Hash-Based Signatures
draft-mcgrew-hash-sigs-01
The information below is for an old version of the document.
| Document | Type |
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| Authors | David McGrew , Michael Curcio | ||
| Last updated | 2013-08-29 (Latest revision 2013-02-25) | ||
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| IETF conflict review | conflict-review-mcgrew-hash-sigs, conflict-review-mcgrew-hash-sigs, conflict-review-mcgrew-hash-sigs, conflict-review-mcgrew-hash-sigs, conflict-review-mcgrew-hash-sigs, conflict-review-mcgrew-hash-sigs | ||
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draft-mcgrew-hash-sigs-01
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
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at http://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on 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 |
| | |
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| 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]
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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]
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+---------+------------+--------------------------------------------+
| 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 |
| | |
McGrew & Curcio Expires August 26, 2013 [Page 42]
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
McGrew & Curcio Expires August 26, 2013 [Page 45]
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]