## Leighton-Micali Hash-Based Signatures

draft-mcgrew-hash-sigs-15

Crypto Forum Research Group D. McGrew Internet-Draft M. Curcio Intended status: Informational Cisco Systems Expires: April 21, 2016 October 19, 2015 Hash-Based Signatures draft-mcgrew-hash-sigs-03 Abstract This note describes a digital signature system based on cryptographic hash functions, following the seminal work in this area of Lamport, Diffie, Winternitz, and Merkle, as adapted by Leighton and Micali in 1995. It specifies a one-time signature scheme and a general signature scheme. These systems provide asymmetric authentication without using large integer mathematics and can achieve a high security level. They are suitable for compact implementations, are relatively simple to implement, and naturally resist side-channel attacks. Unlike most other signature systems, hash-based signatures would still be secure even if it proves feasible for an attacker to build a quantum computer. Status of This Memo 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 April 21, 2016. Copyright Notice Copyright (c) 2015 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (http://trustee.ietf.org/license-info) in effect on the date of McGrew & Curcio Expires April 21, 2016 [Page 1]

Internet-Draft Hash-Based Signatures October 2015 publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Conventions Used In This Document . . . . . . . . . . . . 4 2. Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1. Operators . . . . . . . . . . . . . . . . . . . . . . 5 3.1.2. Strings of w-bit elements . . . . . . . . . . . . . . 5 3.2. Security string . . . . . . . . . . . . . . . . . . . . . 6 3.3. Functions . . . . . . . . . . . . . . . . . . . . . . . . 8 4. LM-OTS One-Time Signatures . . . . . . . . . . . . . . . . . 8 4.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 8 4.2. Hashing Functions . . . . . . . . . . . . . . . . . . . . 9 4.3. Signature Methods . . . . . . . . . . . . . . . . . . . . 9 4.4. Private Key . . . . . . . . . . . . . . . . . . . . . . . 10 4.5. Public Key . . . . . . . . . . . . . . . . . . . . . . . 10 4.6. Checksum . . . . . . . . . . . . . . . . . . . . . . . . 11 4.7. Signature Generation . . . . . . . . . . . . . . . . . . 11 4.8. Signature Verification . . . . . . . . . . . . . . . . . 12 4.9. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.10. Formats . . . . . . . . . . . . . . . . . . . . . . . . . 13 5. Leighton Micali Signatures . . . . . . . . . . . . . . . . . 16 5.1. LMS Private Key . . . . . . . . . . . . . . . . . . . . . 16 5.2. LMS Public Key . . . . . . . . . . . . . . . . . . . . . 17 5.3. LMS Signature . . . . . . . . . . . . . . . . . . . . . . 17 5.3.1. LMS Signature Generation . . . . . . . . . . . . . . 18 5.4. LMS Signature Verification . . . . . . . . . . . . . . . 18 5.5. LMS Formats . . . . . . . . . . . . . . . . . . . . . . . 19 6. Hierarchical signatures . . . . . . . . . . . . . . . . . . . 21 6.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 21 6.2. Signature Generation . . . . . . . . . . . . . . . . . . 21 6.3. Signature Verification . . . . . . . . . . . . . . . . . 22 7. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8. History . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 23 10. Security Considerations . . . . . . . . . . . . . . . . . . . 26 10.1. Stateful signature algorithm . . . . . . . . . . . . . . 26 10.2. Security of LM-OTS Checksum . . . . . . . . . . . . . . 27 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 28 12. References . . . . . . . . . . . . . . . . . . . . . . . . . 28 McGrew & Curcio Expires April 21, 2016 [Page 2]

Internet-Draft Hash-Based Signatures October 2015 12.1. Normative References . . . . . . . . . . . . . . . . . . 28 12.2. Informative References . . . . . . . . . . . . . . . . . 28 Appendix A. LM-OTS Parameter Options . . . . . . . . . . . . . . 29 Appendix B. An iterative algorithm for computing an LMS public key . . . . . . . . . . . . . . . . . . . . . . . . 30 Appendix C. Example implementation . . . . . . . . . . . . . . . 31 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 42 1. Introduction One-time signature systems, and general purpose signature systems built out of one-time signature systems, have been known since 1979 [Merkle79], were well studied in the 1990s [USPTO5432852], and have benefited from renewed attention in the last decade. The characteristics of these signature systems are small private and public keys and fast signature generation and verification, but large signatures and relatively slow key generation. In recent years there has been interest in these systems because of their post-quantum security and their suitability for compact implementations. This note describes the Leighton and Micali adaptation [USPTO5432852] of the original Lamport-Diffie-Winternitz-Merkle one-time signature system [Merkle79] [C:Merkle87][C:Merkle89a][C:Merkle89b] and general signature system [Merkle79] with enough specificity to ensure interoperability between implementations. An example implementation is given in an appendix. A signature system provides asymmetric message authentication. The key generation algorithm produces a public/private key pair. A message is signed by a private key, producing a signature, and a message/signature pair can be verified by a public key. A One-Time Signature (OTS) system can be used to sign exactly one message securely, but cannot securely sign more than one. An N-time signature system can be used to sign N or fewer messages securely. A Merkle tree signature scheme is an N-time signature system that uses an OTS system as a component. In this note we describe the Leighton- Micali Signature (LMS) system, which is a variant of the Merkle scheme. We denote the one-time signature scheme that it incorporates as LM-OTS. This note is structured as follows. Notation is introduced in Section 3. The LM-OTS signature system is described in Section 4, and the LMS N-time signature system is described in Section 5. Sufficient detail is provided to ensure interoperability. The IANA registry for these signature systems is described in Section 9. Security considerations are presented in Section 10. McGrew & Curcio Expires April 21, 2016 [Page 3]

Internet-Draft Hash-Based Signatures October 2015 1.1. Conventions Used In This Document The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119]. 2. Interface The LMS signing algorithm is stateful; once a particular value of the private key is used to sign one message, it MUST NOT be used to sign another. To make this fact explicit in the interface, we use a functional programming approach, in which the key generation, signing, and verification algorithms do not have any side effects. The signing algorithm returns both a signature and a different private key value, which can be used to sign additional messages. The key generation algorithm takes as input an indication of the parameters for the signature system. If it is successful, it returns both a private key and a public key. Otherwise, it returns an indication of failure. The signing algorithm takes as input the message to be signed and the current value of the private key. If successful, it returns a signature and the next value of the private key, if there is such a value. After the private key of an N-time signature system has signed N messages, the signing algorithm returns the signature and an indication that there is no next value of the private key that can be used for signing. If unsuccessful, it returns an indication of failure. The verification algorithm takes as input the public key, a message, and a signature, and returns an indication of whether or not the signature and message pair are valid. A message/signature pair are valid if the signature was returned by the signing algorithm upon input of the message and the private key corresponding to the public key; otherwise, the signature and message pair are not valid with probability very close to one. 3. Notation 3.1. Data Types Bytes and byte strings are the fundamental data types. A single byte is denoted as a pair of hexadecimal digits with a leading "0x". A byte string is an ordered sequence of zero or more bytes and is denoted as an ordered sequence of hexadecimal characters with a leading "0x". For example, 0xe534f0 is a byte string with a length McGrew & Curcio Expires April 21, 2016 [Page 4]

Internet-Draft Hash-Based Signatures October 2015 of three. An array of byte strings is an ordered set, indexed starting at zero, in which all strings have the same length. Unsigned integers are converted into byte strings by representing them in network byte order. To make the number of bytes in the representation explicit, we define the functions uint8str(X), uint16str(X), and uint32str(X), which return one, two, and four byte values, respectively. 3.1.1. Operators When a and b are real numbers, mathematical operators are defined as follows: ^ : a ^ b denotes the result of a raised to the power of b * : a * b denotes the product of a multiplied by b / : a / b denotes the quotient of a divided by b % : a % b denotes the remainder of the integer division of a by b + : a + b denotes the sum of a and b - : a - b denotes the difference of a and b The standard order of operations is used when evaluating arithmetic expressions. If A and B are bytes, then A AND B denotes the bitwise logical and operation. When B is a byte and i is an integer, then B >> i denotes the logical right-shift operation. Similarly, B << i denotes the logical left- shift operation. If S and T are byte strings, then S || T denotes the concatenation of S and T. The i^th byte string in an array A is denoted as A[i]. 3.1.2. Strings of w-bit elements If S is a byte string, then byte(S, i) denotes its i^th byte, where byte(S, 0) is the leftmost byte. In addition, bytes(S, i, j) denotes the range of bytes from the i^th to the j^th byte, inclusive. For example, if S = 0x02040608, then byte(S, 0) is 0x02 and bytes(S, 1, 2) is 0x0406. McGrew & Curcio Expires April 21, 2016 [Page 5]

Internet-Draft Hash-Based Signatures October 2015 A byte string can be considered to be a string of w-bit unsigned integers; the correspondence is defined by the function coef(S, i, w) as follows: If S is a string, i is a positive integer, and w is a member of the set { 1, 2, 4, 8 }, then coef(S, i, w) is the i^th, w-bit value, if S is interpreted as a sequence of w-bit values. That is, coef(S, i, w) = (2^w - 1) AND ( byte(S, floor(i * w / 8)) >> (8 - (w * (i % (8 / w)) + w)) ) For example, if S is the string 0x1234, then coef(S, 7, 1) is 0 and coef(S, 0, 4) is 1. S (represented as bits) +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ | 0| 0| 0| 1| 0| 0| 1| 0| 0| 0| 1| 1| 0| 1| 0| 0| +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ ^ | coef(S, 7, 1) S (represented as four-bit values) +-----------+-----------+-----------+-----------+ | 1 | 2 | 3 | 4 | +-----------+-----------+-----------+-----------+ ^ | coef(S, 0, 4) The return value of coef is an unsigned integer. If i is larger than the number of w-bit values in S, then coef(S, i, w) is undefined, and an attempt to compute that value should raise an error. 3.2. Security string To improve security against attacks that amortize their effort against multiple invocations of the hash function H, Leighton and Micali introduce a "security string" that is distinct for each invocation of H. The following fields can appear in a security string: I - an identifier for the private key. This value is 31 bytes long, and it MUST be distinct from all other such identifiers. It SHOULD be chosen uniformly at random, or via a pseudorandom McGrew & Curcio Expires April 21, 2016 [Page 6]

Internet-Draft Hash-Based Signatures October 2015 process, in order to ensure that it will be distinct with probability close to one, but it MAY be a structured identifier. D - a domain separation parameter, which is a single byte that takes on different values in the different algorithms in which H is invoked. D takes on the following values: D_ITER = 0x00 in the iterations of the LM-OTS algorithms D_PBLC = 0x01 when computing the hash of all of the iterates in the LM-OTS algorithm D_MESG = 0x02 when computing the hash of the message in the LM- OTS algorithms D_LEAF = 0x03 when computing the hash of the leaf of an LMS tree D_INTR = 0x04 when computing the hash of an interior node of an LMS tree C - an n-byte randomizer that is included with the message whenever it is being hashed to improve security. C MUST be chosen uniformly at random, or via a pseudorandom process. i - in the LM-OTS one-time signature scheme, i is the index of the private key element upon which H is being applied. It is represented as a 16-bit (two byte) unsigned integer in network byte order. j - in the LM-OTS one-time signature scheme, j is the iteration number used when the private key element is being iteratively hashed. It is represented as an 8-bit (one byte) unsigned integer. q - in the LM-OTS one-time signature scheme, q is a diversification string provided as input. In the LMS N-time signature scheme, a distinct value of q is provided for each distinct LM-OTS public/private keypair. It is represented as a four byte string. r - in the LMS N-time signature scheme, the node number r associated with a particular node of the hash tree is used as an input to the hash used to compute that node. This value is represented as a 32-bit (four byte) unsigned integer in network byte order. McGrew & Curcio Expires April 21, 2016 [Page 7]

Internet-Draft Hash-Based Signatures October 2015 3.3. Functions If r is a non-negative real number, then we define the following functions: ceil(r) : returns the smallest integer larger than r floor(r) : returns the largest integer smaller than r lg(r) : returns the base-2 logarithm of r 4. LM-OTS One-Time Signatures This section defines LM-OTS signatures. The signature is used to validate the authenticity of a message by associating a secret private key with a shared public key. These are one-time signatures; each private key MUST be used only one time to sign any given message. As part of the signing process, a digest of the original message is computed using the cryptographic hash function H (see Section 4.2), and the resulting digest is signed. In order to facilitate its use in an N-time signature system, the LM- OTS key generation, signing, and verification algorithms all take as input a diversification parameter q. When the LM-OTS signature system is used outside of an N-time signature system, this value SHOULD be set to the all-zero value. 4.1. Parameters The signature system uses the parameters n and w, which are both positive integers. The algorithm description also makes use of the internal parameters p and ls, which are dependent on n and w. These parameters are summarized as follows: n : the number of bytes of the output of the hash function w : the Winternitz parameter; it is a member of the set { 1, 2, 4, 8 } p : the number of n-byte string elements that make up the LM-OTS signature ls : the number of left-shift bits used in the checksum function Cksm (defined in Section 4.6). McGrew & Curcio Expires April 21, 2016 [Page 8]

Internet-Draft Hash-Based Signatures October 2015 The value of n is determined by the functions selected for use as part of the LM-OTS algorithm; the choice of this value has a strong effect on the security of the system. The parameter w can be chosen to set the number of bytes in the signature; it has little effect on security. Note however, that there is a larger computational cost to generate and verify a shorter signature. The values of p and ls are dependent on the choices of the parameters n and w, as described in Appendix A. A table illustrating various combinations of n, w, p, and ls is provided in Table 1. 4.2. Hashing Functions The LM-OTS algorithm uses a hash function H that accepts byte strings of any length, and returns an n-byte string. 4.3. Signature Methods To fully describe a LM-OTS signature method, the parameters n and w, as well as the function H, MUST be specified. This section defines several LM-OTS signature systems, each of which is identified by a name. Values for p and ls are provided as a convenience. +---------------------+-----------+----+---+-----+----+ | Name | H | n | w | p | ls | +---------------------+-----------+----+---+-----+----+ | LMOTS_SHA256_N32_W1 | SHA256 | 32 | 1 | 265 | 7 | | | | | | | | | LMOTS_SHA256_N32_W2 | SHA256 | 32 | 2 | 133 | 6 | | | | | | | | | LMOTS_SHA256_N32_W4 | SHA256 | 32 | 4 | 67 | 4 | | | | | | | | | LMOTS_SHA256_N32_W8 | SHA256 | 32 | 8 | 34 | 0 | | | | | | | | | LMOTS_SHA256_N16_W1 | SHA256-16 | 16 | 1 | 68 | 8 | | | | | | | | | LMOTS_SHA256_N16_W2 | SHA256-16 | 16 | 2 | 68 | 8 | | | | | | | | | LMOTS_SHA256_N16_W4 | SHA256-16 | 16 | 4 | 35 | 4 | | | | | | | | | LMOTS_SHA256_N16_W8 | SHA256-16 | 16 | 8 | 18 | 0 | +---------------------+-----------+----+---+-----+----+ Table 1 Here SHA256 denotes the NIST standard hash function [FIPS180]. SHA256-16 denotes the SHA256 hash function with its final output truncated to return the leftmost 16 bytes. McGrew & Curcio Expires April 21, 2016 [Page 9]

Internet-Draft Hash-Based Signatures October 2015 4.4. Private Key The LM-OTS private key consists of an array of size p containing n-byte strings. Let x denote the private key. This private key must be used to sign one and only one message. It must therefore be unique from all other private keys. The following algorithm shows pseudocode for generating x. Algorithm 0: Generating a Private Key for ( i = 0; i < p; i = i + 1 ) { set x[i] to a uniformly random n-byte string } return x An implementation MAY use a pseudorandom method to compute x[i], as suggested in [Merkle79], page 46. The details of the pseudorandom method do not affect interoperability, but the cryptographic strength MUST match that of the LM-OTS algorithm. 4.5. Public Key The LM-OTS public key is generated from the private key by iteratively applying the function H to each individual element of x, for 2^w - 1 iterations, then hashing all of the resulting values. Each public/private key pair is associated with a single identifier I. This string MUST be 31 bytes long, and be generated as described in Section 3.2. The diversification parameter q is an input to the algorithm, as described in Section 3.2. The following algorithm shows pseudocode for generating the public key, where the array x is the private key. Algorithm 1: Generating a Public Key From a Private Key for ( i = 0; i < p; i = i + 1 ) { tmp = x[i] for ( j = 0; j < 2^w - 1; j = j + 1 ) { tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER) } y[i] = tmp } return H(I || q || y[0] || y[1] || ... || y[p-1] || D_PBLC) McGrew & Curcio Expires April 21, 2016 [Page 10]

Internet-Draft Hash-Based Signatures October 2015 The public key is the string consisting of a four-byte enumeration that identifies the parameters in use, followed by the value returned by Algorithm 1. Section 4.10 specifies the enumeration and more formally defines the format. 4.6. Checksum A checksum is used to ensure that any forgery attempt that manipulates the elements of an existing signature will be detected. The security property that it provides is detailed in Section 10. The checksum function Cksm is defined as follows, where S denotes the byte string that is input to that function, and the value sum is a 16-bit unsigned integer: Algorithm 2: Checksum Calculation sum = 0 for ( i = 0; i < u; i = i + 1 ) { sum = sum + (2^w - 1) - coef(S, i, w) } return (sum << ls) Because of the left-shift operation, the rightmost bits of the result of Cksm will often be zeros. Due to the value of p, these bits will not be used during signature generation or verification. 4.7. Signature Generation The LM-OTS signature of a message is generated by first appending the randomizer C, the identifier string I, and the diversification string q to the message, then using H to compute the hash of the resulting string, concatenating the checksum of the hash to the hash itself, then considering the resulting value as a sequence of w-bit values, and using each of the the w-bit values to determine the number of times to apply the function H to the corresponding element of the private key. The outputs of the function H are concatenated together and returned as the signature. The pseudocode for this procedure is shown below. The identifier string I and diversification string q are the same as in Section 4.5. McGrew & Curcio Expires April 21, 2016 [Page 11]

Internet-Draft Hash-Based Signatures October 2015 Algorithm 3: Generating a Signature From a Private Key and a Message set C to a uniformly random n-byte string set type to the appropriate ots_algorithm_type Q = H(message || C || I || q || D_MESG) for ( i = 0; i < p; i = i + 1 ) { a = coef(Q || Cksm(Q), i, w) tmp = x[i] for ( j = 0; j < a; j = j + 1 ) { tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER) } y[i] = tmp } return type || C || I || 0x00 || q || (y[0] || y[1] || ... || y[p-1]) Note that this algorithm results in a signature whose elements are intermediate values of the elements computed by the public key algorithm in Section 4.5. The signature is the string consisting of a four-byte enumeration that identifies the parameters in use, followed by the value returned by Algorithm 3. Section 4.10 specifies the enumeration and more formally defines the format. 4.8. Signature Verification In order to verify a message with its signature (an array of n-byte strings, denoted as y), the receiver must "complete" the series of applications of H using the w-bit values of the message hash and its checksum. This computation should result in a value that matches the provided public key. McGrew & Curcio Expires April 21, 2016 [Page 12]

Internet-Draft Hash-Based Signatures October 2015 Algorithm 4: Verifying a Signature and Message Using a Public Key parse C, I, q, and y from the signature as follows: type = first 4 bytes C = next n bytes I = next 31 bytes NULL = next byte; this padding value is discarded q = next four bytes y[0] = next n bytes y[1] = next n bytes ... y[p-1] = next n bytes Q = H(message || C || I || q || D_MESG) for ( i = 0; i < p; i = i + 1 ) { a = (2^w - 1) - coef(Q || Cksm(Q), i, w) tmp = y[i] for ( j = a+1; j < 2^w - 1; j = j + 1 ) { tmp = H(tmp || I || q || uint16str(i) || uint8str(j) || D_ITER) } z[i] = tmp } candidate = H(z[0] || z[1] || ... || z[p-1] || I || q || D_PBLC) if (candidate = public_key) return 1 // message/signature pair is valid else return 0 // message/signature pair is invalid 4.9. Notes A future version of this specification may define a method for computing the signature of a very short message in which the hash is not applied to the message during the signature computation. That would allow the signatures to have reduced size. 4.10. Formats The signature and public key formats are formally defined using the External Data Representation (XDR) [RFC4506] in order to provide an unambiguous, machine readable definition. For clarity, we also include a private key format as well, though consistency is not needed for interoperability and an implementation MAY use any private key format. Though XDR is used, these formats are simple and easy to parse without any special tools. The definitions are as follows: /* * ots_algorithm_type identifies a particular signature algorithm */ McGrew & Curcio Expires April 21, 2016 [Page 13]

Internet-Draft Hash-Based Signatures October 2015 enum ots_algorithm_type { ots_reserved = 0, lmots_sha256_m16_w1 = 0x00000001, lmots_sha256_m16_w2 = 0x00000002, lmots_sha256_m16_w4 = 0x00000003, lmots_sha256_m16_w8 = 0x00000004, lmots_sha256_n32_w1 = 0x00000005, lmots_sha256_n32_w2 = 0x00000006, lmots_sha256_n32_w4 = 0x00000007, lmots_sha256_n32_w8 = 0x00000008 }; /* * byte strings (for n=16 and n=32) */ typedef opaque bytestring16[16]; typedef opaque bytestring32[32]; union ots_signature switch (ots_algorithm_type type) { case lmots_sha256_n16_w1: bytestring16 y_n16_p265[265]; case lmots_sha256_n16_w2: bytestring16 y_n16_p133[133]; case lmots_sha256_n16_w4: bytestring16 y_n16_p67[67]; case lmots_sha256_n16_w8: bytestring16 y_n16_p34[34]; case lmots_sha256_n32_w1: bytestring32 y_n32_p265[265]; case lmots_sha256_n32_w2: bytestring32 y_m3_p133[133]; case lmots_sha256_n32_w4: bytestring32 y_n32_y_p67[67]; case lmots_sha256_n32_w8: bytestring32 y_n32_p34[34]; default: void; /* error condition */ }; union ots_public_key switch (ots_algorithm_type type) { case lmots_sha256_n16_w1: case lmots_sha256_n16_w2: case lmots_sha256_n16_w4: case lmots_sha256_n16_w8: case lmots_sha256_n32_w1: case lmots_sha256_n32_w2: case lmots_sha256_n32_w4: case lmots_sha256_n32_w8: McGrew & Curcio Expires April 21, 2016 [Page 14]

Internet-Draft Hash-Based Signatures October 2015 bytestring32 y32; default: void; /* error condition */ }; union ots_private_key switch (ots_algorithm_type type) { case lmots_sha256_m16_w1: case lmots_sha256_m16_w2: case lmots_sha256_m16_w4: case lmots_sha256_m16_w8: bytestring20 x20; case lmots_sha256_n32_w1: case lmots_sha256_n32_w2: case lmots_sha256_n32_w4: case lmots_sha256_n32_w8: bytestring32 x32; default: void; /* error condition */ }; Though the data formats are formally defined by XDR, we include diagrams as well as a convenience to the reader. An example of the format of an lmots_signature is illustrated below, for lmots_sha256_n32_w1. An ots_signature consists of a 32-bit unsigned integer that indicates the ots_algorithm_type, followed by other data, whose format depends only on the ots_algorithm_type. For LM- OTS, that data is an array of equal-length byte strings. The number of bytes in each byte string, and the number of elements in the array, are determined by the ots_algorithm_type field. In the case of lmots_sha256_n32_w1, the array has 265 elements, each of which is a 32-byte string. The XDR array y_n32_p265 denotes the array y as used in the algorithm descriptions above, using the parameters of n=32 and p=265 for lmots_sha256_n32_w1. A verifier MUST check the ots_algorithm_type field, and a verification operation on a signature with an unknown lmots_algorithm_type MUST return FAIL. McGrew & Curcio Expires April 21, 2016 [Page 15]

Internet-Draft Hash-Based Signatures October 2015 +---------------------------------+ | ots_algorithm_type | +---------------------------------+ | | | y_n32_p265[0] | | | +---------------------------------+ | | | y_n32_p265[1] | | | +---------------------------------+ | | ~ .... ~ | | +---------------------------------+ | | | y_n32_p265[264] | | | +---------------------------------+ 5. Leighton Micali Signatures The Leighton Micali Signature (LMS) method can sign a potentially large but fixed number of messages. An LMS system uses two cryptographic components: a one-time signature method and a hash function. Each LMS public/private key pair is associated with a perfect binary tree, each node of which contains an n-byte value. Each leaf of the tree contains the value of the public key of an LM- OTS public/private key pair. The value contained by the root of the tree is the LMS public key. Each interior node is computed by applying the hash function to the concatenation of the values of its children nodes. An LMS system has the following parameters: h : the height (number of levels - 1) in the tree, and n : the number of bytes associated with each node. There are 2^h leaves in the tree. 5.1. LMS Private Key An LMS private key consists of 2^h one-time signature private keys and the leaf number of the next LM-OTS private key that has not yet been used. The leaf number is initialized to zero when the LMS private key is created. McGrew & Curcio Expires April 21, 2016 [Page 16]

Internet-Draft Hash-Based Signatures October 2015 An LMS private key MAY be generated pseudorandomly from a secret value, in which case the secret value MUST be at least n bytes long, be uniformly random, and MUST NOT be used for any other purpose than the generation of the LMS private key. The details of how this process is done do not affect interoperability; that is, the public key verification operation is independent of these details. 5.2. LMS Public Key An LMS public key is defined as follows, where we denote the public key associated with the i^th LM-OTS private key as OTS_PUBKEY[i], with i ranging from 0 to (2^h)-1. Each instance of an LMS public/ private key pair is associated with a perfect binary tree, and the nodes of that tree are indexed from 1 to 2^(h+1)-1. Each node is associated with an n-byte string, and the string for the rth node is denoted as T[r] and is defined as T[r] = / H(OTS_PUBKEY[r-2^h] || I || uint32str(r) || D_LEAF) if r >= 2^h \ H(T[2*r] || T[2*r+1] || I || uint32str(r) || D_INTR) otherwise. The LMS public key is the string consisting of a four-byte enumeration that identifies the parameters in use, followed by the string T[1]. Section 5.5 specifies the enumeration and more formally defines the format. The value T[1] can be computed via recursive application of the above equation, or by any equivalent method. An iterative procedure is outlined in Appendix B. 5.3. LMS Signature An LMS signature consists of a typecode indicating the particular LMS algorithm, an LM-OTS signature, and an array of values that is associated with the path through the tree from the leaf associated with the LM-OTS signature to the root. The array of values contains the siblings of the nodes on the path from the leaf to the root but does not contain the nodes on the path itself. The array for a tree with height h will have h values. The first value is the sibling of the leaf, the next value is the sibling of the parent of the leaf, and so on up the path to the root. McGrew & Curcio Expires April 21, 2016 [Page 17]

Internet-Draft Hash-Based Signatures October 2015 5.3.1. LMS Signature Generation To compute the LMS signature of a message with an LMS private key, the signer first computes the LM-OTS signature of the message using the leaf number of the next unused LM-OTS private key. Before releasing the signature, the leaf number in the LMS private key MUST be incremented to prevent the LM-OTS private key from being used again. The node number in the signature is set to the leaf number of the LMS private key that was used in the signature. Then the signature and the LMS private key are returned. The array of node values in the signature MAY be computed in any way. There are many potential time/storage tradeoffs that can be applied. The fastest alternative is to store all of the nodes of the tree and set the array in the signature by copying them. The least storage intensive alternative is to recompute all of the nodes for each signature. Note that the details of this procedure are not important for interoperability; it is not necessary to know any of these details in order to perform the signature verification operation. The internal nodes of the tree need not be kept secret, and thus a node-caching scheme that stores only internal nodes can sidestep the need for strong protections. One useful time/storage tradeoff is described in Column 19 of [USPTO5432852]. 5.4. LMS Signature Verification An LMS signature is verified by first using the LM-OTS signature verification algorithm to compute the LM-OTS public key from the LM- OTS signature and the message. The value of that public key is then assigned to the associated leaf of the LMS tree, then the root of the tree is computed from the leaf value and the node array (path[]) as described below. If the root value matches the public key, then the signature is valid; otherwise, the signature fails. McGrew & Curcio Expires April 21, 2016 [Page 18]

Internet-Draft Hash-Based Signatures October 2015 Algorithm 6: LMS Signature Verification identify the height h of the tree from the algorithm type determine the leaf number the LM-OTS q value to an integer n = node number = 2^h + leaf_number tmp = candidate public key computed from LM-OTS signature and message tmp = H(tmp || I || uint32str(node_num) || D_LEAF) i = 0 while (node_num > 1) { if (node_num is odd): tmp = H(path[i] || tmp || I || uint32str(node_num/2) || D_INTR) else: tmp = H(tmp || path[i] || I || uint32str(node_num/2) || D_INTR) node_num = node_num/2 i = i + 1 if (tmp == lms_public_key) return 1 // message/signature pair is valid else return 0 // message/signature pair is invalid Upon completion, v contains the value of the root of the LMS tree for comparison. The verifier MAY cache interior node values that have been computed during a successful signature verification for use in subsequent signature verifications. However, any implementation that does so MUST make sure any nodes that are cached during a signature verification process are deleted if that process does not result in a successful match between the root of the tree and the LMS public key. 5.5. LMS Formats LMS signatures and public keys are defined using XDR syntax as follows: enum lms_algorithm_type { lms_reserved = 0x00000000, lms_sha256_n32_h20 = 0x00000001, lms_sha256_n32_h10 = 0x00000002, lms_sha256_n32_h5 = 0x00000003 lms_sha256_n16_h20 = 0x00000004, lms_sha256_n16_h10 = 0x00000005, lms_sha256_n16_h5 = 0x00000006 }; union lms_path switch (lms_algorithm_type type) { case lms_sha256_n32_h20: bytestring32 path_n32_h20[20]; McGrew & Curcio Expires April 21, 2016 [Page 19]

Internet-Draft Hash-Based Signatures October 2015 case lms_sha256_n32_h10: bytestring32 path_n32_h10[10]; case lms_sha256_n32_h5: bytestring32 path_n32_h5[5]; case lms_sha256_n16_h20: bytestring32 path_n16_h20[20]; case lms_sha256_n16_h10: bytestring32 path_n16_h10[10]; case lms_sha256_n16_h5: bytestring32 path_n16_h5[5]; default: void; /* error condition */ }; struct lms_signature { ots_signature ots_sig; lms_path nodes; }; struct lms_public_key_n16 { ots_algorithm_type ots_alg_type; opaque value[16]; /* public key */ }; struct lms_public_key_n64 { ots_algorithm_type ots_alg_type; opaque value[64]; /* public key */ opaque I[31]; /* identity */ }; union lms_public_key switch (lms_algorithm_type type) { case lms_sha256_n32_h20: case lms_sha256_n32_h10: case lms_sha256_n32_h5: lms_public_key_n32 z_n32; case lms_sha256_n16_h20: case lms_sha256_n16_h10: case lms_sha256_n16_h5: lms_public_key_n16 z_n16; default: void; /* error condition */ }; McGrew & Curcio Expires April 21, 2016 [Page 20]

Internet-Draft Hash-Based Signatures October 2015 6. Hierarchical signatures In scenarios where it is necessary to minimize the time taken by the public key generation process, a hierarchical N-time signature scheme can be used. Leighton and Micali describe a scheme in which an LMS public key is used to sign a second LMS public key, which is then distributed along with the signatures generated with the second public key [USPTO5432852]. This hierarchical scheme, which we describe in this section, uses an LMS scheme as a component, and it has two levels. Each level is associated with an LMS public key, private key, and signature. The following notation is used, where i is an integer between 1 and 2 inclusive: prv[i] is the private key of the ith level, pub[i] is the public key of the ith level, and sig[i] is the signature of the ith level. In this section, we say that an N-time private key is exhausted when it has signed all N messages, and thus it can no longer be used for signing. 6.1. Key Generation To generate an HLMS private and public key pair, new LMS private and public keys are generated for prv[i] and pub[i] for i=1,2. These key pairs MUST be generated independently. The public key of the HLMS scheme is pub[1], the public key of the first level. The HLMS private key consists of prv[1] and prv[2]. The values pub[1] and prv[1] do not change, though the values of pub[2] and prv[2] are dynamic, and are changed by the signature generation algorithm. 6.2. Signature Generation To sign a message using the private key prv, the following steps are performed: The message is signed with prv[2], and the value sig[2] is set to that result. The value of the HLMS signature is set to type || pub[2] || sig[1] || sig[2], where type is the typecode for the particular HLMS algorithm. McGrew & Curcio Expires April 21, 2016 [Page 21]

Internet-Draft Hash-Based Signatures October 2015 If prv[2] is exhausted, then a new LMS public and private key pair is generated, and pub[2] and prv[2] are set to those values. pub[2] is signed with prv[1], and sig[1] is set to the resulting value. 6.3. Signature Verification To verify a signature sig and message using the public key pub, the following steps are performed: The signature sig is parsed into its components type, pub[2], sig[1] and sig[2]. The signature sig[2] and message is verified using the public key pub[2]. If verification fails, then an indication of failure is returned. Otherwise, processing continues as follows. The signature sig[1] of the "message" pub[2] is verified using the public key pub. If verification fails, then an indication of failure is returned. Otherwise, an indication of success is returned. 7. Rationale The goal of this note is to describe the LM-OTS and LMS algorithms following the original references and present the modern security analysis of those algorithms. Other signature methods are out of scope and may be interesting follow-on work. We adopt the techniques described by Leighton and Micali to mitigate attacks that amortize their work over multiple invocations of the hash function. The values taken by the identifier I across different LMS public/ private key pairs are required to be distinct in order to improve security. That distinctness ensures the uniqueness of the inputs to H across all of those public/private key pair instances, which is important for provable security in the random oracle model. The length of I is set at 31 bytes so that randomly chosen values of I will be distinct with probability at least 1 - 1/2^128 as long as there are 2^60 or fewer instances of LMS public/private key pairs. The sizes of the parameters in the security string are such that, for n=16, the LM-OTS iterates a 55-byte value (that is, the string that is input to H() during the iteration over j during signature generation and verification is 55 bytes long). Thus, when SHA-256 is used as the function H, only a single invocation of its compression function is needed. McGrew & Curcio Expires April 21, 2016 [Page 22]

Internet-Draft Hash-Based Signatures October 2015 The signature and public key formats are designed so that they are easy to parse. Each format starts with a 32-bit enumeration value that indicates all of the details of the signature algorithm and hence defines all of the information that is needed in order to parse the format. The Checksum Section 4.6 is calculated using a non-negative integer "sum", whose width was chosen to be an integer number of w-bit fields such that it is capable of holding the difference of the total possible number of applications of the function H as defined in the signing algorithm of Section 4.7 and the total actual number. In the worst case (i.e. the actual number of times H is iteratively applied is 0), the sum is (2^w - 1) * ceil(8*n/w). Thus for the purposes of this document, which describes signature methods based on H = SHA256 (n = 32 bytes) and w = { 1, 2, 4, 8 }, the sum variable is a 16-bit non-negative integer for all combinations of n and w. The calculation uses the parameter ls defined in Section 4.1 and calculated in Appendix A, which indicates the number of bits used in the left-shift operation. 8. History This is the third version version of this draft. It has the following changes: It adopts the "security string" approach of Leighton and Micali [USPTO5432852] in order to improve security. It adopts Leighton and Micali's idea of hashing a randomizer string (C, as defined in Section 3.2) with the message, so that finding an arbitrary collision in H will not lead to a forgery. It defines a multi-level signature scheme, again following that described by Leighton and Micali. It eliminates the function F and its iterates; the function H is used in its stead. The adoption of the security string makes this simplification possible. It fixes the branching number at two for simplicity. This section is to be removed by the RFC editor upon publication. 9. IANA Considerations The Internet Assigned Numbers Authority (IANA) is requested to create two registries: one for OTS signatures, which includes all of the LM- OTS signatures as defined in Section 3, and one for Leighton-Micali McGrew & Curcio Expires April 21, 2016 [Page 23]

Internet-Draft Hash-Based Signatures October 2015 Signatures, as defined in Section 4. Additions to these registries require that a specification be documented in an RFC or another permanent and readily available reference in sufficient detail that interoperability between independent implementations is possible. Each entry in the registry contains the following elements: a short name, such as "LMS_SHA256_n32_h10", a positive number, and a reference to a specification that completely defines the signature method test cases that can be used to verify the correctness of an implementation. Requests to add an entry to the registry MUST include the name and the reference. The number is assigned by IANA. These number assignments SHOULD use the smallest available palindromic number. Submitters SHOULD have their requests reviewed by the IRTF Crypto Forum Research Group (CFRG) at cfrg@ietf.org. Interested applicants that are unfamiliar with IANA processes should visit http://www.iana.org. The numbers between 0xDDDDDDDD (decimal 3,722,304,989) and 0xFFFFFFFF (decimal 4,294,967,295) inclusive, will not be assigned by IANA, and are reserved for private use; no attempt will be made to prevent multiple sites from using the same value in different (and incompatible) ways [RFC2434]. The LM-OTS registry is as follows. McGrew & Curcio Expires April 21, 2016 [Page 24]

Internet-Draft Hash-Based Signatures October 2015 +----------------------+-----------+--------------------+ | Name | Reference | Numeric Identifier | +----------------------+-----------+--------------------+ | LMOTS_SHA256_N16_W1 | Section 4 | 0x00000001 | | | | | | LMOTS_SHA256_N16_W2 | Section 4 | 0x00000002 | | | | | | LMOTS_SHA256_N16_W4 | Section 4 | 0x00000003 | | | | | | LMOTS_SHA256_N16_W8 | Section 4 | 0x00000004 | | | | | | LMOTS_SHA256_N32_W1 | Section 4 | 0x00000005 | | | | | | LMOTS_SHA256_N32_W2 | Section 4 | 0x00000006 | | | | | | LMOTS_SHA256_N32_W4 | Section 4 | 0x00000007 | | | | | | LMOTS_SHA256_N32_W8 | Section 4 | 0x00000008 | +----------------------+-----------+--------------------+ Table 2 The LMS registry is as follows. +--------------------+-----------+--------------------+ | Name | Reference | Numeric Identifier | +--------------------+-----------+--------------------+ | LMS_SHA256_N32_H20 | Section 5 | 0x00000001 | | | | | | LMS_SHA256_N32_H10 | Section 5 | 0x00000002 | | | | | | LMS_SHA256_N32_H5 | Section 5 | 0x00000003 | | | | | | LMS_SHA256_N16_H20 | Section 5 | 0x00000004 | | | | | | LMS_SHA256_N16_H10 | Section 5 | 0x00000005 | | | | | | LMS_SHA256_N16_H5 | Section 5 | 0x00000006 | +--------------------+-----------+--------------------+ Table 3 An IANA registration of a signature system does not constitute an endorsement of that system or its security. McGrew & Curcio Expires April 21, 2016 [Page 25]

Internet-Draft Hash-Based Signatures October 2015 10. Security Considerations The security goal of a signature system is to prevent forgeries. A successful forgery occurs when an attacker who does not know the private key associated with a public key can find a message and signature that are valid with that public key (that is, the Signature Verification algorithm applied to that signature and message and public key will return "valid"). Such an attacker, in the strongest case, may have the ability to forge valid signatures for an arbitrary number of other messages. LM-OTS and LMS are provably secure in the random oracle model, as shown by Katz [Katz15]. From Theorem 8 of that reference: For any adversary attacking arbitrarily many instances of the one- time signature scheme, and making at most q hash queries, the probability with which the adversary is able to forge a signature with respect to any of the instances is at most q2^(1-8n). Here n is the number of bytes in the output of the hash function (as defined in Section 4.1). Thus, the security of the algorithms defined in this note can be roughly described as follows. For a security level of roughly 128 bits, assuming that there are no quantum computers, use n=16 by selecting an algorithm identifier with N16 in its name. For a security level of roughly 128 bits, assuming that there are quantum computers that can compute the input to an arbitrary function with computational cost equivalent to the square root of the size of the domain of that function [Grover96], use n=32 by selecting an algorithm identifier with N32 in its name. 10.1. Stateful signature algorithm The LMS signature system, like all N-time signature systems, requires that the signer maintain state across different invocations of the signing algorithm, to ensure that none of the component one-time signature systems are used more than once. This section calls out some important practical considerations around this statefulness. In a typical computing environment, a private key will be stored in non-volatile media such as on a hard drive. Before it is used to sign a message, it will be read into an application's Random Access Memory (RAM). After a signature is generated, the value of the private key will need to be updated by writing the new value of the private key into non-volatile storage. It is essential for security that the application ensure that this value is actually written into that storage, yet there may be one or more memory caches between it and the application. Memory caching is commonly done in the file system, and in a physical memory unit on the hard disk that is McGrew & Curcio Expires April 21, 2016 [Page 26]

Internet-Draft Hash-Based Signatures October 2015 dedicated to that purpose. To ensure that the updated value is written to physical media, the application may need to take several special steps. In a POSIX environment, for instance,the O_SYNC flag (for the open() system call) will cause invocations of the write() system call to block the calling process until the data has been to the underlying hardware. However, if that hardware has its own memory cache, it must be separately dealt with using an operating system or device specific tool such as hdparm to flush the on-drive cache, or turn off write caching for that drive. Because these details vary across different operating systems and devices, this note does not attempt to provide complete guidance; instead, we call the implementer's attention to these issues. When hierarchical signatures are used, an easy way to minimize the private key synchronization issues is to have the private key for the second level resident in RAM only, and never write that value into non-volatile memory. A new second level public/private key pair will be generated whenever the application (re)starts; thus, failures such as a power outage or application crash are automatically accommodated. Implementations SHOULD use this approach wherever possible. 10.2. Security of LM-OTS Checksum To show the security of LM-OTS checksum, we consider the signature y of a message with a private key x and let h = H(message) and c = Cksm(H(message)) (see Section 4.7). To attempt a forgery, an attacker may try to change the values of h and c. Let h' and c' denote the values used in the forgery attempt. If for some integer j in the range 0 to (u-1), inclusive, a' = coef(h', j, w), a = coef(h, j, w), and a' > a then the attacker can compute F^a'(x[j]) from F^a(x[j]) = y[j] by iteratively applying function F to the j^th term of the signature an additional (a' - a) times. However, as a result of the increased number of hashing iterations, the checksum value c' will decrease from its original value of c. Thus a valid signature's checksum will have, for some number k in the range u to (p-1), inclusive, b' = coef(c', k, w), b = coef(c, k, w), and McGrew & Curcio Expires April 21, 2016 [Page 27]

Internet-Draft Hash-Based Signatures October 2015 b' < b Due to the one-way property of F, the attacker cannot easily compute F^b'(x[k]) from F^b(x[k]) = y[k]. 11. Acknowledgements Thanks are due to Chirag Shroff, Andreas Hulsing, Burt Kaliski, Eric Osterweil, Ahmed Kosba, Russ Housley, and Scott Fluhrer for constructive suggestions and valuable detailed review. We esepcially acknowledge Jerry Solinas, Laurie Law, and Kevin Igoe, who pointed out the security benefits of the approach of Leighton and Micali [USPTO5432852] and Jonathan Katz, who gave us security guidance. 12. References 12.1. Normative References [FIPS180] National Institute of Standards and Technology, "Secure Hash Standard (SHS)", FIPS 180-4, March 2012. [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <http://www.rfc-editor.org/info/rfc2119>. [RFC2434] Narten, T. and H. Alvestrand, "Guidelines for Writing an IANA Considerations Section in RFCs", RFC 2434, DOI 10.17487/RFC2434, October 1998, <http://www.rfc-editor.org/info/rfc2434>. [RFC4506] Eisler, M., Ed., "XDR: External Data Representation Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May 2006, <http://www.rfc-editor.org/info/rfc4506>. [USPTO5432852] Leighton, T. and S. Micali, "Large provably fast and secure digital signature schemes from secure hash functions", U.S. Patent 5,432,852, July 1995. 12.2. Informative References [C:Merkle87] Merkle, R., "A Digital Signature Based on a Conventional Encryption Function", Lecture Notes in Computer Science crypto87vol, 1988. McGrew & Curcio Expires April 21, 2016 [Page 28]

Internet-Draft Hash-Based Signatures October 2015 [C:Merkle89a] Merkle, R., "A Certified Digital Signature", Lecture Notes in Computer Science crypto89vol, 1990. [C:Merkle89b] Merkle, R., "One Way Hash Functions and DES", Lecture Notes in Computer Science crypto89vol, 1990. [Grover96] Grover, L., "A fast quantum mechanical algorithm for database search", 28th ACM Symposium on the Theory of Computing p. 212, 1996. [Katz15] Katz, J., "Analysis of a proposed hash-based signature standard", Contribution to IRTF http://www.cs.umd.edu/~jkatz/papers/HashBasedSigs.pdf, 2015. [Merkle79] Merkle, R., "Secrecy, Authentication, and Public Key Systems", Stanford University Information Systems Laboratory Technical Report 1979-1, 1979. Appendix A. LM-OTS Parameter Options A table illustrating various combinations of n and w with the associated values of u, v, ls, and p is provided in Table 4. The parameters u, v, ls, and p are computed as follows: u = ceil(8*n/w) v = ceil((floor(lg((2^w - 1) * u)) + 1) / w) ls = (number of bits in sum) - (v * w) p = u + v Here u and v represent the number of w-bit fields required to contain the hash of the message and the checksum byte strings, respectively. The "number of bits in sum" is defined according to Section 4.6. And as the value of p is the number of w-bit elements of ( H(message) || Cksm(H(message)) ), it is also equivalently the number of byte strings that form the private key and the number of byte strings in the signature. McGrew & Curcio Expires April 21, 2016 [Page 29]

Internet-Draft Hash-Based Signatures October 2015 +---------+------------+-----------+-----------+-------+------------+ | Hash | Winternitz | w-bit | w-bit | Left | Total | | Length | Parameter | Elements | Elements | Shift | Number of | | in | (w) | in Hash | in | (ls) | w-bit | | Bytes | | (u) | Checksum | | Elements | | (n) | | | (v) | | (p) | +---------+------------+-----------+-----------+-------+------------+ | 16 | 1 | 128 | 8 | 8 | 137 | | | | | | | | | 16 | 2 | 64 | 4 | 8 | 68 | | | | | | | | | 16 | 4 | 32 | 3 | 4 | 35 | | | | | | | | | 16 | 8 | 16 | 2 | 0 | 18 | | | | | | | | | 32 | 1 | 256 | 9 | 7 | 265 | | | | | | | | | 32 | 2 | 128 | 5 | 6 | 133 | | | | | | | | | 32 | 4 | 64 | 3 | 4 | 67 | | | | | | | | | 32 | 8 | 32 | 2 | 0 | 34 | +---------+------------+-----------+-----------+-------+------------+ Table 4 Appendix B. An iterative algorithm for computing an LMS public key The LMS public key can be computed using the following algorithm or any equivalent method. The algorithm uses a stack of hashes for data and a separate stack of integers to keep track of the level of the tree. It also makes use of a hash function with the typical init/update/final interface to hash functions; the result of the invocations hash_init(), hash_update(N[1]), hash_update(N[2]), ... , hash_update(N[n]), v = hash_final(), in that order, is identical to that of the invocation of H(N[1] || N[2] || ... || N[n]). McGrew & Curcio Expires April 21, 2016 [Page 30]

Internet-Draft Hash-Based Signatures October 2015 Generating an LMS Public Key From an LMS Private Key for ( i = 0; i < num_lmots_keys; i = i + 2 ) { level = 0; for ( j = 0; j < 2; j = j + 1 ) { r = node number push H(OTS_PUBKEY[i+j] || I || uint32str(r) || D_LEAF) onto data stack push level onto the integer stack } while ( height of the integer stack >= 2 ) { if level of the top 2 elements on the integer stack are equal { hash_init() siblings = "" repeat ( 2 ) { siblings = (pop(data stack) || siblings) level = pop(integer stack) } hash_update(siblings) r = node number hash_update(I || uint32str(r) || D_INTR) push hash_final() onto the data stack push (level + 1) onto the integer stack } } } public_key = pop(data stack) Note that this pseudocode expects that all 2^h leaves of the tree have equal depth. Neither stack ever contains more than h+1 elements. For typical parameters, these stacks will hold around 512 bytes of data. Appendix C. Example implementation # example implementation for Leighton-Micali hash based signatures # Internet draft # # Notes: # # * only a limted set of parameters are supported; in particular, # * w=8 and n=32 # # * HLMS, LMS, and LM-OTS are all implemented # # * uncommenting print statements may be useful for debugging, or # for understanding the mechanics of # # McGrew & Curcio Expires April 21, 2016 [Page 31]

Internet-Draft Hash-Based Signatures October 2015 # LMOTS constants # D_ITER = chr(0x00) # in the iterations of the LM-OTS algorithms D_PBLC = chr(0x01) # when computing the hash of all of the iterates in the LM-OTS algorithm D_MESG = chr(0x02) # when computing the hash of the message in the LMOTS algorithms D_LEAF = chr(0x03) # when computing the hash of the leaf of an LMS tree D_INTR = chr(0x04) # when computing the hash of an interior node of an LMS tree NULL = chr(0) # used as padding for encoding lmots_sha256_n32_w8 = 0x08000008 # typecode for LM-OTS with n=32, w=8 lms_sha256_n32_h10 = 0x02000002 # typecode for LMS with n=32, h=10 hlms_sha256_n32_l2 = 0x01000001 # typecode for two-level HLMS with n=32 # LMOTS parameters # n = 32; p = 34; w = 8; ls = 0 def bytes_in_lmots_sig(): return n*(p+1)+40 # 4 + n + 31 + 1 + 4 + n*p from Crypto.Hash import SHA256 from Crypto import Random # SHA256 hash function # def H(x): # print "hash input: " + stringToHex(x) h = SHA256.new() h.update(x) return h.digest()[0:n] def sha256_iter(x, num): tmp = x for j in range(0, num): tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER) # entropy source # entropySource = Random.new() # integer to string conversion # def uint32ToString(x): c4 = chr(x & 0xff) x = x >> 8 c3 = chr(x & 0xff) McGrew & Curcio Expires April 21, 2016 [Page 32]

Internet-Draft Hash-Based Signatures October 2015 x = x >> 8 c2 = chr(x & 0xff) x = x >> 8 c1 = chr(x & 0xff) return c1 + c2 + c3 + c4 def uint16ToString(x): c2 = chr(x & 0xff) x = x >> 8 c1 = chr(x & 0xff) return c1 + c2 def uint8ToString(x): return chr(x) def stringToUint(x): sum = 0 for c in x: sum = sum * 256 + ord(c) return sum # string-to-hex function needed for debugging # def stringToHex(x): return "".join("{:02x}".format(ord(c)) for c in x) # LM-OTS functions # def encode_lmots_sig(C, I, q, y): result = uint32ToString(lmots_sha256_n32_w8) + C + I + NULL + q for i, e in enumerate(y): result = result + y[i] return result def decode_lmots_sig(sig): if (len(sig) != bytes_in_lmots_sig()): print "error decoding signature; incorrect length (" + str(len(sig)) + " bytes)" typecode = sig[0:4] if (typecode != uint32ToString(lmots_sha256_n32_w8)): print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(lmots_sha256_n32_w8)) return "" C = sig[4:n+4] I = sig[n+4:n+35] q = sig[n+36:n+40] # note: skip over NULL y = list() pos = n+40 for i in range(0, p): y.append(sig[pos:pos+n]) McGrew & Curcio Expires April 21, 2016 [Page 33]

Internet-Draft Hash-Based Signatures October 2015 pos = pos + n return C, I, q, y def print_lmots_sig(sig): C, I, q, y = decode_lmots_sig(sig) print "C:\t" + stringToHex(C) print "I:\t" + stringToHex(I) print "q:\t" + stringToHex(q) for i, e in enumerate(y): print "y[" + str(i) + "]:\t" + stringToHex(e) # Algorithm 0: Generating a Private Key # def lmots_gen_priv(): priv = list() for i in range(0, p): priv.append(entropySource.read(n)) return priv # Algorithm 1: Generating a Public Key From a Private Key # def lmots_gen_pub(private_key, I, q): hash = SHA256.new() hash.update(I + q) for i, x in enumerate(private_key): tmp = x # print "i:" + str(i) + " range: " + str(range(0, 256)) for j in range(0, 256): tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER) hash.update(tmp) hash.update(D_PBLC) return hash.digest() # Algorithm 2: Merkle Checksum Calculation # def checksum(x): sum = 0 for c in x: sum = sum + ord(c) # print format(sum, '04x') c1 = chr(sum >> 8) c2 = chr(sum & 0xff) return c1 + c2 # Algorithm 3: Generating a Signature From a Private Key and a Message # def lmots_gen_sig(private_key, I, q, message): C = entropySource.read(n) McGrew & Curcio Expires April 21, 2016 [Page 34]

Internet-Draft Hash-Based Signatures October 2015 hashQ = H(message + C + I + q + D_MESG) V = hashQ + checksum(hashQ) # print "V: " + stringToHex(V) y = list() for i, x in enumerate(private_key): tmp = x # print "i:" + str(i) + " range: " + str(range(0, ord(V[i]))) for j in range(0, ord(V[i])): tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER) y.append(tmp) return encode_lmots_sig(C, I, q, y) def lmots_sig_to_pub(sig, message): C, I, q, y = decode_lmots_sig(sig) hashQ = H(message + C + I + q + D_MESG) V = hashQ + checksum(hashQ) # print "V: " + stringToHex(V) hash = SHA256.new() hash.update(I + q) for i, y in enumerate(y): tmp = y # print "i:" + str(i) + " range: " + str(range(ord(V[i]), 256)) for j in range(ord(V[i]), 256): tmp = H(tmp + I + q + uint16ToString(i) + uint8ToString(j) + D_ITER) hash.update(tmp) hash.update(D_PBLC) return hash.digest() # Algorithm 4: Verifying a Signature and Message Using a Public Key # def lmots_verify_sig(public_key, sig, message): z = lmots_sig_to_pub(sig, message) # print "z: " + stringToHex(z) if z == public_key: return 1 else: return 0 # LM-OTS test functions # I = entropySource.read(31) q = uint32ToString(0) private_key = lmots_gen_priv() print "LMOTS private key: " for i, x in enumerate(private_key): print "x[" + str(i) + "]:\t" + stringToHex(x) McGrew & Curcio Expires April 21, 2016 [Page 35]

Internet-Draft Hash-Based Signatures October 2015 public_key = lmots_gen_pub(private_key, I, q) print "LMOTS public key: " print stringToHex(public_key) message = "The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no warrants shall issue, but upon probable cause, supported by oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized." print "message: " + message sig = lmots_gen_sig(private_key, I, q, message) print "LMOTS signature byte length: " + str(len(sig)) print "LMOTS signature: " print_lmots_sig(sig) print "verification: " print "true positive test: " if (lmots_verify_sig(public_key, sig, message) == 1): print "passed: message/signature pair is valid as expected" else: print "failed: message/signature pair is invalid" print "false positive test: " if (lmots_verify_sig(public_key, sig, "some other message") == 1): print "failed: message/signature pair is valid (expected failure)" else: print "passed: message/signature pair is invalid as expected" # LMS N-time signatures functions # h = 10 # height (number of levels -1) of tree def encode_lms_sig(lmots_sig, path): result = uint32ToString(lms_sha256_n32_h10) + lmots_sig for i, e in enumerate(path): result = result + path[i] return result def decode_lms_sig(sig): typecode = sig[0:4] if (typecode != uint32ToString(lms_sha256_n32_h10)): print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(lms_sha256_h10)) return "" pos = 4 + bytes_in_lmots_sig() lmots_sig = sig[4:pos] McGrew & Curcio Expires April 21, 2016 [Page 36]

Internet-Draft Hash-Based Signatures October 2015 path = list() for i in range(0,h): # print "sig[" + str(i) + "]:\t" + stringToHex(sig[pos:pos+n]) path.append(sig[pos:pos+n]) pos = pos + n return lmots_sig, path def print_lms_sig(sig): lmots_sig, path = decode_lms_sig(sig) print_lmots_sig(lmots_sig) for i, e in enumerate(path): print "path[" + str(i) + "]:\t" + str(stringToHex(e)) def bytes_in_lms_sig(): return bytes_in_lmots_sig() + h*n + 4 class lms_private_key(object): # Algorithm for computing root and other nodes (alternative to Algorithm 6) # def T(self, j): # print "T(" + str(j) + ")" if (j >= 2**h): self.nodes[j] = H(self.pub[j - 2**h] + self.I + uint32ToString(j) + D_LEAF) return self.nodes[j] else: self.nodes[j] = H(self.T(2*j) + self.T(2*j+1) + self.I + uint32ToString(j) + D_INTR) return self.nodes[j] def __init__(self): self.I = entropySource.read(31) self.priv = list() self.pub = list() for q in range(0, 2**h): # print "generating " + str(q) + "th OTS key" ots_priv = lmots_gen_priv() ots_pub = lmots_gen_pub(ots_priv, self.I, uint32ToString(q)) self.priv.append(ots_priv) self.pub.append(ots_pub) self.leaf_num = 0 self.nodes = {} self.lms_public_key = self.T(1) def num_sigs_remaining(): return 2**h - self.leaf_num def printHex(self): for i, p in enumerate(self.priv): McGrew & Curcio Expires April 21, 2016 [Page 37]

Internet-Draft Hash-Based Signatures October 2015 print "priv[" + str(i) + "]:" for j, x in enumerate(p): print "x[" + str(j) + "]:\t" + stringToHex(x) print "pub[" + str(i) + "]:\t" + stringToHex(self.pub[i]) for t, T in self.nodes.items(): print "T(" + str(t) + "):\t" + stringToHex(T) print "pub: \t" + stringToHex(self.lms_public_key) def get_public_key(self): return self.lms_public_key def get_path(self, leaf_num): node_num = leaf_num + 2**h # print "signing node " + str(node_num) path = list() while node_num > 1: if (node_num % 2): # print "path" + str(node_num - 1) + ": " + stringToHex(self.nodes[node_num - 1]) path.append(self.nodes[node_num - 1]) else: # print "path " + str(node_num + 1) + ": " + stringToHex(self.nodes[node_num + 1]) path.append(self.nodes[node_num + 1]) node_num = node_num/2 return path def sign(self, message): if (self.leaf_num >= 2**h): return "" sig = lmots_gen_sig(self.priv[self.leaf_num], self.I, uint32ToString(self.leaf_num), message) # C, I, q, y = decode_lmots_sig(sig) path = self.get_path(self.leaf_num) leaf_num = self.leaf_num self.leaf_num = self.leaf_num + 1 return encode_lms_sig(sig, path) class lms_public_key(object): def __init__(self, value): self.value = value def verify(self, message, sig): lmots_sig, path = decode_lms_sig(sig) C, I, q, y = decode_lmots_sig(lmots_sig) # note: only q is actually needed here node_num = stringToUint(q) + 2**h # print "verifying node " + str(node_num) pathvalue = iter(path) tmp = lmots_sig_to_pub(lmots_sig, message) McGrew & Curcio Expires April 21, 2016 [Page 38]

Internet-Draft Hash-Based Signatures October 2015 tmp = H(tmp + I + uint32ToString(node_num) + D_LEAF) while node_num > 1: # print "S(" + str(node_num) + "):\t" + stringToHex(tmp) if (node_num % 2): # print "adding node " + str(node_num - 1) tmp = H(pathvalue.next() + tmp + I + uint32ToString(node_num/2) + D_INTR) else: # print "adding node " + str(node_num + 1) tmp = H(tmp + pathvalue.next() + I + uint32ToString(node_num/2) + D_INTR) node_num = node_num/2 # print "pubkey: " + stringToHex(tmp) if (tmp == self.value): return 1 else: return 0 # test LMS signatures # print "LMS test" lms_priv = lms_private_key() lms_pub = lms_public_key(lms_priv.get_public_key()) # lms_priv.printHex() for i in range(0, 2**h): sig = lms_priv.sign(message) print "LMS signature byte length: " + str(len(sig)) # print_lms_sig(sig) print "true positive test" if (lms_pub.verify(message, sig) == 1): print "passed: LMS message/signature pair is valid" else: print "failed: LMS message/signature pair is invalid" print "false positive test" if (lms_pub.verify("other message", sig) == 1): print "failed: LMS message/signature pair is valid (expected failure)" else: print "passed: LMS message/signature pair is invalid as expected" McGrew & Curcio Expires April 21, 2016 [Page 39]

Internet-Draft Hash-Based Signatures October 2015 # Hierarchical LMS signatures (HLMS) def encode_hlms_sig(pub2, sig1, lms_sig): result = uint32ToString(hlms_sha256_n32_l2) result = result + pub2 result = result + sig1 result = result + lms_sig return result def decode_hlms_sig(sig): typecode = sig[0:4] if (typecode != uint32ToString(hlms_sha256_n32_l2)): print "error decoding signature; got typecode " + stringToHex(typecode) + ", expected: " + stringToHex(uint32ToString(hlms_sha256_n32_l2)) return "" pub2 = sig[4:36] lms_sig_len = bytes_in_lms_sig() sig1 = sig[36:36+lms_sig_len] lms_sig = sig[36+lms_sig_len:36+2*lms_sig_len] return pub2, sig1, lms_sig def print_hlms_sig(sig): pub2, sig1, lms_sig = decode_hlms_sig(sig) print "pub2:\t" + stringToHex(pub2) print "sig1: " print_lms_sig(sig1) print "sig2: " print_lms_sig(lms_sig) class hlms_private_key(object): def __init__(self): self.prv1 = lms_private_key() self.init_level_2() def init_level_2(self): self.prv2 = lms_private_key() self.sig1 = self.prv1.sign(self.prv2.get_public_key()) def get_public_key(self): return self.prv1.get_public_key() def sign(self, message): lms_sig = self.prv2.sign(message) if (lms_sig == ""): print "refreshing level 2 public/private key pair" self.init_level_2() lms_sig = self.prv2.sign(message) return encode_hlms_sig(self.prv2.get_public_key(), self.sig1, lms_sig) McGrew & Curcio Expires April 21, 2016 [Page 40]

Internet-Draft Hash-Based Signatures October 2015 class hlms_public_key(object): def __init__(self, value): self.pub1 = lms_public_key(value) def verify(self, message, sig): pub2, sig1, lms_sig = decode_hlms_sig(sig) if (self.pub1.verify(pub2, sig1) == 1): if (lms_public_key(pub2).verify(message, lms_sig) == 1): return 1 else: print "pub2 verification of lms_sig did not pass" else: print "pub1 verification of sig1 did not pass" return 0 print "HLMS testing" hlms_prv = hlms_private_key() hlms_pub = hlms_public_key(hlms_prv.get_public_key()) for i in range(0, 4096): sig = hlms_prv.sign(message) # print_hlms_sig(sig) print "HLMS signature byte length: " + str(len(sig)) print "testing verification (" + str(i) + "th iteration)" print "true positive test" if (hlms_pub.verify(message, sig) == 1): print "passed; HLMS message/signature pair is valid" else: print "failed; HLMS message/signature pair is invalid" print "false positive test" if (hlms_pub.verify("other message", sig) == 1): print "failed; HLMS message/signature pair is valid (expected failure)" else: print "passed; HLMS message/signature pair is invalid as expected" McGrew & Curcio Expires April 21, 2016 [Page 41]

```
Internet-Draft Hash-Based Signatures October 2015
Authors' Addresses
David McGrew
Cisco Systems
13600 Dulles Technology Drive
Herndon, VA 20171
USA
Email: mcgrew@cisco.com
Michael Curcio
Cisco Systems
7025-2 Kit Creek Road
Research Triangle Park, NC 27709-4987
USA
Email: micurcio@cisco.com
McGrew & Curcio Expires April 21, 2016 [Page 42]
```