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Versions: 01 02

Crypto Forum Research Group                                    D. McGrew
Internet-Draft                                                 M. Curcio
Intended status: Informational                             Cisco Systems
Expires: August 26, 2013                               February 22, 2013


                         Hash-Based Signatures
                       draft-mcgrew-hash-sigs-01

Abstract

   This note describes a digital signature system based on cryptographic
   hash functions, following the seminal work in this area.  It
   specifies a one-time signature scheme based on the work of Lamport,
   Diffie, Winternitz, and Merkle (LDWM), and specifies a general
   signature system using a Merkle tree.  These systems provide
   asymmetric authentication without using large integer mathematics and
   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 August 26, 2013.

Copyright Notice

   Copyright (c) 2013 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.













































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Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  5
     1.1.  Conventions Used In This Document  . . . . . . . . . . . .  5

   2.  Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .  6
     2.1.  Data Types . . . . . . . . . . . . . . . . . . . . . . . .  6
       2.1.1.  Strings of w-bit elements  . . . . . . . . . . . . . .  7
     2.2.  Operators  . . . . . . . . . . . . . . . . . . . . . . . .  8
     2.3.  Functions  . . . . . . . . . . . . . . . . . . . . . . . .  9

   3.  LDWM One-Time Signatures . . . . . . . . . . . . . . . . . . . 10
     3.1.  Parameters . . . . . . . . . . . . . . . . . . . . . . . . 10
     3.2.  Hashing Functions  . . . . . . . . . . . . . . . . . . . . 11
     3.3.  Signature Methods  . . . . . . . . . . . . . . . . . . . . 11
     3.4.  Private Key  . . . . . . . . . . . . . . . . . . . . . . . 11
     3.5.  Public Key . . . . . . . . . . . . . . . . . . . . . . . . 12
     3.6.  Checksum . . . . . . . . . . . . . . . . . . . . . . . . . 12
     3.7.  Signature Generation . . . . . . . . . . . . . . . . . . . 13
     3.8.  Signature Verification . . . . . . . . . . . . . . . . . . 14
     3.9.  Notes  . . . . . . . . . . . . . . . . . . . . . . . . . . 14
     3.10. Formats  . . . . . . . . . . . . . . . . . . . . . . . . . 15

   4.  Merkle Tree Signatures . . . . . . . . . . . . . . . . . . . . 17
     4.1.  Private Key  . . . . . . . . . . . . . . . . . . . . . . . 17
     4.2.  MTS Public Key . . . . . . . . . . . . . . . . . . . . . . 17
     4.3.  MTS Signature  . . . . . . . . . . . . . . . . . . . . . . 18
       4.3.1.  MTS Signature Generation . . . . . . . . . . . . . . . 18
     4.4.  MTS Signature Verification . . . . . . . . . . . . . . . . 19
     4.5.  MTS Formats  . . . . . . . . . . . . . . . . . . . . . . . 21

   5.  Rationale  . . . . . . . . . . . . . . . . . . . . . . . . . . 22

   6.  History  . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

   7.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 24

   8.  Security Considerations  . . . . . . . . . . . . . . . . . . . 26
     8.1.  Security of LDWM Checksum  . . . . . . . . . . . . . . . . 26
     8.2.  Security Conjectures . . . . . . . . . . . . . . . . . . . 26
     8.3.  Post-Quantum Security  . . . . . . . . . . . . . . . . . . 27

   9.  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 28

   10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 29
     10.1. Normative References . . . . . . . . . . . . . . . . . . . 29
     10.2. Informative References . . . . . . . . . . . . . . . . . . 29




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   Appendix A.  LDWM Parameter Options  . . . . . . . . . . . . . . . 30

   Appendix B.  Example Data for Testing  . . . . . . . . . . . . . . 32
     B.1.  Parameters . . . . . . . . . . . . . . . . . . . . . . . . 32
     B.2.  Key Generation . . . . . . . . . . . . . . . . . . . . . . 32
     B.3.  Signature Generation . . . . . . . . . . . . . . . . . . . 38
     B.4.  Signature Verification . . . . . . . . . . . . . . . . . . 42
     B.5.  Intermediate Calculation Values  . . . . . . . . . . . . . 42

   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 48









































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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, and have benefited from
   renewed development in the last decade.  The characteristics of these
   signature systems are small private and public keys and low
   computational cost, but large signatures.  In recent years there has
   been interest in these systems because of their post-quantum security
   (see Section 8.3) and their suitability for compact implementations.

   This note describes the original Lamport-Diffie-Winternitz-Merkle
   (LDWM) one-time signature system (following Merkle 1979 but also
   using a technique from Merkle's later work [C:Merkle87][C:
   Merkle89a][C:Merkle89b]) and Merkle tree signature system (following
   Merkle 1979) with enough specificity to ensure interoperability
   between implementations.  While the specification of the algorithms
   follows these early references, the security considerations makes use
   of more recent security analyses (especially Buchmann, Dahmen, Ereth,
   Hulsing, and Ruckert 2011).

   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.  A general signature system can be used to sign multiple
   messages.  The Merkle Tree Signatures (MTS) is a general signature
   system that uses an OTS system as a component.  In principle the MTS
   can be used with any OTS system, but in this note we describe its use
   with the LDWM system.

   This note is structured as follows.  Notation is introduced in
   Section 2.  The LDWM signature system is described in Section 3, and
   the Merkle tree signature system is described in Section 4.
   Sufficient detail is provided to ensure interoperability.  Appendix B
   describes test considerations and contains test cases that can be
   used to validate an implementation.  The IANA registry for these
   signature systems is described in Section 7.  Security considerations
   are presented in Section 8.

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].





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2.  Notation

2.1.  Data Types

   Bytes and byte strings are the fundamental data type.  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
   of three.  An array of byte strings is an ordered set, indexed
   starting at zero, in which all strings have the same length.

   A byte can be considered as a sequence of eight bits, a sequence of
   four duets (two bit unsigned integers), or a sequence of two quartets
   (four bit unsigned integers).  A bit has value 0 or 1.  A duet has a
   value between 0 and 3, and a quartet has value between 0 and 15
   decimal (F hexadecimal).

   The correspondence between bytes, bits, duets, and quartets is shown
   below, where X denotes the value of the byte:

                       Byte (8-bit element)
                    +-----------+-----------+
                    |           X           |
                    +-----------+-----------+

                      Bits (1-bit elements)
                    +--+--+--+--+--+--+--+--+
                    |b0|b1|b2|b3|b4|b5|b6|b7|
                    +--+--+--+--+--+--+--+--+

                     Duets (2-bit elements)
                    +-----+-----+-----+-----+
                    |  d0 |  d1 |  d2 |  d3 |
                    +-----+-----+-----+-----+

                    Quartets (4-bit elements)
                    +-----------+-----------+
                    |     q0    |     q1    |
                    +-----------+-----------+

      X = 128*b0 + 64*b1 + 32*b2 + 16*b3 + 8*b4 + 4*b5 + 2*b6 + b7
      X = 64*d0 + 16*d1 + 4*d2 + d3
      X = 16*q0 + q1







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   For example, the following diagram shows how the byte with decimal
   value 27 (hexadecimal value 0x1b) is represented with bits, duets,
   and quartets.

                       Byte (8-bit element)
                    +-----------+-----------+
                    |          27           |
                    +-----------+-----------+

                     Bits (1-bit elements)
                    +--+--+--+--+--+--+--+--+
                    | 0| 0| 0| 1| 1| 0| 1| 1|
                    +--+--+--+--+--+--+--+--+

                     Duets (2-bit elements)
                    +-----+-----+-----+-----+
                    |  0  |  1  |  2  |  3  |
                    +-----+-----+-----+-----+

                    Quartets (4-bit elements)
                    +-----------+-----------+
                    |     1     |     b     |
                    +-----------+-----------+

2.1.1.  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.  For example, if S = 0x0204ff, then
   byte(S, 0) is 0x02.

   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)) )










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   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 Quartets)
         +-----------+-----------+-----------+-----------+
         |     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.

2.2.  Operators

   When a and b are 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 a divided 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-



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   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].

2.3.  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

   When F is a function that takes m-byte strings (i.e. byte strings of
   length m) as input and returns m-byte strings as output, we denote
   the repeated applications of F with itself a non-negative, integral
   number of times i as F^i.

   Thus for any byte string x,

       F^i(x) = / F( F^(i-1)(x) )  for i > 0
                \ x                for i = 0.

   For example, F^2(x) = F(F(x)).






















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3.  LDWM One-Time Signatures

   This section defines LDWM 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.

   Note that in order to constrain what constitutes the "message" and
   establish fixed sizes for the signature, a digest of the original
   message is computed using a collision-resistant hash function, H (see
   Section 3.2).

3.1.  Parameters

   The signature system uses the parameters m, n, and w; they are all
   positive integers.  The algorithm description also uses the value p.
   These parameters are summarized as follows:

      m : the length in bytes of each element of an LDWM signature

      n : the length in bytes of result of the hash function

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

      p : the number of m-byte string elements that make up the LDWM
      signature; it is equivalent to the number of w-bit elements of
      ( H(message) || C(H(message)) ) as well as the number of n-byte
      strings that form the private key; it can be specified in terms of
      u and v, as shown in Appendix A.

      ls : the number of bits the checksum function C will left-shift
      the sum total of hashing operations required by the verifier
      before returning a result

   The values of m and n are determined by the functions selected for
   use as part of the LDWM algorithm.  They are chosen to ensure an
   appropriate level of security.  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
   a direct result of the choices of n and w.  Appendix A describes how
   p and ls depend on the other parameters.  A table illustrating
   various combinations of n, w, p, and ls is provided in Table 4.






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3.2.  Hashing Functions

   The LDWM algorithm requires a robust one-way function to underpin the
   signature generation and verification.  Let this be defined as a hash
   function H that has inputs of any integral byte length and has n-byte
   outputs.

   In addition, let F be defined as a one-way function that has m-byte
   inputs and m-byte outputs.

3.3.  Signature Methods

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

   +--------------------+--------+-----------+----+----+---+-----+----+
   | Name               | H      | F         | m  | n  | w | p   | ls |
   +--------------------+--------+-----------+----+----+---+-----+----+
   | LDWM_SHA256_M20_W1 | SHA256 | SHA256-20 | 20 | 32 | 1 | 265 | 7  |
   |                    |        |           |    |    |   |     |    |
   | LDWM_SHA256_M20_W2 | SHA256 | SHA256-20 | 20 | 32 | 2 | 133 | 6  |
   |                    |        |           |    |    |   |     |    |
   | LDWM_SHA256_M20_W4 | SHA256 | SHA256-20 | 20 | 32 | 4 | 67  | 4  |
   |                    |        |           |    |    |   |     |    |
   | LDWM_SHA256_M20_W8 | SHA256 | SHA256-20 | 20 | 32 | 8 | 34  | 0  |
   +--------------------+--------+-----------+----+----+---+-----+----+

                                  Table 1

   Here SHA256 denotes the NIST standard hash function.  SHA256-20
   denotes that hash function with its final output truncated to 20
   bytes.

3.4.  Private Key

   The LDWM private key must be 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.









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   Algorithm 0: Generating A Private Key

     for i=0,i<p,i++
       set x[i] to a uniformly random value
     endfor
     return x

   Note that one possible implementation may consist of a single random
   input to a suitable key derivation function.

3.5.  Public Key

   The LDWM public key is generated from the private key through a
   series of hashing operations using the functions F and H. Its value
   is the hash (using H) of the concatenation of the elements of an
   array y.  The content of y is generated by iteratively hashing (using
   F) each element of array x, (2^w - 1) times.  The following algorithm
   shows pseudocode for generating the public key.

   Algorithm 1: Generating a Public Key From a Private Key

     e = 2^w - 1
     for i=0,i<p,i++
       y[i] = F^e(x[i])
     endfor
     return H(y[0] || y[1] || ... || y[p-1])

3.6.  Checksum

   A checksum is used to prevent the manipulation of an existing
   signature in an attempt to produce a new signature for a different
   message outside of the normal signing process.  The security property
   is detailed in Section 8.

   The checksum value is calculated using a non-negative integer, sum,
   whose width is sized 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 F as defined in the signing algorithm of
   Section 3.7 and the total actual number.  In the worst case (i.e. the
   actual number of times F 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 }, let sum be a 16-bit non-negative integer for
   all combinations of n and w.  The checksum function C is defined as
   follows where S is a byte string provided to the function as an
   argument.





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   Algorithm 2: Checksum Calculation

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

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

      Implementation Note: Based on the previous fact, the
      implementation may therefore choose to optimize the width of sum
      to (v * w) bits and set ls to 0.  The rationale for this is given
      that (2^w - 1) * ceil(8*n/w) is the maximum value of sum and the
      value of (2^w - 1) is represented by w bits, the result of
      repeatedly adding two w-bit numbers a total of u = ceil(8*n/w)
      times requires at most (floor(lg(u)) + w) bits.  Dividing by w and
      taking the next largest integer gives the total required number of
      w-bit fields and gives (ceil(floor(lg(u)) / w) + 1), or v.  Thus
      sum requires a minimum width of (v * w) bits and no left-shift
      operation is performed.

3.7.  Signature Generation

   The LDWM signature is generated by using the values of the w-bit
   fields that compose the hash of the message and its checksum to
   determine the number of times to apply the function F to the elements
   of the private key.  This signature is provided by the signer to the
   verifier along with the message and the public key.

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

     V = ( H(message) || C(H(message)) )
     for i=0,i<p,i++
       a = coef(V, i, w)
       y[i] = F^a(x[i])
     endfor
     return ( 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 3.5.







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3.8.  Signature Verification

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

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

     V = ( H(message) || C(H(message)) )
     for i=0,i<p,i++
       a = (2^w - 1) - coef(V, i, w)
       z[i] = F^a(y'[i])
     endfor
     if public key is equal to H(z[0] || z[1] || ... || z[p-1])
       return 1 (message signature is valid)
     else
       return 0 (message signature is invalid)

3.9.  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.

























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3.10.  Formats

   The signature format is defined using XDR [RFC4506] as follows:

   /*
    * ldwm_algorithm_type identifies a particular signature algorithm
    */
   enum ldwm_algorithm_type {
     ldwm_reserved = 0,
     ldwm_sha256_m20_w1  = 1,
     ldwm_sha256_m20_w2  = 2,
     ldwm_sha256_m20_w4  = 3,
     ldwm_sha256_m20_w8  = 4
   };

   /*
    * basic data types; b16 is a string of 16 bytes, and so on
    */
   typedef opaque b16[16];
   typedef opaque b20[20];
   typedef opaque b24[24];
   typedef opaque b28[28];
   typedef opaque b32[32];

   /*
    * arrays
    */
   typedef b20 b20_array_265[265];
   typedef b20 b20_array_133[133];
   typedef b20 b20_array_67[67];
   typedef b20 b20_array_34[34];

   union ldwm_signature switch (ldwm_algorithm_type type) {
    case ldwm_sha256_m20_w1:
      b20_array_265 y265;
    case ldwm_sha256_m20_w2:
      b20_array_133 y133;
    case ldwm_sha256_m20_w4:
      b20_array_67 y67;
    case ldwm_sha256_m20_w8:
      b20_array_34 y34;
    default:
      void;   /* error condition */
   };

   Though the data formats are formally defined by XDR, we describe the
   format as well as a convenience to the reader.  An example of the
   format of an ldwm_signature is illustrated below, for



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   ldwm_sha256_m20_w1, which has 265 elements, each of which is a 20-
   octet string.  An ldwm_signature always consists of a 32-bit unsigned
   integer that indicates the ldwm_algorithm_type followed by an array
   of equal-length octet strings.  The number of octets in each octet
   string, and the number of elements in the array, are determined by
   the ldwm_algorithm_type field.  A receiver MUST check the
   ldwm_algorithm_type field, and a verification operation on a
   signature with an unknown ldwm_algorithm_type MUST return FAIL.

            +---------------------------------+
            |      ldwm_algorithm_type        |
            +---------------------------------+
            |                                 |
            |           y265[0]               |
            |                                 |
            |                                 |
            +---------------------------------+
            |                                 |
            |           y265[1]               |
            |                                 |
            |                                 |
            |                                 |
            +---------------------------------+
            |                                 |
            ~             ....                ~
            |                                 |
            +---------------------------------+
            |                                 |
            |          y265[264]              |
            |                                 |
            |                                 |
            |                                 |
            +---------------------------------+


















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4.  Merkle Tree Signatures

   Merkle Tree Signatures (MTS) are a method for signing a large but
   fixed number of messages.  An MTS system uses two cryptographic
   components: a one-time signature method and a collision-resistant
   hash function.  Each MTS public/private key pair is associated with a
   k-way tree, each node of which contains an n-byte value.  Each leaf
   of the tree contains the value of the public key of an LDWM public/
   private key pair.  The value contained by the root of the tree is the
   MTS public key.  Each interior node is computed by applying the hash
   function to the concatenation of the values of its children nodes.

   An MTS system has the following parameters:

      k : the number of children nodes of an interior node,

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

      n : the number of bytes associated with each node.

   There are k^h leaves in the tree.

4.1.  Private Key

   An MTS private key consists of k^h LDWM private keys and the leaf
   number of the next LDWM private key that has not yet been used.  The
   leaf number is initialized to zero when the MTS private key is
   created.

   An MTS private key MAY be generated pseudorandomly from a secret
   value, in which case the secret value MUST be n bytes long, be
   uniformly random, and MUST NOT be used for any other purpose than the
   generation of the MTS 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.

4.2.  MTS Public Key

   An MTS public key is defined as follows, where we denote the public
   key associated with the i^th LDWM private key as ldwm_public_key(i).

   The MTS public key can be computed using the following algorithm or
   any equivalent method.  The algorithm uses a stack of hashes and a
   separate stack of integers which keeps track of the level of the
   tree.






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   for i = 0 to num_ldwm_keys by steps of k
     level = 0
     for j = 0 to k-1
       push ldwm_public_key(i+j) on the data stack
       push level on the integer stack
     endfor
     while the height of the stack is at least k
       if the top k elements on the integer stack are equal
         pop the top k elements of the data stack
         pop the top k elements of the integer stack,
           and set the variable "level" to be their value
         hash the top k elements of the data stack
         push the hash result on the data stack
         push (level+1) on to the integer stack
       endif
     endwhile
   endfor

   Note that the stack never gets bigger than the logarithm of the
   number of LDWM public keys, so for typical parameters it will have
   something like 20 32-byte elements.

4.3.  MTS Signature

   An MTS signature consists of

      an LDWM signature,

      a node number that identifies the leaf node associated with the
      signature, and

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

   The array of values contains 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 branching number k and
   height h will have (k-1)*h values.  The first (k-1) values are the
   siblings of the leaf, the next (k-1) values are the siblings of the
   parent of the leaf, and so on.

4.3.1.  MTS Signature Generation

   To compute the MTS signature of a message with an MTS private key,
   the signer first computes the LDWM signature of the message using the
   leaf number of the next unused LDWM private key.  Before releasing
   the signature, the leaf number in the MTS private key MUST be
   incremented to prevent the LDWM private key from being used again.



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   The node number in the signature is set to the leaf number of the MTS
   private key that was used in the signature.

   The array of node values MAY be computed in any way.  There are many
   potential time/storage tradeoffs.  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 public
   key verification algorithm.

4.4.  MTS Signature Verification

   An MTS signature is verified by first using the LDWM signature
   verification algorithm to compute the LDWM public key from the LDWM
   signature and the message.  The value of the leaf associated with the
   LDWM signature is assigned to the public key.  Then the root of the
   tree is computed from the leaf value and the node array as described
   below.  If the root value matches the public key, then the signature
   is valid; otherwise, the signature fails.

   An efficient way to compute the root from the leaf and the node array
   is as follows, where n is initially the node number:

     v = leaf
     step = 0
     for i=0 to h-1 by steps of 1
       position = n % k
       hash_init()
       for j=0 to position-1 by steps of 1
         hash_update(node[step + j])
       endfor
       hash_update(v)
       for j=position to (k-1) by steps of 1
         hash_update(node[step + j])
       endfor
       v = hash_final()
       n = floor(n/k)
       step = step + (k-1)
     endfor

   This algorithm uses 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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   This algorithm works because the leaves of the MTS tree are numbered
   starting at zero.  Therefore leaf n is in the position (n % k) in the
   highest level of the tree.

   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 MTS public key.

   A full test example that combines the LDWM OTS and MTS algorithms is
   given in Appendix B.






































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4.5.  MTS Formats

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

   enum mts_algorithm_type {
     mts_reserved      = 0,
     mts_sha256_k2_h20 = 1,
     mts_sha256_k4_h10 = 2,
     mts_sha256_k8_h7  = 3,
     mts_sha256_k16_h5 = 4
   };

   union mts_path switch (mts_algorithm_type type) {
    case mts_sha256_k2_h20:
      b32 t20[20];
    case mts_sha256_k4_h10:
      b32 t30[30];
    case mts_sha256_k8_h7:
      b32 t49[49];
    case mts_sha256_k16_h5:
      b32 t75[75];
    default:
      void;     /* error condition */
   };

   struct mts_signature_t {
     ldwm_signature ldwm_sig;
     unsigned int signature_leaf_number;
     mts_path nodes;
   };

   union mts_public_key switch (mts_algorithm_type type) {
    case mts_sha256_k2_h20:
    case mts_sha256_k4_h10:
    case mts_sha256_k8_h7:
    case mts_sha256_k16_h5:
      b32 z;
    default:
      void;     /* error condition */
   };










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5.  Rationale

   The goal of this note is to describe the LDWM and MTS 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.

   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.







































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6.  History

   This is the initial version of this draft.

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














































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7.  IANA Considerations

   The Internet Assigned Numbers Authority (IANA) is requested to create
   two registries: one for LDWM signatures, as defined in Section 3, and
   one for Merkle Tree 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 "MTS_SHA256_K16_H5", that starts with the
      strings "MTS" and "LDWM" for the MTS and LDWM registries,
      respectively,

      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 positive 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 32,768 (binary 1000000000000000) and 65,535
   (binary 1111111111111111) 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 LDWM registry is as follows.















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          +--------------------+-----------+--------------------+
          | Name               | Reference | Numeric Identifier |
          +--------------------+-----------+--------------------+
          | LDWM_SHA256_M20_W1 | Section 3 |          1         |
          |                    |           |                    |
          | LDWM_SHA256_M20_W2 | Section 3 |          2         |
          |                    |           |                    |
          | LDWM_SHA256_M20_W4 | Section 3 |          3         |
          |                    |           |                    |
          | LDWM_SHA256_M20_W8 | Section 3 |          4         |
          +--------------------+-----------+--------------------+

                                  Table 2

   The MTS registry is as follows.

          +-------------------+-----------+--------------------+
          | Name              | Reference | Numeric Identifier |
          +-------------------+-----------+--------------------+
          | MTS_SHA256_K2_H20 | Section 4 |          1         |
          |                   |           |                    |
          | MTS_SHA256_K4_H10 | Section 4 |          2         |
          |                   |           |                    |
          | MTS_SHA256_K8_H7  | Section 4 |          3         |
          |                   |           |                    |
          | MTS_SHA256_K16_H5 | Section 4 |          4         |
          +-------------------+-----------+--------------------+

                                  Table 3

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



















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8.  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").

   LDWM signatures rely on the fact that, given an m-byte string y, it
   is prohibitively expensive to compute a value x such that F^i(x) = y
   for any i.  Informally, F is said to be a "one-way" function.

8.1.  Security of LDWM Checksum

   To show the security of LDWM checksum, we consider the signature y of
   a message with a private key x and let h = H(message) and
   c = C(H(message)) (see Section 3.7).  To attempt a forgery, an
   attacker can 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 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 this action, the
   checksum will decrease, and 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].

8.2.  Security Conjectures

   LDWM and MTS signatures have a minimum of security conjectures.  In
   particular, their security does not rely on the computational
   difficulty of factoring composites with large prime factors (as does



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   RSA) or the difficulty of computing the discrete logarithm in a
   finite field (as does DSA) or an elliptic curve group (as does
   ECDSA).  All of these signature schemes rely on the security of the
   hash function that they use, but with LDWM and MTS, the security of
   the hash function is sufficient.

8.3.  Post-Quantum Security

   A post-quantum cryptosystem is a system that is secure against
   quantum computers that have more than a trivial number of quantum
   bits.  It is open to conjecture whether or not it is feasible to
   build such a machine.

   The LDWM and Merkle signature systems are post-quantum secure if they
   are used with an appropriate underlying hash function, in which the
   size of m and n are double what they would be otherwise, in order to
   protect against quantum square root attacks due to Grover's
   algorithm.  In contrast, the signature systems in wide use (RSA, DSA,
   and ECDSA) are not post-quantum secure.
































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9.  Acknowledgements

   Thanks are due to Chirag Shroff for constructive feedback.
















































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10.  References

10.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC2434]  Narten, T. and H. Alvestrand, "Guidelines for Writing an
              IANA Considerations Section in RFCs", BCP 26, RFC 2434,
              October 1998.

   [RFC4506]  Eisler, M., "XDR: External Data Representation Standard",
              STD 67, RFC 4506, May 2006.

10.2.  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.

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

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


















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Appendix A.  LDWM Parameter Options

   A table illustrating various combinations of n, w, p, and ls is
   provided in Table 4.

   The parameters p and ls are computed as follows:

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

   Here u and v represent the number of w-bit fields required to contain
   the hash of the message message and the checksum byte strings,
   respectively.  For a further explanation of the values of v and sum,
   see Section 3.6.



































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   +------------+--------------+--------------+----------+-------------+
   |    Hash    |  Winternitz  |     w-bit    |   Left   |  Number of  |
   |  Length in |   Parameter  |  Elements in |   Shift  |   Elements  |
   |  Bytes (n) |      (w)     |   Checksum   |   (ls)   |     (p)     |
   +------------+--------------+--------------+----------+-------------+
   |     20     |       1      |       8      |     8    |     168     |
   |            |              |              |          |             |
   |     20     |       2      |       4      |     8    |      84     |
   |            |              |              |          |             |
   |     20     |       4      |       3      |     4    |      43     |
   |            |              |              |          |             |
   |     20     |       8      |       2      |     0    |      22     |
   |            |              |              |          |             |
   |     32     |       1      |       9      |     7    |     265     |
   |            |              |              |          |             |
   |     32     |       2      |       5      |     6    |     133     |
   |            |              |              |          |             |
   |     32     |       4      |       3      |     4    |      67     |
   |            |              |              |          |             |
   |     32     |       8      |       2      |     0    |      34     |
   |            |              |              |          |             |
   |     48     |       1      |       9      |     7    |     393     |
   |            |              |              |          |             |
   |     48     |       2      |       5      |     6    |     197     |
   |            |              |              |          |             |
   |     48     |       4      |       3      |     4    |      99     |
   |            |              |              |          |             |
   |     48     |       8      |       2      |     0    |      50     |
   |            |              |              |          |             |
   |     64     |       1      |      10      |     6    |     522     |
   |            |              |              |          |             |
   |     64     |       2      |       5      |     6    |     261     |
   |            |              |              |          |             |
   |     64     |       4      |       3      |     4    |     131     |
   |            |              |              |          |             |
   |     64     |       8      |       2      |     0    |      66     |
   +------------+--------------+--------------+----------+-------------+

                                  Table 4












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Appendix B.  Example Data for Testing

   As with all cryptosystems, implementations of LDWM signatures and
   Merkle signatures need to be tested before they are used.  This
   section contains sample data generated from the signing and
   verification operations of software that implements the algorithms
   described in this document.

B.1.  Parameters

   The example contained in this section demonstrates the calculations
   of LDWM_SHA256_M20_W4 using a Merkle Tree Signature of degree 4 and
   height 2.  This corresponds to the following parameter values:

                     +----+----+---+----+---+---+---+
                     |  m |  n | w |  p | l | k | h |
                     +----+----+---+----+---+---+---+
                     | 20 | 32 | 4 | 67 | 4 | 4 | 2 |
                     +----+----+---+----+---+---+---+

                                  Table 5

   The non-standard size of the Merkle tree (h = 2) has been selected
   specifically for this example to reduce the amount of data presented.

B.2.  Key Generation

   The LDWM algorithm does not define a required method of key
   generation.  This is left to the implementer.  The selected method,
   however, must satisfy the requirement that the public/private key
   pairs of the one-time signatures are unique.  In addition, all LDWM
   key pairs must be generated in advance in order to calculate the
   value of the Merkle public key.

   For the test data presented here, a summary of the key generation
   method is as follows:

   1.  MTS Private Key - Set mts_private_key to a pseudorandomly
       generated n-byte value.

   2.  OTS Private Keys - Use the mts_private_key as a key derivation
       key input to some key derivation function, thereby producing n^k
       derived keys.  Then use each derived key as an input to the same
       function again to further derive p elements of n-bytes each.
       This accomplishes the result of Algorithm 0 of Section 3.4 for
       each leaf of the Merkle tree.





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   3.  OTS Public Keys - For each OTS private key, calculate the
       corresponding OTS public key as in Algorithm 1 of Section 3.5.

   4.  MTS Public Key - Each OTS public key is the value of a leaf on
       the Merkle tree.  Calculate the MTS public key using the
       pseudocode algorithm of Section 4.2.

   The above steps result in the following data values associated with
   the first leaf of the Merkle tree, leaf 0.

   +-------------------------------------------------------------------+
   |                          MTS Private Key                          |
   +-------------------------------------------------------------------+
   |                 0x0f677ff1b4cbf10baec89959f051f203                |
   |                   3371492da02f62dd61d6fbd1cee1bd14                |
   +-------------------------------------------------------------------+

                                  Table 6

   +-----------------+-------------------------------------------------+
   |   Key Element   |         OTS Private Key 0 Element (x[i])        |
   |    Index (i)    |                                                 |
   +-----------------+-------------------------------------------------+
   |        0        |        0xbfb757383fb08d324629115a84daf00b       |
   |                 |          188d5695303c83c184e1ec7a501c431f       |
   |                 |                                                 |
   |        1        |        0x7ce628fb82003a2829aab708432787d0       |
   |                 |          fc735a29d671c7d790068b453dc8c913       |
   |                 |                                                 |
   |        2        |        0x8174929461329d15068a4645a34412bd       |
   |                 |          446d4c9e757463a7d5164efd50e05c93       |
   |                 |                                                 |
   |        3        |        0xf283f3480df668de4daa74bb0e4c5531       |
   |                 |          5bc00f7d008bb6311e59a5bbca910fd7       |
   |                 |                                                 |
   |        4        |        0xe62708eaf9c13801622563780302a068       |
   |                 |          0ba9d39c078daa5ebc3160e1d80a1ea7       |
   |                 |                                                 |
   |        5        |        0x1f002efad2bfb4275e376af7138129e3       |
   |                 |          3e88cf7512ec1dcdc7df8d5270bc0fd7       |
   |                 |                                                 |
   |        6        |        0x8ed5a703e9200658d18bc4c05dd0ca8a       |
   |                 |          356448a26f3f4fe4e0418b52bd6750a2       |
   |                 |                                                 |
   |        7        |        0xc74e56d61450c5387e86ddad5a8121c8       |
   |                 |          8b1bc463e64f248a1f1d91d950957726       |
   |                 |                                                 |




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   |        8        |        0x629f18b6a2a4ea65fff4cf758b57333f       |
   |                 |          e1d34af05b1cd7763696899c9869595f       |
   |                 |                                                 |
   |        9        |        0x1741c31fdbb4864712f6b17fadc05d45       |
   |                 |          926c831c7a755b7d7af57ac316ba6c2a       |
   |                 |                                                 |
   |        10       |        0xe59a7b81490c5d1333a9cdd48b9cb364       |
   |                 |          56821517a3a13cb7a8ed381d4d5f3545       |
   |                 |                                                 |
   |        11       |        0x3ba97fe8b2967dd74c8b10f31fc5f527       |
   |                 |          a23b89c1266202a4d7c281e1f41fa020       |
   |                 |                                                 |
   |        12       |        0xa262a9287cc979aaa59225d75df51b82       |
   |                 |          57b92e780d1ab14c4ac3ecdac58f1280       |
   |                 |                                                 |
   |        13       |        0x9dfe0af1a3d9064338d96cb8eae88baa       |
   |                 |          6a69265538873b4c17265fa9d573bcff       |
   |                 |                                                 |
   |        14       |        0xde9c5c6a5c6a274eabe90ed2a8e6148c       |
   |                 |          720196d237a839aaf5868af8da4d0829       |
   |                 |                                                 |
   |        15       |        0x5de81ec17090a82cb722f616362d3808       |
   |                 |          30f04841191e44f1f81b9880164b14cd       |
   |                 |                                                 |
   |        16       |        0xc0d047000604105bad657d9fa2f9ef10       |
   |                 |          1cfd9490f4668b700d738f2fa9e1d11a       |
   |                 |                                                 |
   |        17       |        0xf45297ef310941e1e855f97968129bb1       |
   |                 |          73379193919f7b0fee9c037ae507c2d2       |
   |                 |                                                 |
   |        18       |        0x46ef43a877f023e5e66bbcd4f06b839f       |
   |                 |          3bfb2b64de25cd67d1946b0711989129       |
   |                 |                                                 |
   |        19       |        0x46e2a599861bd9e8722ad1b55b8f0139       |
   |                 |          305fcf8b6077d545d4488c4bcb652f29       |
   |                 |                                                 |
   |        20       |        0xe1ad4d2d296971e4b0b7a57de305779e       |
   |                 |          82319587b58d3ef4daeb08f630bd5684       |
   |                 |                                                 |
   |        21       |        0x7a07fa7aed97cb54ae420a0e6a58a153       |
   |                 |          38110f7743cab8353371f8ca710a4409       |
   |                 |                                                 |
   |        22       |        0x40601f6c4b35362dd4948d5687b5cb6b       |
   |                 |          5ec8b2ec59c2f06fd50f8919ebeaae92       |
   |                 |                                                 |
   |        23       |        0xa061b0ba9f493c4991be5cd3a9d15360       |
   |                 |          a9eb94f6f7adc28dddf174074f3df3c4       |
   |                 |                                                 |



McGrew & Curcio          Expires August 26, 2013               [Page 34]

Internet-Draft            Hash-Based Signatures            February 2013


   |        24       |        0xcf1546a814ff16099cebf1fe0db1ace5       |
   |                 |          1c272fda9846fbb535815924b0077fa4       |
   |                 |                                                 |
   |        25       |        0xcbb06f13155ce4e56c85a32661c90142       |
   |                 |          8b630a4c37ea5c7062156f07f6b3efff       |
   |                 |                                                 |
   |        26       |        0x1181ee7fc03342415094e36191eb450a       |
   |                 |          11cdea9c6f6cdc34de79cee0ba5bf230       |
   |                 |                                                 |
   |        27       |        0xe9f1d429b343bb897881d2a19ef363cd       |
   |                 |          1ab4117cbaad54dc292b74b8af9f5cf2       |
   |                 |                                                 |
   |        28       |        0x87f34b2551ef542f579fa65535c5036f       |
   |                 |          80eb83be4c898266ffc531da2e1a9122       |
   |                 |                                                 |
   |        29       |        0x9b4b467852fe33a03a872572707342fd       |
   |                 |          ddeae64841225186babf353fa2a0cd09       |
   |                 |                                                 |
   |        30       |        0x19d58cd240ab5c80be6ddf5f60d18159       |
   |                 |          2dca2be40118c1fdd46e0f14dffbcc7d       |
   |                 |                                                 |
   |        31       |        0x5c9ad386547ba82939e49c9c74a8eccf       |
   |                 |          1cea60aa327b5d2d0a66b1ca48912d6d       |
   |                 |                                                 |
   |        32       |        0xf49083e502400ffae9273c6de92a301e       |
   |                 |          7bda1537cab085e5adfa9eb746e8eca9       |
   |                 |                                                 |
   |        33       |        0x4074e1812d69543ce3c1ce706f6e0b45       |
   |                 |          f5f26f4ef39b34caa709335fd71e8fc0       |
   |                 |                                                 |
   |        34       |        0x1256612b0ca8398e97b247ae564b74b1       |
   |                 |          3839b3b1cf0a0dd8ba629a2c58355f84       |
   |                 |                                                 |
   |        35       |        0xbab3989f00fd2c327bbfb35a218cc3ce       |
   |                 |          49d6b34cbf8b6e8919e90c4eff400ca9       |
   |                 |                                                 |
   |        36       |        0x96b52a5d395a5615b73dae65586ac5c8       |
   |                 |          7f9dd3b9b3f82dbf509b5881f0643fa8       |
   |                 |                                                 |
   |        37       |        0x5d05ca4c644e1c41ccdaedbd2415d4f0       |
   |                 |          9b4a1b940b51fe823dff7617b8ee8304       |
   |                 |                                                 |
   |        38       |        0xd96aab95ef6248e235d91d0f23b64727       |
   |                 |          a6675adfc64efea72f6f8b4a47996c0d       |
   |                 |                                                 |
   |        39       |        0xfd9c384d52d3ac27c4f4898fcc15e83a       |
   |                 |          c182f97ea63f7d489283e2cc7e6ed180       |
   |                 |                                                 |



McGrew & Curcio          Expires August 26, 2013               [Page 35]

Internet-Draft            Hash-Based Signatures            February 2013


   |        40       |        0xc86eaed6a9e3fbe5b262c1fa1f099f7c       |
   |                 |          35ece71d9e467fab7a371dbcf400b544       |
   |                 |                                                 |
   |        41       |        0xf462b3719a2ed8778155638ff814dbf4       |
   |                 |          2b107bb5246ee3dd82abf97787e6a69e       |
   |                 |                                                 |
   |        42       |        0x014670912e3eb74936ebb64168b447e4       |
   |                 |          2522b57c2540ac4b49b9ae356c01eca6       |
   |                 |                                                 |
   |        43       |        0x2b411096e0ca16587830d3acd673e858       |
   |                 |          863fedc4cea046587cba0556d2bf9884       |
   |                 |                                                 |
   |        44       |        0xa73917c74730582e8e1815b8a07b1896       |
   |                 |          2ac05e500e045676be3f1495fcfa18ca       |
   |                 |                                                 |
   |        45       |        0xa4ab61e6962fe39a255dbf8a46d25110       |
   |                 |          0d127fab08db59512653607bda24302c       |
   |                 |                                                 |
   |        46       |        0x9b910ca516413f376b9eba4b0d571b22       |
   |                 |          253c2a9646131ac9a2af5f615f7322b8       |
   |                 |                                                 |
   |        47       |        0xfc1b4ce627c77ad35a21ea9ded2cce91       |
   |                 |          b3758a758224e35cf2918153a513d64c       |
   |                 |                                                 |
   |        48       |        0xc1902d8e8c02d9442581d7e053a2798a       |
   |                 |          a84d77a74b6e7f2cc5096d50646c890f       |
   |                 |                                                 |
   |        49       |        0xb3f47e2e8e2dcdd890ea00934b9d8234       |
   |                 |          830dbc4a30ac996b144f12b3e463c77f       |
   |                 |                                                 |
   |        50       |        0x8188d1ecfc6ae6118911f2b9b3a6c7a1       |
   |                 |          e5f909aa8b5c0aab8c69f1a7d436c307       |
   |                 |                                                 |
   |        51       |        0xca42d985974c7b870bc76494604eff49       |
   |                 |          2676c942c6cb7c75d4938805885dd054       |
   |                 |                                                 |
   |        52       |        0xbe58851ebe566057e1ee16b8c604a473       |
   |                 |          4c373af622660b2a82357ac6effb4566       |
   |                 |                                                 |
   |        53       |        0xc22d493f7a5642fceba2404dbefa8f95       |
   |                 |          6323fac87fac425f6de8d23c9e8b20ca       |
   |                 |                                                 |
   |        54       |        0x1a76c1ffa467906173fd0245b0cd6639       |
   |                 |          e6013ca79c4ed92426ee69ff5beeac0b       |
   |                 |                                                 |
   |        55       |        0xbc6c0cb7808f379af1b7b7327436ad65       |
   |                 |          c05458f2d0a6923c333e5129c4c99671       |
   |                 |                                                 |



McGrew & Curcio          Expires August 26, 2013               [Page 36]

Internet-Draft            Hash-Based Signatures            February 2013


   |        56       |        0xfbb04488c3c088dc5e63d13e6a701036       |
   |                 |          6109ca4c5f4b0a8d37780187e2e9930e       |
   |                 |                                                 |
   |        57       |        0xaec10811569d4d72e3a1baf71a886b75       |
   |                 |          eba6dc07ed027af0b2beffa71f9b43c8       |
   |                 |                                                 |
   |        58       |        0xf5529be3b7a19212e8baa970d2420bf4       |
   |                 |          123f678267f96c1c3ef26ab610cb0061       |
   |                 |                                                 |
   |        59       |        0x172ba1ba0b701eeafe00692d1eb90181       |
   |                 |          8ccaefaeb8f799395da81711766d1f43       |
   |                 |                                                 |
   |        60       |        0xfe1f8c15825208f3a21346b894b3d94e       |
   |                 |          4f3aa29cbc194a7b2c8a810c4c509042       |
   |                 |                                                 |
   |        61       |        0x2e81c66cc914ea1b0fa5942fe9780d54       |
   |                 |          8c0b330e3bf73f0cb0bda4bc9c9e6ff4       |
   |                 |                                                 |
   |        62       |        0xfc3453aec5cc19a6a4bda4bc25931604       |
   |                 |          704bf4386cd65780c6e73214c1da85ba       |
   |                 |                                                 |
   |        63       |        0x4e8000c587dc917888e7e3d817672c0a       |
   |                 |          ef812788cc8579afa7e9b2e566309003       |
   |                 |                                                 |
   |        64       |        0xba667ca0e44a8601a0fde825d4d2cf1b       |
   |                 |          b9cf467041e04af84c9d0cd9fd8dc784       |
   |                 |                                                 |
   |        65       |        0x4965db75f81c8a596680753ce70a94c6       |
   |                 |          156253bb426947de1d7662dd7e05e9a8       |
   |                 |                                                 |
   |        66       |        0x2c23cc3e5ca37dec279c506101a3d8d9       |
   |                 |          f1e4f99b2a33741b59f8bddba7455419       |
   +-----------------+-------------------------------------------------+

                                  Table 7

   Using the value of the OTS private key above, the corresponding
   public key is given below.  Intermediate values of the SHA-256-20
   function F^(2^w - 1)(x[i]) are provided in Table 13.

   +-------------------------------------------------------------------+
   |                          OTS Public Key 0                         |
   +-------------------------------------------------------------------+
   |                 0x2db55a72075fcfab5aedbef77bf6b371                |
   |                   dfb489d6e61ad2884a248345e6910618                |
   +-------------------------------------------------------------------+

                                  Table 8



McGrew & Curcio          Expires August 26, 2013               [Page 37]

Internet-Draft            Hash-Based Signatures            February 2013


   Following the creation of all OTS public/private key pairs, the OTS
   public keys in Table 14 are used to determine the MTS public key
   below.  Intermediate values of the interior nodes of the Merkle tree
   are provided in Table 15.

   +-------------------------------------------------------------------+
   |                           MTS Public Key                          |
   +-------------------------------------------------------------------+
   |                 0x6610803d9a3546fb0a7895f6a4a0cfed                |
   |                   3a07d45e51d096e204b018e677453235                |
   +-------------------------------------------------------------------+

                                  Table 9

B.3.  Signature Generation

   In order to test signature generation, a text file containing the
   content "Hello world!\n", where '\n' represents the ASCII line feed
   character, was created and signed.  A raw hex dump of the file
   contents is shown in the table below.

   +-------------------------------+-----------------------------------+
   |    Hexadecimal Byte Values    |        ASCII Representation       |
   |                               |      ('.' is substituted for      |
   |                               |      non-printing characters)     |
   +-------------------------------+-----------------------------------+
   | 0x48 0x65 0x6c 0x6c 0x6f 0x20 |           Hello world!.           |
   | 0x77 0x6f 0x72 0x6c 0x64 0x21 |                                   |
   |              0x0a             |                                   |
   +-------------------------------+-----------------------------------+

                                 Table 10

   The SHA256 hash of the text file is provided below.

   +-------------------------------------------------------------------+
   |          SHA256 Hash of Signed File (H("Hello world!\n"))         |
   +-------------------------------------------------------------------+
   |                0x0ba904eae8773b70c75333db4de2f3ac4                |
   |                  5a8ad4ddba1b242f0b3cfc199391dd81c                |
   +-------------------------------------------------------------------+

                                 Table 11

   This value was subsequently used in Algorithm 3 of Section 3.7 to
   create the one-time signature of the message.  Algorithm 2 of
   Section 3.6 was applied to calculate a checksum of 0x1cc.  The
   resulting signature is shown in the following table.



McGrew & Curcio          Expires August 26, 2013               [Page 38]

Internet-Draft            Hash-Based Signatures            February 2013


   +---------+------------+--------------------------------------------+
   |   OTS   |  Function  |       OTS Element (y[i] = F^a(x[i]))       |
   | Element |  Iteration |                                            |
   |  Index  |    Count   |                                            |
   |   (i)   | (a = coef( |                                            |
   |         |  H(msg) || |                                            |
   |         | C(H(msg)), |                                            |
   |         |   i, w ))  |                                            |
   +---------+------------+--------------------------------------------+
   |    0    |      0     | 0xbfb757383fb08d324629115a84daf00b188d5695 |
   |         |            |                                            |
   |    1    |     11     | 0x4af079e885ddfd3245f29778d265e868a3bfeaa4 |
   |         |            |                                            |
   |    2    |     10     | 0xfbad1928bfc57b22bcd949192452293d07d6b9ad |
   |         |            |                                            |
   |    3    |      9     | 0xb98063e184b4cb949a51e1bb76d99d4249c0b448 |
   |         |            |                                            |
   |    4    |      0     | 0xe62708eaf9c13801622563780302a0680ba9d39c |
   |         |            |                                            |
   |    5    |      4     | 0x39343cba3ffa6d75074ce89831b3f3436108318c |
   |         |            |                                            |
   |    6    |     14     | 0xfe08aa73607aec5664188a9dacdc34a295588c9a |
   |         |            |                                            |
   |    7    |     10     | 0xd3346382119552d1ceb92a78597a00c956372bf0 |
   |         |            |                                            |
   |    8    |     14     | 0xf1dd245ec587c0a7a1b754cc327b27c839a6e46a |
   |         |            |                                            |
   |    9    |      8     | 0xa5f158adc1decaf0c1edc1a3a5d8958d726627b5 |
   |         |            |                                            |
   |    10   |      7     | 0x06d2990f62f22f0c943a418473678e3ffdbff482 |
   |         |            |                                            |
   |    11   |      7     | 0xf3390b8d6e5229ae9c5d4c3f45e10455d8241a49 |
   |         |            |                                            |
   |    12   |      3     | 0x22dd5f9d3c89180caa0f695203d8cf90f3c359be |
   |         |            |                                            |
   |    13   |     11     | 0x67999c4043f95de5f07d82b741347a3eb6ac0c25 |
   |         |            |                                            |
   |    14   |      7     | 0xc4ffe472d48adeb37c7360da70711462013b7a4e |
   |         |            |                                            |
   |    15   |      0     | 0x5de81ec17090a82cb722f616362d380830f04841 |
   |         |            |                                            |
   |    16   |     12     | 0x2f892c824af65cc749f912a36dfa8ade2e4c3fd1 |
   |         |            |                                            |
   |    17   |      7     | 0xb644393e8030924403b594fb5cacd8b2d28862e2 |
   |         |            |                                            |
   |    18   |      5     | 0x31b8d2908911dbbf5ba1f479a854808945d9e948 |
   |         |            |                                            |
   |    19   |      3     | 0xa9a02269d24eb8fed6fb86101cbd0d8977219fb1 |



McGrew & Curcio          Expires August 26, 2013               [Page 39]

Internet-Draft            Hash-Based Signatures            February 2013


   |    20   |      3     | 0xe4aae6e6a9fe1b0d5099513f170c111dee95714d |
   |         |            |                                            |
   |    21   |      3     | 0xd79c16e7f2d4dd790e28bab0d562298c864e31e9 |
   |         |            |                                            |
   |    22   |     13     | 0xc29678f0bb4744597e04156f532646c98a0b42e8 |
   |         |            |                                            |
   |    23   |     11     | 0x57b31d75743ff0f9bcf2db39d9b6224110b8d27b |
   |         |            |                                            |
   |    24   |      4     | 0x0a336d93aac081a2d849c612368b8cbb2fa9563a |
   |         |            |                                            |
   |    25   |     13     | 0x917be0c94770a7bb12713a4bae801fb3c1c43002 |
   |         |            |                                            |
   |    26   |     14     | 0x91586feaadcf691b6cb07c16c8a2ed0884666e84 |
   |         |            |                                            |
   |    27   |      2     | 0xdd4e4b720fb2517c4bc6f91ccb8725118e5770c6 |
   |         |            |                                            |
   |    28   |     15     | 0x491f6ec665f54c4b3cffaa02ec594d31e6e26c0e |
   |         |            |                                            |
   |    29   |      3     | 0x4f5a082c9d9c9714701de0bf426e9f893484618c |
   |         |            |                                            |
   |    30   |     10     | 0x11f7017313f0c9549c5d415a8abc25243028514d |
   |         |            |                                            |
   |    31   |     12     | 0x6839a994fccb9cb76241d809146906a3d13f89f1 |
   |         |            |                                            |
   |    32   |      4     | 0x71cd1d9163d7cd563936837c61d97bb1a5337cc0 |
   |         |            |                                            |
   |    33   |      5     | 0x77c9034ffc0f9219841aa8e1edbfb62017ef9fd1 |
   |         |            |                                            |
   |    34   |     10     | 0xad9f6034017d35c338ac35778dd6c4c1abe4472a |
   |         |            |                                            |
   |    35   |      8     | 0x4a1c396b22e4f5cc2428045b36d13737c4007515 |
   |         |            |                                            |
   |    36   |     10     | 0x98cb57b779c5fd3f361cd5debc243303ae5baefd |
   |         |            |                                            |
   |    37   |     13     | 0x29857298f274d6bf595eadc89e5464ccf9608a6c |
   |         |            |                                            |
   |    38   |      4     | 0x95e35a26815a3ae9ad84a24464b174a29364da18 |
   |         |            |                                            |
   |    39   |     13     | 0x4afeb3b95b5b333759c0acdd96ce3f26314bb22b |
   |         |            |                                            |
   |    40   |     13     | 0x325a37ee5e349b22b13b54b24be5145344e7b8f3 |
   |         |            |                                            |
   |    41   |     11     | 0x4f772c93f56fd6958ce135f02847996c67e1f2ef |
   |         |            |                                            |
   |    42   |     10     | 0xd4f6d91c577594060be328b013c9e9b0e8a2e5d8 |
   |         |            |                                            |
   |    43   |      1     | 0x717e1a81c325cdccacb6e9fd9e92dd3e1bb84ae8 |
   |         |            |                                            |



McGrew & Curcio          Expires August 26, 2013               [Page 40]

Internet-Draft            Hash-Based Signatures            February 2013


   |    44   |     11     | 0x1dd363724ec66c090a1228dfa1cd3d9cc806f346 |
   |         |            |                                            |
   |    45   |      2     | 0x64b4110476dd0beea78714c5ab71278818792cfa |
   |         |            |                                            |
   |    46   |      4     | 0xe22290e740056a144af50f0b10962b5bcc18fc82 |
   |         |            |                                            |
   |    47   |      2     | 0x34fd87046a183f4732a52bb7805ce207eebdafc5 |
   |         |            |                                            |
   |    48   |     15     | 0xbd2fdc5e4e8d0ed7c48c1bad9c2f7793fc2c9303 |
   |         |            |                                            |
   |    49   |      0     | 0xb3f47e2e8e2dcdd890ea00934b9d8234830dbc4a |
   |         |            |                                            |
   |    50   |     11     | 0xcd29719c56cdb507030e6132132179e5807e1d3b |
   |         |            |                                            |
   |    51   |      3     | 0xf9edb9b301916217de0d746a0542316bebe9e806 |
   |         |            |                                            |
   |    52   |     12     | 0x7a3801cbfe0cafed863d81210c1ec721eede49e5 |
   |         |            |                                            |
   |    53   |     15     | 0x5caba3ec960efa210f5f3e1c22c567ca475ef3ec |
   |         |            |                                            |
   |    54   |     12     | 0xf911b5d148e1b03fe6983c53411f76ea78772379 |
   |         |            |                                            |
   |    55   |      1     | 0x06da2baa75c6ef752bf59f3812fa042ff8181209 |
   |         |            |                                            |
   |    56   |      9     | 0x2b29f5aa2f34af51a78a5fac586004f749c6e6dc |
   |         |            |                                            |
   |    57   |      9     | 0x55e033ababac0845cc9142e24f9ef0a641c51cbe |
   |         |            |                                            |
   |    58   |      3     | 0xb62d207bb700071fba8a68312ca204ce4d994c33 |
   |         |            |                                            |
   |    59   |      9     | 0x551d5c00fad905bdb99c4f70ec7590a10d3ff8ca |
   |         |            |                                            |
   |    60   |      1     | 0x0d03b1845b5f8838d735142f185f9cf8f8d2db6c |
   |         |            |                                            |
   |    61   |     13     | 0x3b5d9e49e7ede41cd9aa5a09f72a0384fd4ff511 |
   |         |            |                                            |
   |    62   |     13     | 0xa766b0278d14a9b7d32bf0307c0737a8ecf82ab1 |
   |         |            |                                            |
   |    63   |      8     | 0xca85296f354e6e3d2a96ab497c01e5ccd4530cf1 |
   |         |            |                                            |
   |    64   |      1     | 0x7bb29db7dd8aaaf1cd11487cea0d13730edb1df3 |
   |         |            |                                            |
   |    65   |     12     | 0x547ef341b3cf3208753bb1b62d85a4e3fc2cffe0 |
   |         |            |                                            |
   |    66   |     12     | 0xb890e1a99da4b2e0a9dde42f82f92d0946327cee |
   +---------+------------+--------------------------------------------+

                                 Table 12



McGrew & Curcio          Expires August 26, 2013               [Page 41]

Internet-Draft            Hash-Based Signatures            February 2013


   Finally, based on the fact that the message is the first to be signed
   by the Merkle tree (i.e. using leaf node 0), the values of the leaf
   and interior nodes that compose the authentication path from leaf to
   root are determined as described in Section 4.3.  These values are
   marked with an asterisk ('*') in Table 14 and Table 15.

B.4.  Signature Verification

   The signature verification step was provided the following items:

   1.  OTS = (y[0] || y[1] || ... || y[p-1]) - from Table 12.

   2.  Authentication Path = concatenation of (k-1)*h Merkle tree node
       values - from Table 14 and Table 15.

   3.  Message Number = leaf number of Merkle tree.

   4.  Merkle Public Key = root of Merkle tree - from Table 9.

   Using Algorithm 4 of Section 3.8 as a start, the potential OTS public
   key was calculated from the value of the OTS.  Since the actual OTS
   public key was not provided to the verifier, the calculated key was
   checked for validity using the pseudocode algorithm of Section 4.4
   and the provided values of the Authentication Path and Message
   Number.  Since the message was valid, the calculated value of the
   root matched the Merkle public key.  Otherwise, verification would
   have failed.

B.5.  Intermediate Calculation Values

   +----------------------+--------------------------------------------+
   |   Key Element Index  |   SHA256-20 Result for w = 4 (F^15(x[i]))  |
   |          (i)         |                                            |
   +----------------------+--------------------------------------------+
   |           0          | 0x6eff4b0c224874ecc4e4f4500da53dbe2a030e45 |
   |                      |                                            |
   |           1          | 0x58ac2c6c451c7779d67efefdb12e5c3d85475a94 |
   |                      |                                            |
   |           2          | 0xb1f3e42e29c710d69268eed1bbdb7f5a500b7937 |
   |                      |                                            |
   |           3          | 0x51d28e573aac2b84d659abb961c32c465e911b55 |
   |                      |                                            |
   |           4          | 0xa0ed62bccac5888f5000ca6a01e5ffefd442a1c6 |
   |                      |                                            |
   |           5          | 0x44da9e145666322422c1e2b5e21627e05aeb4367 |
   |                      |                                            |
   |           6          | 0x04e7ff9213c2655f28364f659c35d3086d7414e1 |
   |                      |                                            |



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Internet-Draft            Hash-Based Signatures            February 2013


   |           7          | 0x414cdb3215408b9722a02577eeb71f9e016e4251 |
   |                      |                                            |
   |           8          | 0xa3ab06b90a2b20f631175daa9454365a4f408e9e |
   |                      |                                            |
   |           9          | 0xe38acfd3c0a03faa82a0f4aeac1a7c04983fad25 |
   |                      |                                            |
   |          10          | 0xd95a289094ccce8ad9ff1d5f9e38297f9bb306ff |
   |                      |                                            |
   |          11          | 0x593d148b22e33c32f18b66340bdaffceb3ad1a55 |
   |                      |                                            |
   |          12          | 0x16b53fbea11dc7ab70c8336ec3c23881ae5d51bf |
   |                      |                                            |
   |          13          | 0xa639ca0cf871188cadd0020832c4f06e6ebd5f98 |
   |                      |                                            |
   |          14          | 0xe3ab3e0c5ad79d6c8c2a7e9a79856d4380941fe0 |
   |                      |                                            |
   |          15          | 0x8368c2933dabcde69c373867a9bf2dc78df97bea |
   |                      |                                            |
   |          16          | 0xe3609fca11545da156a7779ae565b1e3c87902c0 |
   |                      |                                            |
   |          17          | 0xab029e62c7011772dc0589d79fad01aacf8d2177 |
   |                      |                                            |
   |          18          | 0xa8310f1c27c1aa481192de07d4397b8c4716e25f |
   |                      |                                            |
   |          19          | 0xdbdbb14dbd9a5f03c1849af24b69b9e3f80faca2 |
   |                      |                                            |
   |          20          | 0x1a17399d555dec07d3d4f6d54b2b87d2bcaa398b |
   |                      |                                            |
   |          21          | 0xf81c66cc522bfb203232e44d0003ed65d2462867 |
   |                      |                                            |
   |          22          | 0x202a625b8c5f22de6ea081af6da077cf5c63202f |
   |                      |                                            |
   |          23          | 0x2e080f3591f5ff3d5de39c2698846cc107a09816 |
   |                      |                                            |
   |          24          | 0xa1d9c78c22f9810e3b7db2d59ad9f5fdd259f4d4 |
   |                      |                                            |
   |          25          | 0x658eeb85ebe0f4542c4d32dced2d7226929266b2 |
   |                      |                                            |
   |          26          | 0x67fae1a784f919577afc091504d82d31b4ba9fc7 |
   |                      |                                            |
   |          27          | 0xfc39fb43677fb2d433a6292f19c6e7320279655a |
   |                      |                                            |
   |          28          | 0x491f6ec665f54c4b3cffaa02ec594d31e6e26c0e |
   |                      |                                            |
   |          29          | 0x17cec813a5781409b11d2e4a85f62301c2fd8873 |
   |                      |                                            |
   |          30          | 0xc578eb105454d900c053eb55833db607aa5757e0 |
   |                      |                                            |



McGrew & Curcio          Expires August 26, 2013               [Page 43]

Internet-Draft            Hash-Based Signatures            February 2013


   |          31          | 0xaed094323290a41fd4b546919620e2f6b23916c8 |
   |                      |                                            |
   |          32          | 0x192b5a87b5124dc287e06cdd4ec7c0c11f67dda6 |
   |                      |                                            |
   |          33          | 0x4e9e2bdc1b0204d1ceeb68fb4159e752c40b9608 |
   |                      |                                            |
   |          34          | 0xf34c57ad9ce45d67fd32dc2737e6263bcc5cc61f |
   |                      |                                            |
   |          35          | 0xf73bd27d376186310f83cc66e72060aeaccde371 |
   |                      |                                            |
   |          36          | 0xeea482511acd8be783e9be42b48799653b222db4 |
   |                      |                                            |
   |          37          | 0xa2e53196fec8676065b8f32b3e8498e66a4af3cf |
   |                      |                                            |
   |          38          | 0x670c98185157e1b28d38f7dafb00796b434c8316 |
   |                      |                                            |
   |          39          | 0x441afbb265b93595389aaa66325de792f343f209 |
   |                      |                                            |
   |          40          | 0x7b6c50d20b5edc0bc90eb4b289770514cbc8d547 |
   |                      |                                            |
   |          41          | 0xfde6e862a7ba3534893a3e630e209a24be590b1e |
   |                      |                                            |
   |          42          | 0xc59611200c20b2e73dfb24c84cedf4792d6daf10 |
   |                      |                                            |
   |          43          | 0x66e3527bee88373d18f91b230b53b569361f0a15 |
   |                      |                                            |
   |          44          | 0xd0fd79c7116198e689275fec9b4c46f4aac73293 |
   |                      |                                            |
   |          45          | 0x65f07406ad4241e7cf4174c5f284267292cdbc32 |
   |                      |                                            |
   |          46          | 0x7b1b5535d45f46542e2b876245b66ea83cde3d8f |
   |                      |                                            |
   |          47          | 0x7a11620934eb0eb17e10e4a8bbd52aa4b020da0e |
   |                      |                                            |
   |          48          | 0xbd2fdc5e4e8d0ed7c48c1bad9c2f7793fc2c9303 |
   |                      |                                            |
   |          49          | 0x00432602437252a0622a30676dbaaef3023328b9 |
   |                      |                                            |
   |          50          | 0x09a9c4b25034466a5acd7ff681af1c27e8f97577 |
   |                      |                                            |
   |          51          | 0x4b31481d52aa5e1a261064bbd87ea46479a6be23 |
   |                      |                                            |
   |          52          | 0xaca2ad4aa1264618ab633bf11cbca3cc8fa43091 |
   |                      |                                            |
   |          53          | 0x5caba3ec960efa210f5f3e1c22c567ca475ef3ec |
   |                      |                                            |
   |          54          | 0x353e3ffcedfd9500141921cf2aebc2e111364dad |
   |                      |                                            |



McGrew & Curcio          Expires August 26, 2013               [Page 44]

Internet-Draft            Hash-Based Signatures            February 2013


   |          55          | 0xe1c498c32169c869174ccf2f1e71e7202f45fba7 |
   |                      |                                            |
   |          56          | 0x5b8519a40d4305813936c7c00a96f5b4ceb603f1 |
   |                      |                                            |
   |          57          | 0x3b942ae6a6bd328d08804ade771a0775bb3ff8f8 |
   |                      |                                            |
   |          58          | 0x6f3be60ee1c34372599b8d634be72e168453bf10 |
   |                      |                                            |
   |          59          | 0xf700c70bac24db0aab1257940661f5b57da6e817 |
   |                      |                                            |
   |          60          | 0x85ccf60624b13663a290fa808c6bbecaf89523cd |
   |                      |                                            |
   |          61          | 0xd049be55ab703c44f42167d5d9e939c830df960f |
   |                      |                                            |
   |          62          | 0xd27a178ccc3b364c7e03d2266093a0d1dfdd9d51 |
   |                      |                                            |
   |          63          | 0xd73c53fdddbe196b9ab56fcc5c9a4a57ad868cd1 |
   |                      |                                            |
   |          64          | 0xb59a70a7372f0c121fa71727baaf6588eccec400 |
   |                      |                                            |
   |          65          | 0x9b5bf379f989f9a499799c12a3202db58b084eed |
   |                      |                                            |
   |          66          | 0xccabf40f3c1dacf114b5e5f98a73103b4c1f9b55 |
   +----------------------+--------------------------------------------+

                                 Table 13

























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Internet-Draft            Hash-Based Signatures            February 2013


   +-----------+------------------------------------+------------------+
   |  MTS Leaf |           OTS Public Key           |     Member of    |
   | (Level 2) | (H(x[0] || x[1] || ... || x[p-1])) |  Authentication  |
   |    Node   |                                    |      Path of     |
   |   Number  |                                    |     Message 0    |
   +-----------+------------------------------------+------------------+
   |     0     | 0x2db55a72075fcfab5aedbef77bf6b371 |                  |
   |           |   dfb489d6e61ad2884a248345e6910618 |                  |
   |           |                                    |                  |
   |     1     | 0x8c6c6a1215bfe7fda10b7754e73cd984 |         *        |
   |           |   a64823b1ab9d5f50feda6b151c0fee6d |                  |
   |           |                                    |                  |
   |     2     | 0xc1fb91de68b3059c273e53596108ec7c |         *        |
   |           |   f39923757597fe86439e91ce1c25fc84 |                  |
   |           |                                    |                  |
   |     3     | 0x1b511189baee50251335695b74d67c40 |         *        |
   |           |   5a04eddaa79158a9090cc7c3eb204cbf |                  |
   |           |                                    |                  |
   |     4     | 0xf3bcf088ccf9d00338b6c87e8f822da6 |                  |
   |           |   8ec471f88d1561193b3c017d20b3c971 |                  |
   |           |                                    |                  |
   |     5     | 0x40584c059e6cc72fb61f7bd1b9c28e73 |                  |
   |           |   c689551e6e7de6b0b9b730fab9237531 |                  |
   |           |                                    |                  |
   |     6     | 0x1b1d09de1ca16ca890036e018d7e73de |                  |
   |           |   b39b07de80c19dcc5e55a699f021d880 |                  |
   |           |                                    |                  |
   |     7     | 0x83a82632acaac5418716f4f357f5007f |                  |
   |           |   719d604525dbe1831c09a2ead9400a52 |                  |
   |           |                                    |                  |
   |     8     | 0xccb8b2a1d60f731b5f51910eb427e211 |                  |
   |           |   96090d5cd2a077f33968b425301e3fbd |                  |
   |           |                                    |                  |
   |     9     | 0x616767ebf3c1f3ec662d8c57c630c6ae |                  |
   |           |   b31853fd40a18c3d831f5490610c1f16 |                  |
   |           |                                    |                  |
   |     10    | 0x5a4b3e157b66327c75d7f01304d188e2 |                  |
   |           |   cecd1b6168240c11a01775d581b01fb6 |                  |
   |           |                                    |                  |
   |     11    | 0xf25744b8a1c2184ba38521801bf4727c |                  |
   |           |   407b85eb5aef8884d8fbb1c12e2f6108 |                  |
   |           |                                    |                  |
   |     12    | 0xaf8189f51874999162890f72e0ef25e6 |                  |
   |           |   f76b4ab94dc53569bdd66507f5ab0d8e |                  |
   |           |                                    |                  |
   |     13    | 0x96251e396756686645f35cd059da329f |                  |
   |           |   7083838d56c9ccacebbaf8486af18844 |                  |
   |           |                                    |                  |



McGrew & Curcio          Expires August 26, 2013               [Page 46]

Internet-Draft            Hash-Based Signatures            February 2013


   |     14    | 0x773d5206e40065d3553c3c2ed2500122 |                  |
   |           |   e3ee6fd2c91f35a57f084dc839aab1fc |                  |
   |           |                                    |                  |
   |     15    | 0xcda7fae67ce2c3ed29ce426fdcd3f2a8 |                  |
   |           |   eb699e47a67a52f1c94e89726ffe97fa |                  |
   +-----------+------------------------------------+------------------+

                                 Table 14

   +------------+------------------------------------+-----------------+
   |     MTS    |             Node Value             |    Member of    |
   |  Interior  |   (H(child_0 || child_1 || ... ||  |  Authentication |
   |  (Level 1) |             child_k-1))            |     Path of     |
   |    Node    |                                    |    Message 0    |
   |   Number   |                                    |                 |
   +------------+------------------------------------+-----------------+
   |      0     | 0xb6a310deb55ed48004133ece2aebb25e |                 |
   |            |   d74defb77ebd8d63c79a42b5b4191b0c |                 |
   |            |                                    |                 |
   |      1     | 0x71a0c8b767ade2c97ebac069383e4dfb |        *        |
   |            |   a1c06d5fd3f69a775711ea6470747664 |                 |
   |            |                                    |                 |
   |      2     | 0x91109fa97662dc88ae63037391ac2650 |        *        |
   |            |   f6c664ac2448b54800a1df748953af31 |                 |
   |            |                                    |                 |
   |      3     | 0xd277fb8c89689525f90de567068d6c93 |        *        |
   |            |   565df3588b97223276ef8e9495468996 |                 |
   +------------+------------------------------------+-----------------+

                                 Table 15





















McGrew & Curcio          Expires August 26, 2013               [Page 47]

Internet-Draft            Hash-Based Signatures            February 2013


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

































McGrew & Curcio          Expires August 26, 2013               [Page 48]


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