## Hash-Based Signatures

draft-mcgrew-hash-sigs-00

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Crypto Forum Research Group D. McGrew Internet-Draft M. Curcio Intended status: Informational Cisco Systems Expires: August 22, 2013 February 18, 2013 Hash-Based Signatures draft-mcgrew-hash-sigs-00 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 22, 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 McGrew & Curcio Expires August 22, 2013 [Page 1] Internet-Draft Hash-Based Signatures February 2013 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. McGrew & Curcio Expires August 22, 2013 [Page 2] Internet-Draft Hash-Based Signatures February 2013 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 . . . . . . . . . . . . . . . . . . 26 9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 28 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 29 10.1. Normative References . . . . . . . . . . . . . . . . . . . 29 10.2. Informative References . . . . . . . . . . . . . . . . . . 29 McGrew & Curcio Expires August 22, 2013 [Page 3] Internet-Draft Hash-Based Signatures February 2013 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 McGrew & Curcio Expires August 22, 2013 [Page 4] Internet-Draft Hash-Based Signatures February 2013 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 keypair. 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]. McGrew & Curcio Expires August 22, 2013 [Page 5] Internet-Draft Hash-Based Signatures February 2013 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 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 McGrew & Curcio Expires August 22, 2013 [Page 6] Internet-Draft Hash-Based Signatures February 2013 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, i*w/8) >> (8 - (w * (i % (8 / w)) + w))) McGrew & Curcio Expires August 22, 2013 [Page 7] Internet-Draft Hash-Based Signatures February 2013 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 between zero and seven inclusive, then B >> i denotes the logical right-shift operation. McGrew & Curcio Expires August 22, 2013 [Page 8] Internet-Draft Hash-Based Signatures February 2013 Similarly, B << i denotes the logical left-shift operation. If S and T are byte strings, then S || T denotes the concatenation of S and T. The i-th byte string in an array A is denoted as A[i]. 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)). McGrew & Curcio Expires August 22, 2013 [Page 9] Internet-Draft Hash-Based Signatures February 2013 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(message) ) and the number of n-byte strings that form the private key 'ls' : the number of bits the checksum function C will left-shift the sum 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. A table illustrating various combinations of n, w, p, and ls is provided in Appendix A, Table 2, and that section also describes how p and ls depend on the other parameters. McGrew & Curcio Expires August 22, 2013 [Page 10] Internet-Draft Hash-Based Signatures February 2013 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 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. McGrew & Curcio Expires August 22, 2013 [Page 11] Internet-Draft Hash-Based Signatures February 2013 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 manipulation of an existing signature for the purposes of producing a signature for a 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. McGrew & Curcio Expires August 22, 2013 [Page 12] Internet-Draft Hash-Based Signatures February 2013 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. McGrew & Curcio Expires August 22, 2013 [Page 13] Internet-Draft Hash-Based Signatures February 2013 3.8. Signature Verification In order to verify a message with its signature, 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(x[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. McGrew & Curcio Expires August 22, 2013 [Page 14] Internet-Draft Hash-Based Signatures February 2013 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 McGrew & Curcio Expires August 22, 2013 [Page 15] Internet-Draft Hash-Based Signatures February 2013 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] | | | | | | | +---------------------------------+ McGrew & Curcio Expires August 22, 2013 [Page 16] Internet-Draft Hash-Based Signatures February 2013 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 is associated with an LDWM public/private key pair, and the value contained by a leaf is the LDWM public key. The value contained by the root of the tree is the MTS public key. Each interior node is computed from all of its children nodes, by applying the hash function to the concatenation of the values of those 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. McGrew & Curcio Expires August 22, 2013 [Page 17] Internet-Draft Hash-Based Signatures February 2013 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 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 and 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 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 McGrew & Curcio Expires August 22, 2013 [Page 18] Internet-Draft Hash-Based Signatures February 2013 incremented. 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]). McGrew & Curcio Expires August 22, 2013 [Page 19] Internet-Draft Hash-Based Signatures February 2013 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. McGrew & Curcio Expires August 22, 2013 [Page 20] Internet-Draft Hash-Based Signatures February 2013 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 */ }; McGrew & Curcio Expires August 22, 2013 [Page 21] Internet-Draft Hash-Based Signatures February 2013 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 all of the information that is needed in order to parse the format. McGrew & Curcio Expires August 22, 2013 [Page 22] Internet-Draft Hash-Based Signatures February 2013 6. History This is the initial version of this draft. This section is to be removed by the RFC editor upon publication. McGrew & Curcio Expires August 22, 2013 [Page 23] Internet-Draft Hash-Based Signatures February 2013 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. +--------------------+-----------+--------------------+ | 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 | +--------------------+-----------+--------------------+ McGrew & Curcio Expires August 22, 2013 [Page 24] Internet-Draft Hash-Based Signatures February 2013 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 | +-------------------+-----------+--------------------+ An IANA registration of a signature system does not constitute an endorsement of that system or its security. McGrew & Curcio Expires August 22, 2013 [Page 25] Internet-Draft Hash-Based Signatures February 2013 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 of a message with a private key x, and let h = H(message) and c = C(message); the signature is thus h || c. 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 coef(h', j, w) > coef(h, j, w) for some j, the attacker can compute F^a'(x[j]) from F^a(x[j]), where a' = coef(h', j, w) and a = coef(h, j, w). However, the checksum will increase, and thus a valid signature will have coef(c', k, w) > coef(c, k, w) for some 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 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 McGrew & Curcio Expires August 22, 2013 [Page 26] Internet-Draft Hash-Based Signatures February 2013 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. McGrew & Curcio Expires August 22, 2013 [Page 27] Internet-Draft Hash-Based Signatures February 2013 9. Acknowledgements Thanks are due to Chirag Shroff for constructive feedback. McGrew & Curcio Expires August 22, 2013 [Page 28] Internet-Draft Hash-Based Signatures February 2013 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 Infomation Systems Laboratory Technical Report 1979-1, 1979. McGrew & Curcio Expires August 22, 2013 [Page 29] Internet-Draft Hash-Based Signatures February 2013 Appendix A. LDWM Parameter Options A table illustrating various combinations of n, w, p, and ls is provided in Table 2. 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. McGrew & Curcio Expires August 22, 2013 [Page 30] Internet-Draft Hash-Based Signatures February 2013 +------------+--------------+--------------+----------+-------------+ | Hash | Winternitz | w-bit | Left | Number of | | Length in | Parameter | Elements in | Shift | Elements | | Bytes (n) | (w) | Count | (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 2 McGrew & Curcio Expires August 22, 2013 [Page 31] Internet-Draft Hash-Based Signatures February 2013 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 +----+----+---+----+---+---+---+ | m | n | w | p | l | k | h | +----+----+---+----+---+---+---+ | 20 | 32 | 4 | 67 | 4 | 4 | 2 | +----+----+---+----+---+---+---+ Table 3 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. 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 McGrew & Curcio Expires August 22, 2013 [Page 32] Internet-Draft Hash-Based Signatures February 2013 pseudocode given in the first 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 4 +-----------------+-------------------------------------------------+ | 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 | | | | | 8 | 0x629f18b6a2a4ea65fff4cf758b57333f | | | e1d34af05b1cd7763696899c9869595f | | | | | 9 | 0x1741c31fdbb4864712f6b17fadc05d45 | | | 926c831c7a755b7d7af57ac316ba6c2a | | | | McGrew & Curcio Expires August 22, 2013 [Page 33] Internet-Draft Hash-Based Signatures February 2013 | 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 | | | | | 24 | 0xcf1546a814ff16099cebf1fe0db1ace5 | | | 1c272fda9846fbb535815924b0077fa4 | | | | | 25 | 0xcbb06f13155ce4e56c85a32661c90142 | | | 8b630a4c37ea5c7062156f07f6b3efff | | | | McGrew & Curcio Expires August 22, 2013 [Page 34] Internet-Draft Hash-Based Signatures February 2013 | 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 | | | | | 40 | 0xc86eaed6a9e3fbe5b262c1fa1f099f7c | | | 35ece71d9e467fab7a371dbcf400b544 | | | | | 41 | 0xf462b3719a2ed8778155638ff814dbf4 | | | 2b107bb5246ee3dd82abf97787e6a69e | | | | McGrew & Curcio Expires August 22, 2013 [Page 35] Internet-Draft Hash-Based Signatures February 2013 | 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 | | | | | 56 | 0xfbb04488c3c088dc5e63d13e6a701036 | | | 6109ca4c5f4b0a8d37780187e2e9930e | | | | | 57 | 0xaec10811569d4d72e3a1baf71a886b75 | | | eba6dc07ed027af0b2beffa71f9b43c8 | | | | McGrew & Curcio Expires August 22, 2013 [Page 36] Internet-Draft Hash-Based Signatures February 2013 | 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 5 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 11. +-------------------------------------------------------------------+ | OTS Public Key 0 | +-------------------------------------------------------------------+ | 0x2db55a72075fcfab5aedbef77bf6b371 | | dfb489d6e61ad2884a248345e6910618 | +-------------------------------------------------------------------+ Table 6 Following the creation of all OTS public/private key pairs, the OTS public keys in Table 12 are used to determine the MTS public key below. Intermediate values of the interior nodes of the Merkle tree are provided in Table 13. McGrew & Curcio Expires August 22, 2013 [Page 37] Internet-Draft Hash-Based Signatures February 2013 +-------------------------------------------------------------------+ | MTS Public Key | +-------------------------------------------------------------------+ | 0x6610803d9a3546fb0a7895f6a4a0cfed | | 3a07d45e51d096e204b018e677453235 | +-------------------------------------------------------------------+ Table 7 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 8 The SHA256 hash of the text file is provided below. +-------------------------------------------------------------------+ | SHA256 Hash of Signed File (H("Hello world!\n")) | +-------------------------------------------------------------------+ | 0x0ba904eae8773b70c75333db4de2f3ac4 | | 5a8ad4ddba1b242f0b3cfc199391dd81c | +-------------------------------------------------------------------+ Table 9 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 22, 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(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 22, 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 22, 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 10 McGrew & Curcio Expires August 22, 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 as described in Section 4.2 are determined. These values are marked with an asterisk ('*') in Table 12 and Table 13. B.4. Signature Verification The signature verification step was provided the following items: 1. OTS = (y[0] || y[1] || ... || y[p-1]) - from Table 10. 2. Authentication Path = concatenation of (k-1)*h Merkle tree node values - from Table 12 and Table 13. 3. Message Number = leaf number of Merkle tree. 4. Merkle Public Key = root of Merkle tree - from Table 7. 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 second pseudocode algorithm of Section 4.2 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 22, 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 22, 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 22, 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 11 McGrew & Curcio Expires August 22, 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 22, 2013 [Page 46] Internet-Draft Hash-Based Signatures February 2013 | 14 | 0x773d5206e40065d3553c3c2ed2500122 | | | | e3ee6fd2c91f35a57f084dc839aab1fc | | | | | | | 15 | 0xcda7fae67ce2c3ed29ce426fdcd3f2a8 | | | | eb699e47a67a52f1c94e89726ffe97fa | | +-----------+------------------------------------+------------------+ Table 12 +------------+------------------------------------+-----------------+ | 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 13 McGrew & Curcio Expires August 22, 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 22, 2013 [Page 48]