October 2015


Authors' Addresses

   David McGrew
   Cisco Systems
   13600 Dulles Technology Drive
   Herndon, VA  20171
   USA

   Email: mcgrew@cisco.com


   Michael Curcio
   Cisco Systems
   7025-2 Kit Creek Road
   Research Triangle Park, NC  27709-4987
   USA

   Email: micurcio@cisco.com

































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