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Crypto Forum Research Group                                    D. McGrew
Internet-Draft                                                 M. Curcio
Intended status: Informational                                S. Fluhrer
Expires: May 4, 2017                                       Cisco Systems
                                                        October 31, 2016


                         Hash-Based Signatures
                       draft-mcgrew-hash-sigs-05

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

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

   Internet-Drafts are working documents of the Internet Engineering
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   This Internet-Draft will expire on May 4, 2017.

Copyright Notice

   Copyright (c) 2016 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents



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

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  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 . . . . . . . . . . . . . .   6
     3.2.  Security string . . . . . . . . . . . . . . . . . . . . .   7
     3.3.  Functions . . . . . . . . . . . . . . . . . . . . . . . .   8
     3.4.  Typecodes . . . . . . . . . . . . . . . . . . . . . . . .   8
   4.  LM-OTS One-Time Signatures  . . . . . . . . . . . . . . . . .   8
     4.1.  Parameters  . . . . . . . . . . . . . . . . . . . . . . .   9
     4.2.  Hashing Functions . . . . . . . . . . . . . . . . . . . .   9
     4.3.  Signature Methods . . . . . . . . . . . . . . . . . . . .   9
     4.4.  Private Key . . . . . . . . . . . . . . . . . . . . . . .  10
     4.5.  Public Key  . . . . . . . . . . . . . . . . . . . . . . .  11
     4.6.  Checksum  . . . . . . . . . . . . . . . . . . . . . . . .  11
     4.7.  Signature Generation  . . . . . . . . . . . . . . . . . .  12
     4.8.  Signature Verification  . . . . . . . . . . . . . . . . .  13
   5.  Leighton Micali Signatures  . . . . . . . . . . . . . . . . .  15
     5.1.  Parameters  . . . . . . . . . . . . . . . . . . . . . . .  16
     5.2.  LMS Private Key . . . . . . . . . . . . . . . . . . . . .  16
     5.3.  LMS Public Key  . . . . . . . . . . . . . . . . . . . . .  17
     5.4.  LMS Signature . . . . . . . . . . . . . . . . . . . . . .  17
       5.4.1.  LMS Signature Generation  . . . . . . . . . . . . . .  18
     5.5.  LMS Signature Verification  . . . . . . . . . . . . . . .  19
   6.  Hierarchical signatures . . . . . . . . . . . . . . . . . . .  21
     6.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  21
     6.2.  Signature Generation  . . . . . . . . . . . . . . . . . .  22
     6.3.  Signature Verification  . . . . . . . . . . . . . . . . .  22
   7.  Formats . . . . . . . . . . . . . . . . . . . . . . . . . . .  23
   8.  Rationale . . . . . . . . . . . . . . . . . . . . . . . . . .  26
   9.  History . . . . . . . . . . . . . . . . . . . . . . . . . . .  27
   10. IANA Considerations . . . . . . . . . . . . . . . . . . . . .  28
   11. Intellectual Property . . . . . . . . . . . . . . . . . . . .  29
     11.1.  Disclaimer . . . . . . . . . . . . . . . . . . . . . . .  29
   12. Security Considerations . . . . . . . . . . . . . . . . . . .  30
     12.1.  Stateful signature algorithm . . . . . . . . . . . . . .  31



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     12.2.  Security of LM-OTS Checksum  . . . . . . . . . . . . . .  32
   13. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  32
   14. References  . . . . . . . . . . . . . . . . . . . . . . . . .  33
     14.1.  Normative References . . . . . . . . . . . . . . . . . .  33
     14.2.  Informative References . . . . . . . . . . . . . . . . .  33
   Appendix A.  Pseudorandom Key Generation  . . . . . . . . . . . .  34
   Appendix B.  LM-OTS Parameter Options . . . . . . . . . . . . . .  35
   Appendix C.  An iterative algorithm for computing an LMS public
                key  . . . . . . . . . . . . . . . . . . . . . . . .  36
   Appendix D.  Example implementation . . . . . . . . . . . . . . .  36
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  37

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 verifier 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, and a Hierarchical Signature
   System (HSS) built on top of it that can efficiently scale to larger
   numbers of signatures.  We denote the one-time signature scheme
   incorporate in LMS 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 and HSS N-time signature systems



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   are described in Section 5 and Section 6, respectively.  Sufficient
   detail is provided to ensure interoperability.  The IANA registry for
   these signature systems is described in Section 10.  Security
   considerations are presented in Section 12.

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.

      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



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   leading "0x".  For example, 0xe534f0 is a byte string with a length
   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 u8str(X), u16str(X),
   and u32str(X), which take a nonnegative integer X as input and return
   one, two, and four byte strings, respectively.  We also make use of
   the functions strTou8(S), strTou16(S), and strTou32(S), which take a
   one, two, or four byte string S as input and return a nonnegative
   integer; these functions are such that u8str(strTou8(S)) = S,
   u16str(strTou16(S)) = S, and u32str(strTou326(S)) for all values of S
   that are in the appropriate range.

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].





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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.

   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.







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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 LMS public/private keypair.  The length
      of this value varies based on the LMS parameter set and it MUST be
      chosen uniformly at random, or via a pseudorandom process, at the
      time that a key pair is generated, in order to ensure that it will
      be distinct from the identifier of any other LMS private key with
      probability close to one.

      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

         D_I = 0x05 when computing the I value for a nonroot LM tree in
         the HSS system

         D_PRG = 0x06 in the recommended pseudorandom process for
         generating LMS private keys

      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.

      r - in the LMS N-time signature scheme, the node number r
      associated with a particular node of a 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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      q - in the LMS N-time signature scheme, each LM-OTS signature is
      associated with the leaf of a hash tree, and q is set to the leaf
      number.  This ensures that a distinct value of q is used for each
      distinct LM-OTS public/private keypair.  This value is represented
      as a 32-bit (four byte) unsigned integer in network byte order.

      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.

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

3.4.  Typecodes

   A typecode is an unsigned integer that is associated with a
   particular data format.  The format of the LM-OTS, LMS, and HSS
   signatures and public keys all begin with a typecode that indicates
   the precise details used in that format.  These typecodes are
   represented as four-byte unsigned integers in network byte order;
   equivalently, they are XDR enumerations (see Section 7).

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.




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   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 width (number of bits) of the Winternitz coefficients; 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).

   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 determines the
   length of the Winternitz chains computed as a part of the OTS
   signature (which involve 2^w-1 invocations of the hash function); it
   has little effect on security.  Increasing w will shorten the
   signature, but at a cost of a larger computation to generate and
   verify a signature.  The values of p and ls are dependent on the
   choices of the parameters n and w, as described in Appendix B.  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,
   the length LenS of the security string S, 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.



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        +---------------------+--------+----+---+------+-----+----+
        | Name                | H      | n  | w | LenS | p   | ls |
        +---------------------+--------+----+---+------+-----+----+
        | LMOTS_SHA256_N32_W1 | SHA256 | 32 | 1 | 68   | 265 | 7  |
        |                     |        |    |   |      |     |    |
        | LMOTS_SHA256_N32_W2 | SHA256 | 32 | 2 | 68   | 133 | 6  |
        |                     |        |    |   |      |     |    |
        | LMOTS_SHA256_N32_W4 | SHA256 | 32 | 4 | 68   | 67  | 4  |
        |                     |        |    |   |      |     |    |
        | LMOTS_SHA256_N32_W8 | SHA256 | 32 | 8 | 68   | 34  | 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; that is, immediately after
   computing the SHA256 hash, the 32 bit hash output is truncated to be
   the leftmost 16 bytes.

4.4.  Private Key

   The LM-OTS private key consists of a typecode indicating the
   particular LM-OTS algorithm, an array x[] containing p n-byte
   strings, and a LenS-bygte security string S.  This private key MUST
   be used to sign (at most) one message.  The following algorithm shows
   pseudocode for generating a private key.

   Algorithm 0: Generating a Private Key

    1. set type to the typecode of the algorithm

    2. if no security string S has been provided as input, then set S to
       a LenS-byte string generated uniformly at random

    3. set n and p according to the typecode and Table 1

    4. compute the array x as follows:
       for ( i = 0; i < p; i = i + 1 ) {
         set x[i] to a uniformly random n-byte string
       }

    5. return u32str(type) || S || x[0] || x[1] || ... || x[p-1]

   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




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   MUST match that of the LM-OTS algorithm.  Appendix A provides an
   example of a pseudorandom method for computing LM-OTS private key.

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.

   The public key is generated from the private key using the following
   algorithm, or any equivalent process.

   Algorithm 1: Generating a One Time Signature Public Key From a
   Private Key

   1. set type to the typecode of the algorithm

   2. set the integers n, p, and w according to the typecode and Table 1

   3. determine x and S from the private key

   3. compute the string K as follows:
      for ( i = 0; i < p; i = i + 1 ) {
        tmp = x[i]
        for ( j = 0; j < 2^w - 1; j = j + 1 ) {
           tmp = H(S || tmp || u16str(i) || u8str(j) || D_ITER)
        }
        y[i] = tmp
      }
      K = H(S || y[0] || ... || y[p-1] || D_PBLC)

   4. return u32str(type) || S || K

   The public key is the value returned by Algorithm 1.

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 12.
   The checksum function Cksm is defined as follows, where S denotes the
   n-byte string that is input to that function, and the value sum is a
   16-bit unsigned integer:








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   Algorithm 2: Checksum Calculation

     sum = 0
     for ( i = 0; i < (n*8/w); 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 prepending
   the randomizer C and the security string S to the message, then
   appending D_MESG to the resulting string then computing its hash,
   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 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 One Time Signature From a Private Key and a
   Message

     1. set type to the typecode of the algorithm

     2. set n, p, and w according to the typecode and Table 1

     3. determine x and S from the private key

     3. set C to a uniformly random n-byte string

     4. compute the array y as follows:
        Q = H(S || C || message || 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(S || tmp || u16str(i) || u8str(j) || D_ITER)
          }
          y[i] = tmp
        }

      5. return u32str(type) || C || y[0] || ... || 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 returned by Algorithm 3.  Section 7
   specifies the typecode and more formally defines the encoding and
   decoding of the string.

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 chain of
   iterations of H using the w-bit coefficients of the string resulting
   from the concatenation 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 4a: Verifying a Signature and Message Using a Public Key

    1. if the public key is not at least four bytes long, return INVALID

    2. parse pubtype, S, and K from the public key as follows:
       a. pubtype = strTou32(first 4 bytes of public key)

       b. if pubtype is not equal to sigtype, return INVALID

       c. if the public key is not exactly 4 + LenS + n bytes long,
          return INVALID

       c. S = next LenS bytes of public key

       d. K = next n bytes of public key

    3. compute the public key candidate Kc from the signature,
       message, and the security string S obtained from the
       public key, using Algorithm 4b.

    4. if Kc is equal to K, return VALID; otherwise, return INVALID






























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   Algorithm 4b: Computing a Public Key Candidate Kc from a Signature,
   Message, Signature Typecode Type , and a Security String S

    1. if the signature is not at least four bytes long, return INVALID

    2. parse sigtype, C, and y from the signature as follows:
       a. sigtype = strTou32(first 4 bytes of signature)

       b. if sigtype is not equal to Type, return INVALID

       c. set n and p according to the sigtype and Table 1;  if the
       signature is not exactly 8 + n * (p+1) bytes long, return INVALID

       d. C = next n bytes of signature

       e.  y[0] = next n bytes of signature
           y[1] = next n bytes of signature
           ...
         y[p-1] = next n bytes of signature

    3. compute the string Kc as follows
       Q = H(S || C || message || D_MESG)
       for ( i = 0; i < p; i = i + 1 ) {
         a = coef(Q || Cksm(Q), i, w)
         tmp = y[i]
         for ( j = a; j < 2^w - 1; j = j + 1 ) {
            tmp = H(S || tmp || u16str(i) || u8str(j) || D_ITER)
         }
         z[i] = tmp
       }
       Kc = H(S || z[0] || z[1] || ... || z[p-1] || D_PBLC)

    4. return Kc

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.





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   Each node of the tree is associated with a node number, an unsigned
   integer that is denoted as node_num in the algorithms below, which is
   computed as follows.  The root node has node number 1; for each node
   with node number N, its left child has node number 2*N, while its
   right child has node number 2*N+1.  The result of this is that each
   node within the tree will have a unique node number, and the leaves
   will have node numbers 2^h, (2^h)+1, (2^h)+2, ..., (2^h)+(2^h)-1.  In
   general, the jth node at level L has node number 2^L + j.  The node
   number can conveniently be computed when it is needed in the LMS
   algorithms, as described in those algorithms.

5.1.  Parameters

   An LMS system has the following parameters:

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

      m : the number of bytes associated with each node.

   There are 2^h leaves in the tree.  The parameter m MAY have any value
   in principle, but it SHOULD be equal to the parameter n from
   Section 4.1 so that the security level of LMS is comparable to that
   of LM-OTS.

                 +--------------------+--------+----+----+
                 | Name               | H      | m  | h  |
                 +--------------------+--------+----+----+
                 | LMS_SHA256_M32_H5  | SHA256 | 32 | 5  |
                 |                    |        |    |    |
                 | LMS_SHA256_M32_H10 | SHA256 | 32 | 10 |
                 |                    |        |    |    |
                 | LMS_SHA256_M32_H15 | SHA256 | 32 | 15 |
                 |                    |        |    |    |
                 | LMS_SHA256_M32_H20 | SHA256 | 32 | 20 |
                 +--------------------+--------+----+----+

                                  Table 2

5.2.  LMS Private Key

   An LMS private key consists of an array OTS_PRIV[] of 2^h LM-OTS
   private keys, and the leaf number q of the next LM-OTS private key
   that has not yet been used.  The qth element of OTS_PRIV[] is
   generated using Algorithm 0 with the security string S = I || q.  The
   leaf number q is initialized to zero when the LMS private key is
   created.  The process is as follows:





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   Algorithm 5: Computing an LMS Private Key.

      1. determine h and m from the typecode and Table 2.

      2. compute the array OTS_PRIV[] as follows:
         for ( q = 0; q < 2^h; q = q + 1) {
            S = I || q
            OTS_PRIV[q] = LM-OTS private key with security string S
          }

      3. q = 0

   An LMS private key MAY be generated pseudorandomly from a secret
   value, in which case the secret value MUST be at least m 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.
   Appendix A provides an example of a pseudorandom method for computing
   an LMS private key.

5.3.  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_PUB[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 m-byte string, and the string for the rth node is denoted as
   T[r] and is defined as

 T[r] = / H(I || OTS_PUB[r-2^h]  || u32str(r) || D_LEAF)    if r >= 2^h,
        \ H(I || T[2*r] || T[2*r+1] || u32str(r) || D_INTR) otherwise.

   The LMS public key is the string u32str(type) || I || T[1].
   Section 7 specifies the format of the type variable.  The value I is
   the private key identifier (whose length is denoted by the parameter
   set), and is the value used for all computations for the same LMS
   tree.  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 C.

5.4.  LMS Signature

   An LMS signature consists of

      a typecode indicating the particular LMS algorithm,




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      the number q of the leaf associated with the LM-OTS signature, as
      a four-byte unsigned integer in network byte order,

      an LM-OTS signature, and

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

   Symbolically, the signature can be represented as u32str(type) ||
   u32str(q) || ots_signature || path[0] || path[1] || ... || path[h-1].
   Section 7 specifies the typecode and more formally defines the
   format.  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 themselves.  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.

5.4.1.  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.  The leaf
   number q in the signature is set to the leaf number of the LMS
   private key that was used in the signature.  Before releasing the
   signature, the leaf number q in the LMS private key MUST be
   incremented, to prevent the LM-OTS private key from being used again.
   If the LMS private key is maintained in nonvolatile memory, then the
   implementation MUST ensure that the incremented value has been stored
   before releasing the signature.

   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.

   Several useful time/storage tradeoffs are described in the 'Small-
   Memory LM Schemes' section of [USPTO5432852].





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5.5.  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 array path[] as
   described in Algorithm 6 below.  If the root value matches the public
   key, then the signature is valid; otherwise, the signature fails.

   Algorithm 6: LMS Signature Verification

     1. if the public key is not at least four bytes long, return
        INVALID

     2. parse pubtype, I, and T[1] from the public key as follows:
        a. pubtype = strTou32(first 4 bytes of public key)

        b. if the public key is not exactly 4 + LenI + m bytes
           long, return INVALID

        c. I = next LenI bytes of the public key

        d. T[1] = next m bytes of the public key

     6. compute the candidate LMS root value Tc from the signature,
        message, identifier and pubtype using Algorithm 6b.

     7. if Tc is equal to T[1], return VALID; otherwise, return INVALID






















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   Algorithm 6b: Computing an LMS Public Key Candidate from a Signature,
   Message, Identifier, and algorithm typecode

    1. if the signature is not at least eight bytes long, return INVALID

    2. parse sigtype, q, ots_signature, and path from the signature as
       follows:
      a. sigtype = strTou32(first 4 bytes of signature)

      b. if pubtype is not equal to sigtype, return INVALID

      c. set m, h, and LenI according to sigtype and Table 2;

      d. q = strTou32(next 4 bytes of signature)

      e. otssigtype = strTou32(next 4 bytes of signature)

      f. set n and p according to otssigtype and Table 1; if the
      signature does string is not at least 12 + n * (p + 1) + m * h
      bytes long, return INVALID

      g. ots_signature = bytes 8 through 8 + n * (p + 1) of signature

      h. set path as follows:
            path[0] = next m bytes of signature
            path[1] = next m bytes of signature
            ...
            path[h-1] = next m bytes of signature

    5. Kc = candidate public key computed by applying Algorithm 4b
       to the signature ots_signature, the message, and the
       security string S = I || q

    6. compute the candidate LMS root value Tc as follows:
       tmp = H(I || Kc || u32str(node_num) || D_LEAF)
       i = 0
       node_num = 2^h + q
       while (node_num > 1) {
         if (node_num is odd):
           tmp = H(I || path[i] || tmp || u32str(node_num/2) || D_INTR)
         else:
           tmp = H(I || tmp || path[i] || u32str(node_num/2) || D_INTR)
        node_num = node_num/2
        i = i + 1

    7. return Tc





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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 System
   (HSS) 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.  It also
   makes use of an hbs_key_info structure, which contains the typecode
   of the LMS algorithm and the typecode of the LM-OTS algorithm.

   Each level of the hierarchy is associated with a distinct LMS public
   key, private key, signature, and identifier.  The number of levels is
   denoted L, and is between two and eight, inclusive.  The following
   notation is used, where i is an integer between 0 and L-1 inclusive,
   and the root of the hierarchy is level 0:

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

      pub[i] is the LMS public key of the ith level (which includes the
      identifier I as well as the key value K),

      sig[i] is the LMS signature of the ith level,

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

6.1.  Key Generation

   When an HSS keypair is generated, the keypair for each level has its
   own identifier.

   To generate an HSS private and public key pair, new LMS private and
   public keys are generated for prv[i] and pub[i] for i=0, ..., L-1.
   These key pairs, and their identifiers, MUST be generated
   independently.  All of the information of the leaf level L-1,
   including the private key, MUST NOT be stored in nonvolatile memory.
   Letting nv denote the lowest level for which prv[nv] is stored in
   nonvolatile memory, there are nv nonvolatile levels, and L-nv
   volatile levels.  For security, nv should be as close to zero as
   possible (see Section 12.1).

   The public key of the HSS scheme is pub[1], the public key of the
   first level, followed by an array info[] containing the hbs_key_info
   structures for the remaining levels 1, ..., L-1.




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   The HSS private key consists of prv[0], ... , prv[L-1].  The values
   pub[0] and prv[0] do not change, though the values of pub[i] and
   prv[i] are dynamic for i > 1, 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:

      If prv[L-1] is exhausted, then determine the smallest integer d
      such that all of the private keys prv[d], prv[d+1], ... , prv[L-1]
      are exhausted.  If d is equal to one, then the HSS keypair is
      exhausted, and it MUST NOT generate any more signatures.
      Otherwise, the keypairs for levels d through L-1 must be
      regenerated during the signature generation process, as follows.
      For i from d to L-1, a new LMS public and private key pair with a
      new identifier is generated, pub[i] and prv[i] are set to those
      values, then the public key pub[i] is signed with prv[i-1], and
      sig[i-1] is set to the resulting value.

      The message is signed with prv[L-1], and the value sig[L-1] is set
      to that result.

      The value of the HSS signature is set as follows.  We let
      signed_pub_key denote an array of strings, where signed_pub_key[i]
      = sig[i] || pub[i+1], for i between 0 and L-2, inclusive.  Then
      the HSS signature is u32str(L-1) || signed_pub_key[0] || ... ||
      signed_pub_key[L-2] || sig[L-1].

      Note that the number of signed_pub_key elements in the signature
      is indicated by the value L-1 that appears in the initial four
      bytes of the signature.

6.3.  Signature Verification

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













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      The signature S is parsed into its components as follows:

      L' = strToU32(first four bytes of S)
      for (i = 0; i < L'; i = i + 1) {
         siglist[0] = next LMS signature parsed from S
         publist[1] = next LMS public key parsed from S
      }
      siglist[L-1] = next LMS signature parsed from S

      key = pub
      for (i =0; i < L'; i = i + 1) {
         sig = siglist[i]
         msg = publist[i]
         if (lms_verify(msg, key, sig) != VALID):
             return INVALID
         key = msg
      return lms_verify(message, key, siglist[L-1])


   Since the length of an LMS signature cannot be known without parsing
   it, the HSS signature verification algorithm makes use of an LMS
   signature parsing routine that takes as input a string consisting of
   an LMS signature with an arbitrary string appended to it, and returns
   both the LMS signature and the appended string.  The latter is passed
   on for futher processing.

7.  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.  An illustration of the layout of
   data in these objects is provided below.  The definitions are as
   follows:


   /* one-time signatures */

   enum ots_algorithm_type {
     ots_reserved         = 0,
     lmots_sha256_n32_w1  = 1,
     lmots_sha256_n32_w2  = 2,
     lmots_sha256_n32_w4  = 3,
     lmots_sha256_n32_w8  = 4
   };



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   typedef opaque bytestring16[16];
   typedef opaque bytestring32[32];

   struct lmots_signature_n32_p265 {
     bytestring32 C;
     bytestring32 y[265];
   };

   struct lmots_signature_n32_p133 {
     bytestring32 C;
     bytestring32 y[133];
   };

   struct lmots_signature_n32_p67 {
     bytestring32 C;
     bytestring32 y[67];
   };

   struct lmots_signature_n32_p34 {
     bytestring32 C;
     bytestring32 y[34];
   };

   union ots_signature switch (ots_algorithm_type type) {
    case lmots_sha256_n32_w1:
      lmots_signature_n32_p265 sig_n32_p265;
    case lmots_sha256_n32_w2:
      lmots_signature_n32_p133 sig_n32_p133;
    case lmots_sha256_n32_w4:
      lmots_signature_n32_p67  sig_n32_p67;
    case lmots_sha256_n32_w8:
      lmots_signature_n32_p34  sig_n32_p34;
    default:
      void;   /* error condition */
   };

   union ots_private_key switch (ots_algorithm_type type) {
    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 */
   };

   /* hash based signatures (hbs) */




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   enum hbs_algorithm_type {
     hbs_reserved       = 0,
     lms_sha256_n32_h20 = 1,
     lms_sha256_n32_h10 = 2,
     lms_sha256_n32_h5  = 3,
   };

   /* leighton mical signatures (lms) */

   union lms_path switch (hbs_algorithm_type type) {
    case lms_sha256_n32_h20:
      bytestring32 path_n32_h20[20];
   case lms_sha256_n32_h15:
      bytestring32 path_n32_h15[15];
    case lms_sha256_n32_h10:
      bytestring32 path_n32_h10[10];
    case lms_sha256_n32_h5:
      bytestring32 path_n32_h5[5];
    default:
      void;     /* error condition */
   };

   struct lms_signature {
     unsigned int q;
     ots_signature lmots_sig;
     lms_path nodes;
   };

   struct lms_key_n32 {
     ots_algorithm_type ots_alg_type;
     opaque I[64];
     opaque K[32];
   };

   union hbs_public_key switch (hbs_algorithm_type type) {
    case lms_sha256_n32_h20:
    case lms_sha256_n32_h10:
    case lms_sha256_n32_h5:
         lms_key_n32 z_n32;
    default:
      void;     /* error condition */
   };

   /* hierarchical signature system (hss)  */

   struct hss_public_key {
     unsigned int levels;
     hbs_public_key pub;



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   };

   struct signed_public_key {
     hbs_signature sig;
     hbs_public_key pub;
   }

   struct hss_signature {
     signed_public_key signed_keys<L-1>;
     hbs_signature sig_of_message;
   };

   Many of the objects start with a typecode.  A verifier MUST check
   each of these typecodes, and a verification operation on a signature
   with an unknown type, or a type that does not correspond to the type
   within the public key MUST return INVALID.  The expected length of a
   variable-length object can be determined from its typecode, and if an
   object has a different length, then any signature computed from the
   object is INVALID.

8.  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 or 64 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
   relatively easy to parse.  Each format starts with a 32-bit
   enumeration value that indicates the details of the signature
   algorithm and provides 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
   case that the number of times H is applied is 0, the sum is (2^w - 1)
   * (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 B, which indicates
   the number of bits used in the left-shift operation.

   A future version of this specification may support hash functions
   other than SHA-256.

9.  History

   This is the fifth version of this draft.  It has the following
   changes from previous versions:

   Version 04

      Specified that, in the HSS method, the I value was computed from
      the I value of the parent LM tree.  Previous versions had the I
      value extracted from the public key (which meant that all LM trees
      of a particular level and public key used the same I value)

      Changed the length of the I field based on the parameter set.  As
      noted in the Rationale section, this allows an implemetation to
      compute N=32 based parameter sets significantly faster.

      Modified the XDR of an HSS signature not to use an array of LM
      signatures; LM signatures are variable length, and XDR doesn't
      support arrays of variable length structures.

      Changed the LMS registry to be in a consistent order with the LM-
      OTS paramter sets.  Also, added LMS parameter sets with height 15
      trees

   Previous versions




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      In Algorithms 3 and 4, the message was moved from the initial
      position of the input to the function H to the final position, in
      the computation of the intermediate variable Q.  This was done to
      improve security by preventing an attacker that can find a
      collision in H from taking advantage of that fact via the forward
      chaining property of Merkle-Damgard.

      The Hierarchical Signature Scheme was generalized slightly so that
      it can use more than two levels.

      Several points of confusion were corrected; these had resulted
      from incomplete or inconsistent changes from the Merkle approach
      of the earlier draft to the Leighton-Micali approach.

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

10.  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
   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.  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_N32_W1  | Section 4 |     0x00000001     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W2  | Section 4 |     0x00000002     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W4  | Section 4 |     0x00000003     |
         |                      |           |                    |
         | LMOTS_SHA256_N32_W8  | Section 4 |     0x00000004     |
         +----------------------+-----------+--------------------+

                                  Table 3

   The LMS registry is as follows.

          +--------------------+-----------+--------------------+
          | Name               | Reference | Numeric Identifier |
          +--------------------+-----------+--------------------+
          | LMS_SHA256_M32_H5  | Section 5 |     0x00000005     |
          |                    |           |                    |
          | LMS_SHA256_M32_H10 | Section 5 |     0x00000006     |
          |                    |           |                    |
          | LMS_SHA256_M32_H15 | Section 5 |     0x00000007     |
          |                    |           |                    |
          | LMS_SHA256_M32_H20 | Section 5 |     0x00000008     |
          +--------------------+-----------+--------------------+

                                  Table 4

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

11.  Intellectual Property

   This draft is based on U.S. patent 5,432,852, which issued over
   twenty years ago and is thus expired.

11.1.  Disclaimer

   This document is not intended as legal advice.  Readers are advised
   to consult with their own legal advisers if they would like a legal
   interpretation of their rights.

   The IETF policies and processes regarding intellectual property and
   patents are outlined in [RFC3979] and [RFC4879] and at
   https://datatracker.ietf.org/ipr/about.




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12.  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 is 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, even 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.

   The format of the inputs to H() have the property that each
   invocation of that function has an input that is distinct from all
   others, with high probability.  This property is important for a
   proof of security in the random oracle model.  The formats used
   during key generation and signing are

      S || tmp || u16str(i) || u8str(j) || D_ITER
      S || y[0] || ... || y[p-1] || D_PBLC
      S || C || message || D_MESG
      I || OTS_PUB[r-2^h]  || u32str(r) || D_LEAF
      I || T[2*r] || T[2*r+1] || u32str(r) || D_INTR
      I || u32str(q) || x_q[j-1] || u16str(j) || D_PRG

   Because the suffixes D_ITER, D_PBLC, D_LEAF, D_INTR, and D_PRG are
   distinct, the input formats ending with different suffixes are all
   distinct.  It remains to show the distinctness of the inputs for each
   suffix.






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   The values of I and C are chosen uniformly at random from the set of
   all n*8 bit strings.  For n=32, it is highly likely that each value
   of I and C will be distinct, even when 2^96 such values are chosen.

   For D_ITER, D_PBLC, and D_MESG, the value of S = I || u32str(q) is
   distinct for each LMS leaf (or equivalently, for each q value).  For
   D_ITER, the value of u16str(i) || u8str(j) is distinct for each
   invocation of H for a given leaf.  For D_PBLC and D_MESG, the input
   format is used only once for each value of S, and thus distinctness
   is assured.  The formats for D_INTR and D_LEAF are used exactly once
   for each value of r, which ensures their distinctness.  For D_PRG,
   for a given value of I, q and j are distinct for each invocation of H
   (note that x_q[0] = SEED when j=0).

12.1.  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
   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



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   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.

12.2.  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, where u = ceil(8*n/w) is the size of the range
   that the checksum value can over), 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

      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].

13.  Acknowledgements

   Thanks are due to Chirag Shroff, Andreas Huelsing, Burt Kaliski, Eric
   Osterweil, Ahmed Kosba, and Russ Housley for constructive suggestions
   and valuable detailed review.  We especially 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.





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14.  References

14.1.  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>.

   [RFC3979]  Bradner, S., Ed., "Intellectual Property Rights in IETF
              Technology", BCP 79, RFC 3979, DOI 10.17487/RFC3979, March
              2005, <http://www.rfc-editor.org/info/rfc3979>.

   [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>.

   [RFC4879]  Narten, T., "Clarification of the Third Party Disclosure
              Procedure in RFC 3979", BCP 79, RFC 4879,
              DOI 10.17487/RFC4879, April 2007,
              <http://www.rfc-editor.org/info/rfc4879>.

   [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.

14.2.  Informative References

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

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






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   [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.  Pseudorandom Key Generation

   An implementation MAY use the following pseudorandom process for
   generating an LMS private key.

      SEED is an m-byte value that is generated uniformly at random at
      the start of the process,

      I is LMS keypair identifier,

      q denotes the LMS leaf number of an LM-OTS private key,

      x_q denotes the x array of private elements in the LM-OTS private
      key with leaf number q,

      j is an index of the private key element,

      D_PRG is a diversification constant, and

      H is the hash function used in LM-OTS.

   The elements of the LM-OTS private keys are computed as follows:

x_q[j] = / H(I || u32str(q) || SEED || u16str(j) || D_PRG)     if j = 0,
         \ H(I || u32str(q) || x_q[j-1] || u16str(j) || D_PRG) otherwise

   This process stretches the m-byte random value SEED into a (much
   larger) set of pseudorandom values, using both output feedback and a
   unique counter in each invocation of H.  The format of the inputs to



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   H are chosen so that they are distinct from all other uses of H in
   LMS and LM-OTS.

Appendix B.  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 5.

   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.

   +---------+------------+-----------+-----------+-------+------------+
   |   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 5



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Appendix C.  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]).

   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_num
       push H(I || OTS_PUBKEY[i+j] || u32str(r) || D_LEAF) on 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_num
         hash_update(I || u32str(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 D.  Example implementation







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        --- Pending Revision ---



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


   Scott Fluhrer
   Cisco Systems
   170 West Tasman Drive
   San Jose, CA
   USA

   Email: sfluhrer@cisco.com




















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