LeightonMicali HashBased Signatures
RFC 8554
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(April 2019; No errata)
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Internet Research Task Force (IRTF) D. McGrew Request for Comments: 8554 M. Curcio Category: Informational S. Fluhrer ISSN: 20701721 Cisco Systems April 2019 LeightonMicali HashBased Signatures Abstract This note describes a digitalsignature 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 onetime 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 are naturally resistant to side channel attacks. Unlike many other signature systems, hashbased signatures would still be secure even if it proves feasible for an attacker to build a quantum computer. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. This has been reviewed by many researchers, both in the research group and outside of it. The Acknowledgements section lists many of them. Status of This Memo This document is not an Internet Standards Track specification; it is published for informational purposes. This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internetrelated research and development activities. These results might not be suitable for deployment. This RFC represents the consensus of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not candidates for any level of Internet Standard; see Section 2 of RFC 7841. Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at https://www.rfceditor.org/info/rfc8554. McGrew, et al. Informational [Page 1] RFC 8554 LMS HashBased Signatures April 2019 Copyright Notice Copyright (c) 2019 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 (https://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. CFRG Note on PostQuantum Cryptography . . . . . . . . . 5 1.2. Intellectual Property . . . . . . . . . . . . . . . . . . 6 1.2.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . 6 1.3. Conventions Used in This Document . . . . . . . . . . . . 6 2. Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1. Operators . . . . . . . . . . . . . . . . . . . . . . 7 3.1.2. Functions . . . . . . . . . . . . . . . . . . . . . . 8 3.1.3. Strings of wBit Elements . . . . . . . . . . . . . . 8 3.2. Typecodes . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3. Notation and Formats . . . . . . . . . . . . . . . . . . 9 4. LMOTS OneTime Signatures . . . . . . . . . . . . . . . . . 12 4.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 13 4.2. Private Key . . . . . . . . . . . . . . . . . . . . . . . 14 4.3. Public Key . . . . . . . . . . . . . . . . . . . . . . . 15 4.4. Checksum . . . . . . . . . . . . . . . . . . . . . . . . 15 4.5. Signature Generation . . . . . . . . . . . . . . . . . . 16 4.6. Signature Verification . . . . . . . . . . . . . . . . . 17 5. LeightonMicali Signatures . . . . . . . . . . . . . . . . . 19 5.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 19 5.2. LMS Private Key . . . . . . . . . . . . . . . . . . . . . 20 5.3. LMS Public Key . . . . . . . . . . . . . . . . . . . . . 21 5.4. LMS Signature . . . . . . . . . . . . . . . . . . . . . . 22 5.4.1. LMS Signature Generation . . . . . . . . . . . . . . 23 5.4.2. LMS Signature Verification . . . . . . . . . . . . . 24 6. Hierarchical Signatures . . . . . . . . . . . . . . . . . . . 26 6.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 29 6.2. Signature Generation . . . . . . . . . . . . . . . . . . 30 6.3. Signature Verification . . . . . . . . . . . . . . . . . 32 6.4. Parameter Set Recommendations . . . . . . . . . . . . . . 32 7. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . 34 7.1. Security String . . . . . . . . . . . . . . . . . . . . . 35 McGrew, et al. Informational [Page 2] RFC 8554 LMS HashBased Signatures April 2019 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 36 9. Security Considerations . . . . . . . . . . . . . . . . . . . 38 9.1. Hash Formats . . . . . . . . . . . . . . . . . . . . . . 39 9.2. Stateful Signature Algorithm . . . . . . . . . . . . . . 40 9.3. Security of LMOTS Checksum . . . . . . . . . . . . . . . 41 10. Comparison with Other Work . . . . . . . . . . . . . . . . . 42 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 43 11.1. Normative References . . . . . . . . . . . . . . . . . . 43 11.2. Informative References . . . . . . . . . . . . . . . . . 43 Appendix A. Pseudorandom Key Generation . . . . . . . . . . . . 45 Appendix B. LMOTS Parameter Options . . . . . . . . . . . . . . 45 Appendix C. An Iterative Algorithm for Computing an LMS Public Key . . . . . . . . . . . . . . . . . . . . . . . . 47 Appendix D. Method for Deriving Authentication Path for a Signature . . . . . . . . . . . . . . . . . . . . . 48 Appendix E. Example Implementation . . . . . . . . . . . . . . . 49 Appendix F. Test Cases . . . . . . . . . . . . . . . . . . . . . 49 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 60 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 61 1. Introduction Onetime signature systems, and generalpurpose signature systems built out of onetime 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 moderately slow key generation (in comparison with RSA and ECDSA (Elliptic Curve Digital Signature Algorithm)). Private keys can be made very small by appropriate key generation, for example, as described in Appendix A. In recent years, there has been interest in these systems because of their postquantum security and their suitability for compact verifier implementations. This note describes the Leighton and Micali adaptation [USPTO5432852] of the original LamportDiffieWinternitzMerkle onetime signature system [Merkle79] [C:Merkle87] [C:Merkle89a] [C:Merkle89b] and general signature system [Merkle79] with enough specificity to ensure interoperability between implementations. A signature system provides asymmetric message authentication. The keygeneration 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 OneTime Signature (OTS) system can be used to sign one message securely but will become insecure if more than one is signed with the same public/ McGrew, et al. Informational [Page 3] RFC 8554 LMS HashBased Signatures April 2019 private key pair. An Ntime signature system can be used to sign N or fewer messages securely. A Merkletree signature scheme is an Ntime signature system that uses an OTS system as a component. In the Merkle scheme, a binary tree of height h is used to hold 2^h OTS key pairs. Each interior node of the tree holds a value that is the hash of the values of its two child nodes. The public key of the tree is the value of the root node (a recursive hash of the OTS public keys), while the private key of the tree is the collection of all the OTS private keys, together with the index of the next OTS private key to sign the next message with. In this note, we describe the LeightonMicali Signature (LMS) system (a variant of the Merkle scheme) with the Hierarchical Signature System (HSS) built on top of it that allows it to efficiently scale to larger numbers of signatures. In order to support signing a large number of messages on resourceconstrained systems, the Merkle tree can be subdivided into a number of smaller trees. Only the bottommost tree is used to sign messages, while trees above that are used to sign the public keys of their children. For example, in the simplest case with two levels with both levels consisting of height h trees, the root tree is used to sign 2^h trees with 2^h OTS key pairs, and each secondlevel tree has 2^h OTS key pairs, for a total of 2^(2h) bottomlevel key pairs, and so can sign 2^(2h) messages. The advantage of this scheme is that only the active trees need to be instantiated, which saves both time (for key generation) and space (for key storage). On the other hand, using a multilevel signature scheme increases the size of the signature as well as the signature verification time. This note is structured as follows. Notes on postquantum cryptography are discussed in Section 1.1. Intellectual property issues are discussed in Section 1.2. The notation used within this note is defined in Section 3, and the public formats are described in Section 3.3. The LeightonMicali OneTime Signature (LMOTS) system is described in Section 4, and the LMS and HSS Ntime signature systems are described in Sections 5 and 6, respectively. Sufficient detail is provided to ensure interoperability. The rationale for the design decisions is given in Section 7. The IANA registry for these signature systems is described in Section 8. Security considerations are presented in Section 9. Comparison with another hashbased signature algorithm (eXtended Merkle Signature Scheme (XMSS)) is in Section 10. This document represents the rough consensus of the CFRG. McGrew, et al. Informational [Page 4] RFC 8554 LMS HashBased Signatures April 2019 1.1. CFRG Note on PostQuantum Cryptography All postquantum algorithms documented by the Crypto Forum Research Group (CFRG) are today considered ready for experimentation and further engineering development (e.g., to establish the impact of performance and sizes on IETF protocols). However, at the time of writing, we do not have significant deployment experience with such algorithms. Many of these algorithms come with specific restrictions, e.g., change of classical interface or less cryptanalysis of proposed parameters than established schemes. The CFRG has consensus that all documents describing postquantum technologies include the above paragraph and a clear additional warning about any specific restrictions, especially as those might affect use or deployment of the specific scheme. That guidance may be changed over time via document updates. Additionally, for LMS: CFRG consensus is that we are confident in the cryptographic security of the signature schemes described in this document against quantum computers, given the current state of the research community's knowledge about quantum algorithms. Indeed, we are confident that the security of a significant part of the Internet could be made dependent on the signature schemes defined in this document, if developers take care of the following. In contrast to traditional signature schemes, the signature schemes described in this document are stateful, meaning the secret key changes over time. If a secret key state is used twice, no cryptographic security guarantees remain. In consequence, it becomes feasible to forge a signature on a new message. This is a new property that most developers will not be familiar with and requires careful handling of secret keys. Developers should not use the schemes described here except in systems that prevent the reuse of secret key states. Note that the fact that the schemes described in this document are stateful also implies that classical APIs for digital signatures cannot be used without modification. The API MUST be able to handle a dynamic secret key state; that is, the API MUST allow the signaturegeneration algorithm to update the secret key state. McGrew, et al. Informational [Page 5] RFC 8554 LMS HashBased Signatures April 2019 1.2. Intellectual Property This document is based on U.S. Patent 5,432,852, which was issued over twenty years ago and is thus expired. 1.2.1. Disclaimer This document is not intended as legal advice. Readers are advised to consult with their own legal advisers if they would like a legal interpretation of their rights. The IETF policies and processes regarding intellectual property and patents are outlined in [RFC8179] and at <https://datatracker.ietf.org/ipr/about>. 1.3. Conventions Used in This Document The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here. 2. Interface The LMS signing algorithm is stateful; it modifies and updates the private key as a side effect of generating a signature. Once a particular value of the private key is used to sign one message, it MUST NOT be used to sign another. The keygeneration 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 Ntime 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; it returns an indication of whether or not the signatureandmessage pair is valid. McGrew, et al. Informational [Page 6] RFC 8554 LMS HashBased Signatures April 2019 A message/signature pair is 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 is 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 of three. An array of byte strings is an ordered set, indexed starting at zero, in which all strings have the same length. Unsigned integers are converted into byte strings by representing them in network byte order. To make the number of bytes in the representation explicit, we define the functions u8str(X), u16str(X), and u32str(X), which take a nonnegative integer X as input and return one, two, and fourbyte strings, respectively. We also make use of the function strTou32(S), which takes a fourbyte string S as input and returns a nonnegative integer; the identity u32str(strTou32(S)) = S holds for any fourbyte string S. 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 (with a and b being restricted to integers in this case) + : a + b denotes the sum of a and b  : a  b denotes the difference of a and b AND : a AND b denotes the bitwise AND of the two nonnegative integers a and b (represented in binary notation) McGrew, et al. Informational [Page 7] RFC 8554 LMS HashBased Signatures April 2019 The standard order of operations is used when evaluating arithmetic expressions. When B is a byte and i is an integer, then B >> i denotes the logical rightshift operation by i bit positions. Similarly, B << i denotes the logical leftshift operation. If S and T are byte strings, then S  T denotes the concatenation of S and T. If S and T are equallength byte strings, then S AND T denotes the bitwise logical and operation. The ith element in an array A is denoted as A[i]. 3.1.2. Functions If r is a nonnegative real number, then we define the following functions: ceil(r) : returns the smallest integer greater than or equal to r floor(r) : returns the largest integer less than or equal to r lg(r) : returns the base2 logarithm of r 3.1.3. Strings of wBit Elements If S is a byte string, then byte(S, i) denotes its ith byte, where the index starts at 0 at the left. Hence, byte(S, 0) is the leftmost byte of S, byte(S, 1) is the second byte from the left, and (assuming S is n bytes long) byte(S, n1) is the rightmost byte of S. In addition, bytes(S, i, j) denotes the range of bytes from the ith to the jth byte, inclusive. For example, if S = 0x02040608, then byte(S, 0) is 0x02 and bytes(S, 1, 2) is 0x0406. A byte string can be considered to be a string of wbit 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 ith, wbit value, if S is interpreted as a sequence of wbit values. That is, coef(S, i, w) = (2^w  1) AND ( byte(S, floor(i * w / 8)) >> (8  (w * (i % (8 / w)) + w)) ) McGrew, et al. Informational [Page 8] RFC 8554 LMS HashBased Signatures April 2019 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 fourbit 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 wbit values in S, then coef(S, i, w) is undefined, and an attempt to compute that value MUST raise an error. 3.2. Typecodes A typecode is an unsigned integer that is associated with a particular data format. The format of the LMOTS, LMS, and HSS signatures and public keys all begin with a typecode that indicates the precise details used in that format. These typecodes are represented as fourbyte unsigned integers in network byte order; equivalently, they are External Data Representation (XDR) enumerations (see Section 3.3). 3.3. Notation and Formats The signature and public key formats are formally defined in XDR to provide an unambiguous, machinereadable definition [RFC4506]. The private key format is not included as it is not needed for interoperability and an implementation MAY use any private key format. However, for clarity, we include an example of private key data in Test Case 2 of Appendix F. Though XDR is used, these formats McGrew, et al. Informational [Page 9] RFC 8554 LMS HashBased Signatures April 2019 are simple and easy to parse without any special tools. An illustration of the layout of data in these objects is provided below. The definitions are as follows: /* onetime signatures */ enum lmots_algorithm_type { lmots_reserved = 0, lmots_sha256_n32_w1 = 1, lmots_sha256_n32_w2 = 2, lmots_sha256_n32_w4 = 3, lmots_sha256_n32_w8 = 4 }; typedef opaque bytestring32[32]; struct lmots_signature_n32_p265 { bytestring32 C; bytestring32 y[265]; }; struct lmots_signature_n32_p133 { bytestring32 C; bytestring32 y[133]; }; struct lmots_signature_n32_p67 { bytestring32 C; bytestring32 y[67]; }; struct lmots_signature_n32_p34 { bytestring32 C; bytestring32 y[34]; }; union lmots_signature switch (lmots_algorithm_type type) { case lmots_sha256_n32_w1: lmots_signature_n32_p265 sig_n32_p265; case lmots_sha256_n32_w2: lmots_signature_n32_p133 sig_n32_p133; case lmots_sha256_n32_w4: lmots_signature_n32_p67 sig_n32_p67; case lmots_sha256_n32_w8: lmots_signature_n32_p34 sig_n32_p34; default: void; /* error condition */ }; McGrew, et al. Informational [Page 10] RFC 8554 LMS HashBased Signatures April 2019 /* hashbased signatures (hbs) */ enum lms_algorithm_type { lms_reserved = 0, lms_sha256_n32_h5 = 5, lms_sha256_n32_h10 = 6, lms_sha256_n32_h15 = 7, lms_sha256_n32_h20 = 8, lms_sha256_n32_h25 = 9 }; /* leightonmicali signatures (lms) */ union lms_path switch (lms_algorithm_type type) { case lms_sha256_n32_h5: bytestring32 path_n32_h5[5]; case lms_sha256_n32_h10: bytestring32 path_n32_h10[10]; case lms_sha256_n32_h15: bytestring32 path_n32_h15[15]; case lms_sha256_n32_h20: bytestring32 path_n32_h20[20]; case lms_sha256_n32_h25: bytestring32 path_n32_h25[25]; default: void; /* error condition */ }; struct lms_signature { unsigned int q; lmots_signature lmots_sig; lms_path nodes; }; struct lms_key_n32 { lmots_algorithm_type ots_alg_type; opaque I[16]; opaque K[32]; }; union lms_public_key switch (lms_algorithm_type type) { case lms_sha256_n32_h5: case lms_sha256_n32_h10: case lms_sha256_n32_h15: case lms_sha256_n32_h20: case lms_sha256_n32_h25: lms_key_n32 z_n32; McGrew, et al. Informational [Page 11] RFC 8554 LMS HashBased Signatures April 2019 default: void; /* error condition */ }; /* hierarchical signature system (hss) */ struct hss_public_key { unsigned int L; lms_public_key pub; }; struct signed_public_key { lms_signature sig; lms_public_key pub; }; struct hss_signature { signed_public_key signed_keys<7>; lms_signature sig_of_message; }; 4. LMOTS OneTime Signatures This section defines LMOTS 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 onetime signatures; each private key MUST be used at most 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 4.1), and the resulting digest is signed. In order to facilitate its use in an Ntime signature system, the LMOTS key generation, signing, and verification algorithms all take as input parameters I and q. The parameter I is a 16byte string that indicates which Merkle tree this LMOTS is used with. The parameter q is a 32bit integer that indicates the leaf of the Merkle tree where the OTS public key appears. These parameters are used as part of the security string, as listed in Section 7.1. When the LMOTS signature system is used outside of an Ntime signature system, the value I MAY be used to differentiate this onetime signature from others; however, the value q MUST be set to the all zero value. McGrew, et al. Informational [Page 12] RFC 8554 LMS HashBased Signatures April 2019 4.1. Parameters The signature system uses the parameters n and w, which are both positive integers. The algorithm description also makes use of the internal parameters p and ls, which are dependent on n and w. These parameters are summarized as follows: n : the number of bytes of the output of the hash function. w : the width (in bits) of the Winternitz coefficients; that is, the number of bits from the hash or checksum that is used with a single Winternitz chain. It is a member of the set { 1, 2, 4, 8 }. p : the number of nbyte string elements that make up the LMOTS signature. This is a function of n and w; the values for the defined parameter sets are listed in Table 1; it can also be computed by the algorithm given in Appendix B. ls : the number of leftshift bits used in the checksum function Cksm (defined in Section 4.4). H : a secondpreimageresistant cryptographic hash function that accepts byte strings of any length and returns an nbyte string. For more background on the cryptographic security requirements for H, see Section 9. The value of n is determined by the hash function selected for use as part of the LMOTS algorithm; the choice of this value has a strong effect on the security of the system. The parameter w determines the length of the Winternitz chains computed as a part of the OTS signature (which involve 2^w  1 invocations of the hash function); it has little effect on security. Increasing w will shorten the signature, but at a cost of a larger computation to generate and verify a signature. The values of p and ls are dependent on the choices of the parameters n and w, as described in Appendix B. Table 1 illustrates various combinations of n, w, p and ls, along with the resulting signature length. The value of w describes a space/time tradeoff; increasing the value of w will cause the signature to shrink (by decreasing the value of p) while increasing the amount of time needed to perform operations with it: generate the public key and generate and verify the signature. In general, the LMOTS signature is 4+n*(p+1) bytes long, and public key generation will take p*(2^w  1) + 1 hash computations (and signature generation and verification will take approximately half that on average). McGrew, et al. Informational [Page 13] RFC 8554 LMS HashBased Signatures April 2019 ++++++++  Parameter Set Name  H  n  w  p  ls  sig_len  ++++++++  LMOTS_SHA256_N32_W1  SHA256  32  1  265  7  8516           LMOTS_SHA256_N32_W2  SHA256  32  2  133  6  4292           LMOTS_SHA256_N32_W4  SHA256  32  4  67  4  2180           LMOTS_SHA256_N32_W8  SHA256  32  8  34  0  1124  ++++++++ Table 1 Here SHA256 denotes the SHA256 hash function defined in NIST standard [FIPS180]. 4.2. Private Key The format of the LMOTS private key is an internal matter to the implementation, and this document does not attempt to define it. One possibility is that the private key may consist of a typecode indicating the particular LMOTS algorithm, an array x[] containing p nbyte strings, and the 16byte string I and the 4byte string q. This private key MUST be used to sign (at most) one message. The following algorithm shows pseudocode for generating a private key. Algorithm 0: Generating a Private Key 1. Retrieve the values of q and I (the 16byte identifier of the LMS public/private key pair) from the LMS tree that this LMOTS private key will be used with 2. Set type to the typecode of the algorithm 3. Set n and p according to the typecode and Table 1 4. Compute the array x as follows: for ( i = 0; i < p; i = i + 1 ) { set x[i] to a uniformly random nbyte string } 5. Return u32str(type)  I  u32str(q)  x[0]  x[1]  ...  x[p1] 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 McGrew, et al. Informational [Page 14] RFC 8554 LMS HashBased Signatures April 2019 MUST match that of the LMOTS algorithm. Appendix A provides an example of a pseudorandom method for computing the LMOTS private key. 4.3. Public Key The LMOTS public key is generated from the private key by iteratively applying the function H to each individual element of x, for 2^w  1 iterations, then hashing all of the resulting values. The public key is generated from the private key using the following algorithm, or any equivalent process. Algorithm 1: Generating a OneTime Signature Public Key From a Private Key 1. Set type to the typecode of the algorithm 2. Set the integers n, p, and w according to the typecode and Table 1 3. Determine x, I, and q from the private key 4. Compute the string K as follows: for ( i = 0; i < p; i = i + 1 ) { tmp = x[i] for ( j = 0; j < 2^w  1; j = j + 1 ) { tmp = H(I  u32str(q)  u16str(i)  u8str(j)  tmp) } y[i] = tmp } K = H(I  u32str(q)  u16str(D_PBLC)  y[0]  ...  y[p1]) 5. Return u32str(type)  I  u32str(q)  K where D_PBLC is the fixed twobyte value 0x8080, which is used to distinguish the last hash from every other hash in this system. The public key is the value returned by Algorithm 1. 4.4. Checksum A checksum is used to ensure that any forgery attempt that manipulates the elements of an existing signature will be detected. This checksum is needed because an attacker can freely advance any of the Winternitz chains. That is, if this checksum were not present, then an attacker who could find a hash that has every digit larger than the valid hash could replace it (and adjust the Winternitz McGrew, et al. Informational [Page 15] RFC 8554 LMS HashBased Signatures April 2019 chains). The security property that the checksum provides is detailed in Section 9. The checksum function Cksm is defined as follows, where S denotes the nbyte string that is input to that function, and the value sum is a 16bit unsigned integer: Algorithm 2: Checksum Calculation sum = 0 for ( i = 0; i < (n*8/w); i = i + 1 ) { sum = sum + (2^w  1)  coef(S, i, w) } return (sum << ls) ls is the parameter that shifts the significant bits of the checksum into the positions that will actually be used by the coef function when encoding the digits of the checksum. The actual ls parameter is a function of the n and w parameters; the values for the currently defined parameter sets are shown in Table 1. It is calculated by the algorithm given in Appendix B. Because of the leftshift 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. 4.5. Signature Generation The LMOTS signature of a message is generated by doing the following in sequence: prepending the LMS key identifier I, the LMS leaf identifier q, the value D_MESG (0x8181), and the randomizer C to the message; computing the hash; concatenating the checksum of the hash to the hash itself; considering the resulting value as a sequence of wbit values; and using each of the wbit 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. Algorithm 3: Generating a OneTime Signature From a Private Key and a Message 1. Set type to the typecode of the algorithm 2. Set n, p, and w according to the typecode and Table 1 3. Determine x, I, and q from the private key 4. Set C to a uniformly random nbyte string McGrew, et al. Informational [Page 16] RFC 8554 LMS HashBased Signatures April 2019 5. Compute the array y as follows: Q = H(I  u32str(q)  u16str(D_MESG)  C  message) 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(I  u32str(q)  u16str(i)  u8str(j)  tmp) } y[i] = tmp } 6. Return u32str(type)  C  y[0]  ...  y[p1] Note that this algorithm results in a signature whose elements are intermediate values of the elements computed by the public key algorithm in Section 4.3. The signature is the string returned by Algorithm 3. Section 3.3 formally defines the structure of the string as the lmots_signature union. 4.6. Signature Verification In order to verify a message with its signature (an array of nbyte strings, denoted as y), the receiver must "complete" the chain of iterations of H using the wbit coefficients of the string resulting from the concatenation of the message hash and its checksum. This computation should result in a value that matches the provided public key. Algorithm 4a: Verifying a Signature and Message Using a Public Key 1. If the public key is not at least four bytes long, return INVALID. 2. Parse pubtype, I, q, and K from the public key as follows: a. pubtype = strTou32(first 4 bytes of public key) b. Set n according to the pubkey and Table 1; if the public key is not exactly 24 + n bytes long, return INVALID. c. I = next 16 bytes of public key d. q = strTou32(next 4 bytes of public key) e. K = next n bytes of public key McGrew, et al. Informational [Page 17] RFC 8554 LMS HashBased Signatures April 2019 3. Compute the public key candidate Kc from the signature, message, pubtype, and the identifiers I and q obtained from the public key, using Algorithm 4b. If Algorithm 4b returns INVALID, then return INVALID. 4. If Kc is equal to K, return VALID; otherwise, return INVALID. Algorithm 4b: Computing a Public Key Candidate Kc from a Signature, Message, Signature Typecode pubtype, and Identifiers I, q 1. If the signature is not at least four bytes long, return INVALID. 2. Parse sigtype, C, and y from the signature as follows: a. sigtype = strTou32(first 4 bytes of signature) b. If sigtype is not equal to pubtype, return INVALID. c. Set n and p according to the pubtype and Table 1; if the signature is not exactly 4 + n * (p+1) bytes long, return INVALID. d. C = next n bytes of signature e. y[0] = next n bytes of signature y[1] = next n bytes of signature ... y[p1] = next n bytes of signature 3. Compute the string Kc as follows: Q = H(I  u32str(q)  u16str(D_MESG)  C  message) for ( i = 0; i < p; i = i + 1 ) { a = coef(Q  Cksm(Q), i, w) tmp = y[i] for ( j = a; j < 2^w  1; j = j + 1 ) { tmp = H(I  u32str(q)  u16str(i)  u8str(j)  tmp) } z[i] = tmp } Kc = H(I  u32str(q)  u16str(D_PBLC)  z[0]  z[1]  ...  z[p1]) 4. Return Kc. McGrew, et al. Informational [Page 18] RFC 8554 LMS HashBased Signatures April 2019 5. LeightonMicali Signatures The LeightonMicali Signature (LMS) method can sign a potentially large but fixed number of messages. An LMS system uses two cryptographic components: a onetime 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 mbyte value, where m is the output length of the hash function. Each leaf of the tree contains the value of the public key of an LMOTS 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. Each node of the tree is associated with a node number, an unsigned integer that is denoted as node_num in the algorithms below, which is computed as follows. The root node has node number 1; for each node with node number N < 2^h (where h is the height of the tree), its left child has node number 2*N, while its right child has node number 2*N + 1. The result of this is that each node within the tree will have a unique node number, and the leaves will have node numbers 2^h, (2^h)+1, (2^h)+2, ..., (2^h)+(2^h)1. In general, the jth node at level i has node number 2^i + j. The node number can conveniently be computed when it is needed in the LMS algorithms, as described in those algorithms. 5.1. Parameters An LMS system has the following parameters: h : the height of the tree m : the number of bytes associated with each node H : a secondpreimageresistant cryptographic hash function that accepts byte strings of any length and returns an mbyte string. There are 2^h leaves in the tree. The overall strength of LMS signatures is governed by the weaker of the hash function used within the LMOTS and the hash function used within the LMS system. In order to minimize the risk, these two hash functions SHOULD be the same (so that an attacker could not take advantage of the weaker hash function choice). McGrew, et al. Informational [Page 19] RFC 8554 LMS HashBased Signatures April 2019 +++++  Name  H  m  h  +++++  LMS_SHA256_M32_H5  SHA256  32  5        LMS_SHA256_M32_H10  SHA256  32  10        LMS_SHA256_M32_H15  SHA256  32  15        LMS_SHA256_M32_H20  SHA256  32  20        LMS_SHA256_M32_H25  SHA256  32  25  +++++ Table 2 5.2. LMS Private Key The format of the LMS private key is an internal matter to the implementation, and this document does not attempt to define it. One possibility is that it may consist of an array OTS_PRIV[] of 2^h LMOTS private keys and the leaf number q of the next LMOTS private key that has not yet been used. The qth element of OTS_PRIV[] is generated using Algorithm 0 with the identifiers I, q. The leaf number q is initialized to zero when the LMS private key is created. The process is as follows: Algorithm 5: Computing an LMS Private Key. 1. Determine h and m from the typecode and Table 2. 2. Set I to a uniformly random 16byte string. 3. Compute the array OTS_PRIV[] as follows: for ( q = 0; q < 2^h; q = q + 1) { OTS_PRIV[q] = LMOTS private key with identifiers I, q } 4. q = 0 An LMS private key MAY be generated pseudorandomly from a secret value; in this case, the secret value MUST be at least m bytes long and uniformly random and MUST NOT be used for any other purpose than the generation of the LMS private key. The details of how this process is done do not affect interoperability; that is, the public key verification operation is independent of these details. Appendix A provides an example of a pseudorandom method for computing an LMS private key. McGrew, et al. Informational [Page 20] RFC 8554 LMS HashBased Signatures April 2019 The signaturegeneration logic uses q as the next leaf to use; hence, step 4 starts it off at the leftmost leaf. Because the signature process increments q after the signature operation, the first signature will have q=0. 5.3. LMS Public Key An LMS public key is defined as follows, where we denote the public key final hash value (namely, the K value computed in Algorithm 1) associated with the ith LMOTS private key as OTS_PUB_HASH[i], with i ranging from 0 to (2^h)1. Each instance of an LMS public/private key pair is associated with a balanced binary tree, and the nodes of that tree are indexed from 1 to 2^(h+1)1. Each node is associated with an mbyte string. The string for the rth node is denoted as T[r] and defined as if r >= 2^h: H(Iu32str(r)u16str(D_LEAF)OTS_PUB_HASH[r2^h]) else H(Iu32str(r)u16str(D_INTR)T[2*r]T[2*r+1]) where D_LEAF is the fixed twobyte value 0x8282 and D_INTR is the fixed twobyte value 0x8383, both of which are used to distinguish this hash from every other hash in this system. When we have r >= 2^h, then we are processing a leaf node (and thus hashing only a single LMOTS public key). When we have r < 2^h, then we are processing an internal node  that is, a node with two child nodes that we need to combine. The LMS public key can be represented as the byte string u32str(type)  u32str(otstype)  I  T[1] Section 3.3 specifies the format of the type variable. The value otstype is the parameter set for the LMOTS public/private key pairs used. The value I is the private key identifier and is the value used for all computations for the same LMS tree. The value T[1] can be computed via recursive application of the above equation or by any equivalent method. An iterative procedure is outlined in Appendix C. McGrew, et al. Informational [Page 21] RFC 8554 LMS HashBased Signatures April 2019 5.4. LMS Signature An LMS signature consists of the number q of the leaf associated with the LMOTS signature, as a fourbyte unsigned integer in network byte order, an LMOTS signature, a typecode indicating the particular LMS algorithm, an array of h mbyte values that is associated with the path through the tree from the leaf associated with the LMOTS signature to the root. Symbolically, the signature can be represented as u32str(q)  lmots_signature  u32str(type)  path[0]  path[1]  path[2]  ...  path[h1] Section 3.3 formally defines the format of the signature as the lms_signature structure. The array for a tree with height h will have h values and contains the values of the siblings of (that is, is adjacent to) the nodes on the path from the leaf to the root, where the sibling to node A is the other node that shares node A's parent. In the signature, 0 is counted from the bottom level of the tree, and so path[0] is the value of the node adjacent to leaf node q; path[1] is the secondlevel node that is adjacent to leaf node q's parent, and so on up the tree until we get to path[h1], which is the value of the nexttothetoplevel node whose branch the leaf node q does not reside in. Below is a simple example of the authentication path for h=3 and q=2. The leaf marked OTS is the onetime signature that is used to sign the actual message. The nodes on the path from the OTS public key to the root are marked with a *, while the nodes that are used within the path array are marked with **. The values in the path array are those nodes that are siblings of the nodes on the path; path[0] is the leaf** node that is adjacent to the OTS public key (which is the start of the path); path[1] is the T[4]** node that is the sibling of the second node T[5]* on the path, and path[2] is the T[3]** node that is the sibling of the third node T[2]* on the path. McGrew, et al. Informational [Page 22] RFC 8554 LMS HashBased Signatures April 2019 Root     T[2]* T[3]**         T[4]** T[5]* T[6] T[7]                 leaf leaf OTS leaf** leaf leaf leaf leaf The idea behind this authentication path is that it allows us to validate the OTS hash with using h path array values and hash computations. What the verifier does is recompute the hashes up the path; first, it hashes the given OTS and path[0] value, giving a tentative T[5]' value. Then, it hashes its path[1] and tentative T[5]' value to get a tentative T[2]' value. Then, it hashes that and the path[2] value to get a tentative Root' value. If that value is the known public key of the Merkle tree, then we can assume that the value T[2]' it got was the correct T[2] value in the original tree, and so the T[5]' value it got was the correct T[5] value in the original tree, and so the OTS public key is the same as in the original and, hence, is correct. 5.4.1. LMS Signature Generation To compute the LMS signature of a message with an LMS private key, the signer first computes the LMOTS signature of the message using the leaf number of the next unused LMOTS private key. The leaf number q in the signature is set to the leaf number of the LMS private key that was used in the signature. Before releasing the signature, the leaf number q in the LMS private key MUST be incremented to prevent the LMOTS private key from being used again. If the LMS private key is maintained in nonvolatile memory, then the implementation MUST ensure that the incremented value has been stored before releasing the signature. The issue this tries to prevent is a scenario where a) we generate a signature using one LMOTS private key and release it to the application, b) before we update the nonvolatile memory, we crash, and c) we reboot and generate a second signature using the same LMOTS private key. With two different signatures using the same LMOTS private key, an attacker could potentially generate a forged signature of a third message. McGrew, et al. Informational [Page 23] RFC 8554 LMS HashBased Signatures April 2019 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; pseudocode to do so appears in Appendix D. The least storageintensive 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 signatureverification operation. The internal nodes of the tree need not be kept secret, and thus a nodecaching scheme that stores only internal nodes can sidestep the need for strong protections. Several useful time/storage tradeoffs are described in the "Small Memory LM Schemes" section of [USPTO5432852]. 5.4.2. LMS Signature Verification An LMS signature is verified by first using the LMOTS signature verification algorithm (Algorithm 4b) to compute the LMOTS public key from the LMOTS signature and the message. The value of that public key is then assigned to the associated leaf of the LMS tree, and then the root of the tree is computed from the leaf value and the array path[] as described in Algorithm 6 below. If the root value matches the public key, then the signature is valid; otherwise, the signature verification fails. Algorithm 6: LMS Signature Verification 1. If the public key is not at least eight bytes long, return INVALID. 2. Parse pubtype, I, and T[1] from the public key as follows: a. pubtype = strTou32(first 4 bytes of public key) b. ots_typecode = strTou32(next 4 bytes of public key) c. Set m according to pubtype, based on Table 2. d. If the public key is not exactly 24 + m bytes long, return INVALID. e. I = next 16 bytes of the public key f. T[1] = next m bytes of the public key McGrew, et al. Informational [Page 24] RFC 8554 LMS HashBased Signatures April 2019 3. Compute the LMS Public Key Candidate Tc from the signature, message, identifier, pubtype, and ots_typecode, using Algorithm 6a. 4. If Tc is equal to T[1], return VALID; otherwise, return INVALID. Algorithm 6a: Computing an LMS Public Key Candidate from a Signature, Message, Identifier, and Algorithm Typecodes 1. If the signature is not at least eight bytes long, return INVALID. 2. Parse sigtype, q, lmots_signature, and path from the signature as follows: a. q = strTou32(first 4 bytes of signature) b. otssigtype = strTou32(next 4 bytes of signature) c. If otssigtype is not the OTS typecode from the public key, return INVALID. d. Set n, p according to otssigtype and Table 1; if the signature is not at least 12 + n * (p + 1) bytes long, return INVALID. e. lmots_signature = bytes 4 through 7 + n * (p + 1) of signature f. sigtype = strTou32(bytes 8 + n * (p + 1)) through 11 + n * (p + 1) of signature) g. If sigtype is not the LM typecode from the public key, return INVALID. h. Set m, h according to sigtype and Table 2. i. If q >= 2^h or the signature is not exactly 12 + n * (p + 1) + m * h bytes long, return INVALID. j. Set path as follows: path[0] = next m bytes of signature path[1] = next m bytes of signature ... path[h1] = next m bytes of signature McGrew, et al. Informational [Page 25] RFC 8554 LMS HashBased Signatures April 2019 3. Kc = candidate public key computed by applying Algorithm 4b to the signature lmots_signature, the message, and the identifiers I, q 4. Compute the candidate LMS root value Tc as follows: node_num = 2^h + q tmp = H(I  u32str(node_num)  u16str(D_LEAF)  Kc) i = 0 while (node_num > 1) { if (node_num is odd): tmp = H(Iu32str(node_num/2)u16str(D_INTR)path[i]tmp) else: tmp = H(Iu32str(node_num/2)u16str(D_INTR)tmppath[i]) node_num = node_num/2 i = i + 1 } Tc = tmp 5. Return Tc. 6. Hierarchical Signatures In scenarios where it is necessary to minimize the time taken by the public key generation process, the Hierarchical Signature System (HSS) can be used. This hierarchical scheme, which we describe in this section, uses the LMS scheme as a component. In HSS, we have a sequence of L LMS trees, where the public key for the first LMS tree is included in the public key of the HSS system, each LMS private key signs the next LMS public key, and the last LMS private key signs the actual message. For example, if we have a threelevel hierarchy (L=3), then to sign a message, we would have: The first LMS private key (level 0) signs a level 1 LMS public key. The second LMS private key (level 1) signs a level 2 LMS public key. The third LMS private key (level 2) signs the message. The root of the level 0 LMS tree is contained in the HSS public key. To verify the LMS signature, we would verify all the signatures: We would verify that the level 1 LMS public key is correctly signed by the level 0 signature. McGrew, et al. Informational [Page 26] RFC 8554 LMS HashBased Signatures April 2019 We would verify that the level 2 LMS public key is correctly signed by the level 1 signature. We would verify that the message is correctly signed by the level 2 signature. We would accept the HSS signature only if all the signatures validated. During the signaturegeneration process, we sign messages with the lowest (level L1) LMS tree. Once we have used all the leafs in that tree to sign messages, we would discard it, generate a fresh LMS tree, and sign it with the next (level L2) LMS tree (and when that is used up, recursively generate and sign a fresh level L2 LMS tree). HSS, in essence, utilizes a tree of LMS trees. There is a single LMS tree at level 0 (the root). Each LMS tree (actually, the private key corresponding to the LMS tree) at level i is used to sign 2^h objects (where h is the height of trees at level i). If i < L1, then each object will be another LMS tree (actually, the public key) at level i+1; if i = L1, we've reached the bottom of the HSS tree, and so each object will be a message from the application. The HSS public key contains the public key of the LMS tree at the root, and an HSS signature is associated with a path from the root of the HSS tree to the leaf. Compared to LMS, HSS has a much reduced public key generation time, as only the root tree needs to be generated prior to the distribution of the HSS public key. For example, an L=3 tree (with h=10 at each level) would have one level 0 LMS tree, 2^10 level 1 LMS trees (with each such level 1 public key signed by one of the 1024 level 0 OTS public keys), and 2^20 level 2 LMS trees. Only 1024 OTS public keys need to be computed to generate the HSS public key (as you need to compute only the level 0 LMS tree to compute that value; you can, of course, decide to compute the initial level 1 and level 2 LMS trees). In addition, the 2^20 level 2 LMS trees can jointly sign a total of over a billion messages. In contrast, a single LMS tree that could sign a billion messages would require a billion OTS public keys to be computed first (if h=30 were allowed in a supported parameter set). Each LMS tree within the hierarchy is associated with a distinct LMS public key, private key, signature, and identifier. The number of levels is denoted as L and is between one and eight, inclusive. The following notation is used, where i is an integer between 0 and L1 inclusive, and the root of the hierarchy is level 0: prv[i] is the current LMS private key of the ith level. McGrew, et al. Informational [Page 27] RFC 8554 LMS HashBased Signatures April 2019 pub[i] is the current LMS public key of the ith level, as described in Section 5.3. sig[i] is the LMS signature of public key pub[i+1] generated using the private key prv[i]. It is expected that the above arrays are maintained for the course of the HSS key. The contents of the prv[] array MUST be kept private; the pub[] and sig[] array may be revealed should the implementation find that convenient. In this section, we say that an Ntime private key is exhausted when it has generated N signatures; thus, it can no longer be used for signing. For i > 0, the values prv[i], pub[i], and (for all values of i) sig[i] will be updated over time as private keys are exhausted and replaced by newer keys. When these key pairs are updated (or initially generated before the first message is signed), then the LMS key generation processes outlined in Sections 5.2 and 5.3 are performed. If the generated key pairs are for level i of the HSS hierarchy, then we store the public key in pub[i] and the private key in prv[i]. In addition, if i > 0, then we sign the generated public key with the LMS private key at level i1, placing the signature into sig[i1]. When the LMS key pair is generated, the key pair and the corresponding identifier MUST be generated independently of all other key pairs. HSS allows L=1, in which case the HSS public key and signature formats are essentially the LMS public key and signature formats, prepended by a fixed field. Since HSS with L=1 has very little overhead compared to LMS, all implementations MUST support HSS in order to maximize interoperability. We specifically allow different LMS levels to use different parameter sets. For example, the 0th LMS public key (the root) may use the LMS_SHA256_M32_H15 parameter set, while the 1th public key may use LMS_SHA256_M32_H10. There are practical reasons to allow this; for one, the signer may decide to store parts of the 0th LMS tree (that it needs to construct while computing the public key) to accelerate later operations. As the 0th tree is never updated, these internal nodes will never need to be recomputed. In addition, during the signaturegeneration operation, almost all the operations involved with updating the authentication path occur with the bottom (L1th) LMS public key; hence, it may be useful to select the parameter set for that public key to have a shorter LMS tree. McGrew, et al. Informational [Page 28] RFC 8554 LMS HashBased Signatures April 2019 A close reading of the HSS verification pseudocode shows that it would allow the parameters of the nontop LMS public keys to change over time; for example, the signer might initially have the 1th LMS public key use the LMS_SHA256_M32_H10 parameter set, but when that tree is exhausted, the signer might replace it with an LMS public key that uses the LMS_SHA256_M32_H15 parameter set. While this would work with the example verification pseudocode, the signer MUST NOT change the parameter sets for a specific level. This prohibition is to support verifiers that may keep state over the course of several signature verifications. 6.1. Key Generation The public key of the HSS scheme consists of the number of levels L, followed by pub[0], the public key of the top level. The HSS private key consists of prv[0], ... , prv[L1], along with the associated pub[0], ... pub[L1] and sig[0], ..., sig[L2] values. As stated earlier, the values of the pub[] and sig[] arrays need not be kept secret and may be revealed. The value of pub[0] does not change (and, except for the index q, the value of prv[0] need not change); however, the values of pub[i] and prv[i] are dynamic for i > 0 and are changed by the signaturegeneration algorithm. During the key generation, the public and private keys are initialized. Here is some pseudocode that explains the key generation logic: Algorithm 7: Generating an HSS Key Pair 1. Generate an LMS key pair, as specified in Sections 5.2 and 5.3, placing the private key into priv[0], and the public key into pub[0] 2. For i = 1 to L1 do { generate an LMS key pair, placing the private key into priv[i] and the public key into pub[i] sig[i1] = lms_signature( pub[i], priv[i1] ) } 3. Return u32str(L)  pub[0] as the public key and the priv[], pub[], and sig[] arrays as the private key In the above algorithm, each LMS public/private key pair generated MUST be generated independently. McGrew, et al. Informational [Page 29] RFC 8554 LMS HashBased Signatures April 2019 Note that the value of the public key does not depend on the execution of step 2. As a result, an implementation may decide to delay step 2 until later  for example, during the initial signaturegeneration operation. 6.2. Signature Generation To sign a message using an HSS key pair, the following steps are performed: If prv[L1] is exhausted, then determine the smallest integer d such that all of the private keys prv[d], prv[d+1], ... , prv[L1] are exhausted. If d is equal to zero, then the HSS key pair is exhausted, and it MUST NOT generate any more signatures. Otherwise, the key pairs for levels d through L1 must be regenerated during the signaturegeneration process, as follows. For i from d to L1, a new LMS public and private key pair with a new identifier is generated, pub[i] and prv[i] are set to those values, then the public key pub[i] is signed with prv[i1], and sig[i1] is set to the resulting value. The message is signed with prv[L1], and the value sig[L1] is set to that result. The value of the HSS signature is set as follows. We let signed_pub_key denote an array of octet strings, where signed_pub_key[i] = sig[i]  pub[i+1], for i between 0 and Nspk1, inclusive, where Nspk = L1 denotes the number of signed public keys. Then the HSS signature is u32str(Nspk)  signed_pub_key[0]  ...  signed_pub_key[Nspk1]  sig[Nspk]. Note that the number of signed_pub_key elements in the signature is indicated by the value Nspk that appears in the initial four bytes of the signature. Here is some pseudocode of the above logic: Algorithm 8: Generating an HSS signature 1. If the messagesigning key prv[L1] is exhausted, regenerate that key pair, together with any parent key pairs that might be necessary. If the root key pair is exhausted, then the HSS key pair is exhausted and MUST NOT generate any more signatures. McGrew, et al. Informational [Page 30] RFC 8554 LMS HashBased Signatures April 2019 d = L while (prv[d1].q == 2^(prv[d1].h)) { d = d  1 if (d == 0) return FAILURE } while (d < L) { create lms key pair pub[d], prv[d] sig[d1] = lms_signature( pub[d], prv[d1] ) d = d + 1 } 2. Sign the message. sig[L1] = lms_signature( msg, prv[L1] ) 3. Create the list of signed public keys. i = 0; while (i < L1) { signed_pub_key[i] = sig[i]  pub[i+1] i = i + 1 } 4. Return u32str(L1)  signed_pub_key[0]  ...  signed_pub_key[L2]  sig[L1] In the specific case of L=1, the format of an HSS signature is u32str(0)  sig[0] In the general case, the format of an HSS signature is u32str(Nspk)  signed_pub_key[0]  ...  signed_pub_key[Nspk1]  sig[Nspk] which is equivalent to u32str(Nspk)  sig[0]  pub[1]  ...  sig[Nspk1]  pub[Nspk]  sig[Nspk] McGrew, et al. Informational [Page 31] RFC 8554 LMS HashBased Signatures April 2019 6.3. Signature Verification To verify a signature S and message using the public key pub, perform the following steps: The signature S is parsed into its components as follows: Nspk = strTou32(first four bytes of S) if Nspk+1 is not equal to the number of levels L in pub: return INVALID for (i = 0; i < Nspk; i = i + 1) { siglist[i] = next LMS signature parsed from S publist[i] = next LMS public key parsed from S } siglist[Nspk] = next LMS signature parsed from S key = pub for (i = 0; i < Nspk; i = i + 1) { sig = siglist[i] msg = publist[i] if (lms_verify(msg, key, sig) != VALID): return INVALID key = msg } return lms_verify(message, key, siglist[Nspk]) Since the length of an LMS signature cannot be known without parsing it, the HSS signature verification algorithm makes use of an LMS signature parsing routine that takes as input a string consisting of an LMS signature with an arbitrary string appended to it and returns both the LMS signature and the appended string. The latter is passed on for further processing. 6.4. Parameter Set Recommendations As for guidance as to the number of LMS levels and the size of each, any discussion of performance is implementation specific. In general, the sole drawback for a single LMS tree is the time it takes to generate the public key; as every LMOTS public key needs to be generated, the time this takes can be substantial. For a twolevel tree, only the toplevel LMS tree and the initial bottomlevel LMS tree need to be generated initially (before the first signature is generated); this will in general be significantly quicker. To give a general idea of the tradeoffs available, we include some measurements taken with the LMS implementation available at <https://github.com/cisco/hashsigs>, taken on a 3.3 GHz Xeon processor with threading enabled. We tried various parameter sets, McGrew, et al. Informational [Page 32] RFC 8554 LMS HashBased Signatures April 2019 all with W=8 (which minimizes signature size, while increasing time). These are here to give a guideline as to what's possible; for the computational time, your mileage may vary, depending on the computing resources you have. The machine these tests were performed on does not have the SHA256 extensions; you could possibly do significantly better. +++++  ParmSet  KeyGenTime  SigSize  KeyLifetime  +++++  15  6 sec  1616  30 seconds        20  3 min  1776  16 minutes        25  1.5 hour  1936  9 hours        15/10  6 sec  3172  9 hours        15/15  6 sec  3332  12 days        20/10  3 min  3332  12 days        20/15  3 min  3492  1 year        25/10  1.5 hour  3492  1 year        25/15  1.5 hour  3652  34 years  +++++ Table 3 ParmSet: this is the height of the Merkle tree(s); parameter sets listed as a single integer have L=1 and consist of a single Merkle tree of that height; parameter sets with L=2 are listed as x/y, with x being the height of the toplevel Merkle tree and y being the bottom level. KeyGenTime: the measured keygeneration time; that is, the time needed to generate the public/private key pair. SigSize: the size of a signature (in bytes) KeyLifetime: the lifetime of a key, assuming we generated 1000 signatures per second. In practice, we're not likely to get anywhere close to 1000 signatures per second sustained; if you have a more appropriate figure for your scenario, this column is easy to recompute. McGrew, et al. Informational [Page 33] RFC 8554 LMS HashBased Signatures April 2019 As for signature generation or verification times, those are moderately insensitive to the above parameter settings (except for the Winternitz setting and the number of Merkle trees for verification). Tests on the same machine (without multithreading) gave approximately 4 msec to sign a short message, 2.6 msec to verify; these tests used a twolevel ParmSet; a single level would approximately halve the verification time. All times can be significantly improved (by perhaps a factor of 8) by using a parameter set with W=4; however, that also about doubles the signature size. 7. Rationale The goal of this note is to describe the LMOTS, LMS, and HSS 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 followon 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 chosen randomly in order to improve security. The analysis of this method in [Fluhrer17] shows that we do not need uniqueness to ensure security; we do need to ensure that we don't have a large number of private keys that use the same I value. By randomly selecting 16byte I values, the chance that, out of 2^64 private keys, 4 or more of them will use the same I value is negligible (that is, has probability less than 2^128). The reason 16byte I values were selected was to optimize the Winternitz hashchain operation. With the current settings, the value being hashed is exactly 55 bytes long (for a 32byte hash function), which SHA256 can hash in a single hashcompression operation. Other hash functions may be used in future specifications; all the ones that we will be likely to support (SHA512/256 and the various SHA3 hashes) would work well with a 16byte I value. The signature and public key formats are designed so that they are relatively easy to parse. Each format starts with a 32bit enumeration value that indicates the details of the signature algorithm and provides all of the information that is needed in order to parse the format. McGrew, et al. Informational [Page 34] RFC 8554 LMS HashBased Signatures April 2019 The Checksum (Section 4.4) is calculated using a nonnegative integer "sum" whose width was chosen to be an integer number of wbit 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 4.5) and the total actual number. In the case that the number of times H is applied is 0, the sum is (2^w  1) * (8*n/w). Thus, for the purposes of this document, which describes signature methods based on H = SHA256 (n = 32 bytes) and w = { 1, 2, 4, 8 }, the sum variable is a 16bit nonnegative integer for all combinations of n and w. The calculation uses the parameter ls defined in Section 4.1 and calculated in Appendix B, which indicates the number of bits used in the leftshift operation. 7.1. Security String To improve security against attacks that amortize their effort against multiple invocations of the hash function, Leighton and Micali introduced a "security string" that is distinct for each invocation of that function. Whenever this process computes a hash, the string being hashed will start with a string formed from the fields below. These fields will appear in fixed locations in the value we compute the hash of, and so we list where in the hash these fields would be present. The fields that make up this string are as follows: I A 16byte identifier for the LMS public/private key pair. It MUST be chosen uniformly at random, or via a pseudorandom process, at the time that a key pair is generated, in order to minimize the probability that any specific value of I be used for a large number of different LMS private keys. This is always bytes 015 of the value being hashed. r In the LMS Ntime signature scheme, the node number r associated with a particular node of a hash tree is used as an input to the hash used to compute that node. This value is represented as a 32bit (four byte) unsigned integer in network byte order. Either r or q (depending on the domainseparation parameter) will be bytes 1619 of the value being hashed. q In the LMS Ntime signature scheme, each LMOTS signature is associated with the leaf of a hash tree, and q is set to the leaf number. This ensures that a distinct value of q is used for each distinct LMOTS public/private key pair. This value is represented as a 32bit (four byte) unsigned integer in network byte order. Either r or q (depending on the domain separation parameter) will be bytes 1619 of the value being hashed. McGrew, et al. Informational [Page 35] RFC 8554 LMS HashBased Signatures April 2019 D A domainseparation parameter, which is a twobyte identifier that takes on different values in the different contexts in which the hash function is invoked. D occurs in bytes 20 and 21 of the value being hashed and takes on the following values: D_PBLC = 0x8080 when computing the hash of all of the iterates in the LMOTS algorithm D_MESG = 0x8181 when computing the hash of the message in the LMOTS algorithms D_LEAF = 0x8282 when computing the hash of the leaf of an LMS tree D_INTR = 0x8383 when computing the hash of an interior node of an LMS tree i A value between 0 and 264; this is used in the LMOTS scheme when either computing the iterations of the Winternitz chain or using the suggested LMOTS private key generation process. It is represented as a 16bit (twobyte) unsigned integer in network byte order. If present, it occurs at bytes 20 and 21 of the value being hashed. j In the LMOTS scheme, j is the iteration number used when the private key element is being iteratively hashed. It is represented as an 8bit (one byte) unsigned integer and is present if i is a value between 0 and 264. If present, it occurs at bytes 22 to 21+n of the value being hashed. C An nbyte randomizer that is included with the message whenever it is being hashed to improve security. C MUST be chosen uniformly at random or via another unpredictable process. It is present if D=D_MESG, and it occurs at bytes 22 to 21+n of the value being hashed. 8. IANA Considerations IANA has created two registries: "LMOTS Signatures", which includes all of the LMOTS signatures as defined in Section 4, and "Leighton Micali Signatures (LMS)" for LMS as defined in Section 5. 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 [RFC8126]. IANA MUST verify that all applications for additions to these registries have first been reviewed by the IRTF Crypto Forum Research Group (CFRG). McGrew, et al. Informational [Page 36] RFC 8554 LMS HashBased Signatures April 2019 Each entry in either of the registries contains the following elements: a short name (Name), such as "LMS_SHA256_M32_H10", a positive number (Numeric Identifier), and a Reference to a specification that completely defines the signaturemethod test cases that can be used to verify the correctness of an implementation. 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 [RFC8126]. The initial contents of the "LMOTS Signatures" registry are as follows. ++++  Name  Reference  Numeric Identifier  ++++  Reserved   0x00000000       LMOTS_SHA256_N32_W1  Section 4  0x00000001       LMOTS_SHA256_N32_W2  Section 4  0x00000002       LMOTS_SHA256_N32_W4  Section 4  0x00000003       LMOTS_SHA256_N32_W8  Section 4  0x00000004       Unassigned   0x00000005  0xDDDDDDDC       Reserved for Private Use   0xDDDDDDDD  0xFFFFFFFF  ++++ Table 4 McGrew, et al. Informational [Page 37] RFC 8554 LMS HashBased Signatures April 2019 The initial contents of the "Leighton Micali Signatures (LMS)" registry are as follows. ++++  Name  Reference  Numeric Identifier  ++++  Reserved   0x0  0x4       LMS_SHA256_M32_H5  Section 5  0x00000005       LMS_SHA256_M32_H10  Section 5  0x00000006       LMS_SHA256_M32_H15  Section 5  0x00000007       LMS_SHA256_M32_H20  Section 5  0x00000008       LMS_SHA256_M32_H25  Section 5  0x00000009       Unassigned   0x0000000A  0xDDDDDDDC       Reserved for Private Use   0xDDDDDDDD  0xFFFFFFFF  ++++ Table 5 An IANA registration of a signature system does not constitute an endorsement of that system or its security. Currently, the two registries assign a disjoint set of values to the defined parameter sets. This coincidence is a historical accident; the correctness of the system does not depend on this. IANA is not required to maintain this situation. 9. Security Considerations The hash function H MUST have second preimage resistance: it must be computationally infeasible for an attacker that is given one message M to be able to find a second message M' such that H(M) = H(M'). 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 (distinct from all previously signed ones) and signature that is 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. McGrew, et al. Informational [Page 38] RFC 8554 LMS HashBased Signatures April 2019 LMS is provably secure in the random oracle model, as shown by [Katz16]. In addition, further analysis is done by [Fluhrer17], where the hash compression function (rather than the entire hash function) is considered to be a random oracle. Corollary 1 of the latter paper states: If we have no more than 2^64 randomly chosen LMS private keys, allow the attacker access to a signing oracle and a SHA256 hash compression oracle, and allow a maximum of 2^120 hash compression computations, then the probability of an attacker being able to generate a single forgery against any of those LMS keys is less than 2^129. Many of the objects within the public key and the signature start with a typecode. A verifier MUST check each of these typecodes, and a verification operation on a signature with an unknown type, or a type that does not correspond to the type within the public key, MUST return INVALID. The expected length of a variablelength object can be determined from its typecode; if an object has a different length, then any signature computed from the object is INVALID. 9.1. Hash Formats The format of the inputs to the hash function H has the property that each invocation of that function has an input that is repeated by a small bounded number of other inputs (due to potential repeats of the I value). In particular, it will vary somewhere in the first 23 bytes of the value being hashed. This property is important for a proof of security in the random oracle model. The formats used during key generation and signing (including the recommended pseudorandom keygeneration procedure in Appendix A) are as follows: I  u32str(q)  u16str(i)  u8str(j)  tmp I  u32str(q)  u16str(D_PBLC)  y[0]  ...  y[p1] I  u32str(q)  u16str(D_MESG)  C  message I  u32str(r)  u16str(D_LEAF)  OTS_PUB_HASH[r2^h] I  u32str(r)  u16str(D_INTR)  T[2*r]  T[2*r+1] I  u32str(q)  u16str(i)  u8str(0xff)  SEED Each hash type listed is distinct; at locations 20 and 21 of the value being hashed, there exists either a fixed value D_PBLC, D_MESG, D_LEAF, D_INTR, or a 16bit value i. These fixed values are distinct from each other and are large (over 32768), while the 16bit values of i are small (currently no more than 265; possibly being slightly larger if larger hash functions are supported); hence, the range of possible values of i will not collide any of the D_PBLC, D_MESG, McGrew, et al. Informational [Page 39] RFC 8554 LMS HashBased Signatures April 2019 D_LEAF, D_INTR identifiers. The only other collision possibility is the Winternitz chain hash colliding with the recommended pseudorandom keygeneration process; here, at location 22 of the value being hashed, the Winternitz chain function has the value u8str(j), where j is a value between 0 and 254, while location 22 of the recommended pseudorandom keygeneration process has value 255. For the Winternitz chaining function, D_PBLC, and D_MESG, the value of I  u32str(q) is distinct for each LMS leaf (or equivalently, for each q value). For the Winternitz chaining function, the value of u16str(i)  u8str(j) is distinct for each invocation of H for a given leaf. For D_PBLC and D_MESG, the input format is used only once for each value of q and, thus, distinctness is assured. The formats for D_INTR and D_LEAF are used exactly once for each value of r, which ensures their distinctness. For the recommended pseudorandom keygeneration process, for a given value of I, q and j are distinct for each invocation of H. The value of I is chosen uniformly at random from the set of all 128bit strings. If 2^64 public keys are generated (and, hence, 2^64 random I values), there is a nontrivial probability of a duplicate (which would imply duplicate prefixes). However, there will be an extremely high probability there will not be a fourway collision (that is, any I value used for four distinct LMS keys; probability < 2^132), and, hence, the number of repeats for any specific prefix will be limited to at most three. This is shown (in [Fluhrer17]) to have only a limited effect on the security of the system. 9.2. Stateful Signature Algorithm The LMS signature system, like all Ntime signature systems, requires that the signer maintain state across different invocations of the signing algorithm to ensure that none of the component onetime signature systems are used more than once. This section calls out some important practical considerations around this statefulness. These issues are discussed in greater detail in [STMGMT]. In a typical computing environment, a private key will be stored in nonvolatile media such as on a hard drive. Before it is used to sign a message, it will be read into an application's RandomAccess 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 nonvolatile storage. It is essential for security that the application ensures that this value is actually written into that storage, yet there may be one or more memory caches between it and the application. Memory caching is commonly done in the file system and in a physical memory unit on the hard disk that is dedicated to that purpose. To ensure that the updated value is written to McGrew, et al. Informational [Page 40] RFC 8554 LMS HashBased Signatures April 2019 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 written to the underlying hardware. However, if that hardware has its own memory cache, it must be separately dealt with using an operating system or devicespecific tool such as hdparm to flush the ondrive 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 secondlevel resident in RAM only and never write that value into nonvolatile memory. A new secondlevel 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. 9.3. Security of LMOTS Checksum To show the security of LMOTS 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 4.5). To attempt a forgery, an attacker may try to change the values of h and c. Let h' and c' denote the values used in the forgery attempt. If for some integer j in the range 0 to u, where u = ceil(8*n/w) is the size of the range that the checksum value can cover, inclusive, a' = coef(h', j, w), a = coef(h, j, w), and a' > a McGrew, et al. Informational [Page 41] RFC 8554 LMS HashBased Signatures April 2019 then the attacker can compute F^a'(x[j]) from F^a(x[j]) = y[j] by iteratively applying function F to the jth 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 (p1), inclusive, b' = coef(c', k, w), b = coef(c, k, w), and b' < b Due to the oneway property of F, the attacker cannot easily compute F^b'(x[k]) from F^b(x[k]) = y[k]. 10. Comparison with Other Work The eXtended Merkle Signature Scheme (XMSS) is similar to HSS in several ways [XMSS][RFC8391]. Both are stateful hashbased signature schemes, and both use a hierarchical approach, with a Merkle tree at each level of the hierarchy. XMSS signatures are slightly shorter than HSS signatures, for equivalent security and an equal number of signatures. HSS has several advantages over XMSS. HSS operations are roughly four times faster than the comparable XMSS ones, when SHA256 is used as the underlying hash. This occurs because the hash operation done as a part of the Winternitz iterations dominates performance, and XMSS performs four compressionfunction invocations (two for the PRF, two for the F function) where HSS only needs to perform one. Additionally, HSS is somewhat simpler (as each hash invocation is just a prefix followed by the data being hashed). McGrew, et al. Informational [Page 42] RFC 8554 LMS HashBased Signatures April 2019 11. References 11.1. Normative References [FIPS180] National Institute of Standards and Technology, "Secure Hash Standard (SHS)", FIPS PUB 1804, DOI 10.6028/NIST.FIPS.1804, 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, <https://www.rfceditor.org/info/rfc2119>. [RFC4506] Eisler, M., Ed., "XDR: External Data Representation Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May 2006, <https://www.rfceditor.org/info/rfc4506>. [RFC8126] Cotton, M., Leiba, B., and T. Narten, "Guidelines for Writing an IANA Considerations Section in RFCs", BCP 26, RFC 8126, DOI 10.17487/RFC8126, June 2017, <https://www.rfceditor.org/info/rfc8126>. [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, <https://www.rfceditor.org/info/rfc8174>. [RFC8179] Bradner, S. and J. Contreras, "Intellectual Property Rights in IETF Technology", BCP 79, RFC 8179, DOI 10.17487/RFC8179, May 2017, <https://www.rfceditor.org/info/rfc8179>. [USPTO5432852] Leighton, T. and S. Micali, "Large provably fast and secure digital signature schemes based on secure hash functions", U.S. Patent 5,432,852, July 1995. 11.2. Informative References [C:Merkle87] Merkle, R., "A Digital Signature Based on a Conventional Encryption Function", in Advances in Cryptology  CRYPTO '87 Proceedings, Lecture Notes in Computer Science Vol. 293, DOI 10.1007/3540481842_32, 1988. McGrew, et al. Informational [Page 43] RFC 8554 LMS HashBased Signatures April 2019 [C:Merkle89a] Merkle, R., "A Certified Digital Signature", in Advances in Cryptology  CRYPTO '89 Proceedings, Lecture Notes in Computer Science Vol. 435, DOI 10.1007/0387348050_21, 1990. [C:Merkle89b] Merkle, R., "One Way Hash Functions and DES", in Advances in Cryptology  CRYPTO '89 Proceedings, Lecture Notes in Computer Science Vol. 435, DOI 10.1007/0387348050_40, 1990. [Fluhrer17] Fluhrer, S., "Further Analysis of a Proposed HashBased Signature Standard", Cryptology ePrint Archive Report 2017/553, 2017, <https://eprint.iacr.org/2017/553>. [Katz16] Katz, J., "Analysis of a Proposed HashBased Signature Standard", in SSR 2016: Security Standardisation Research (SSR) pp. 261273, Lecture Notes in Computer Science Vol. 10074, DOI 10.1007/9783319491004_12, 2016. [Merkle79] Merkle, R., "Secrecy, Authentication, and Public Key Systems", Technical Report No. 19791, Information Systems Laboratory, Stanford University, 1979, <http://www.merkle.com/papers/Thesis1979.pdf>. [RFC8391] Huelsing, A., Butin, D., Gazdag, S., Rijneveld, J., and A. Mohaisen, "XMSS: eXtended Merkle Signature Scheme", RFC 8391, DOI 10.17487/RFC8391, May 2018, <https://www.rfceditor.org/info/rfc8391>. [STMGMT] McGrew, D., Kampanakis, P., Fluhrer, S., Gazdag, S., Butin, D., and J. Buchmann, "State Management for Hash Based Signatures.", in SSR 2016: Security Standardisation Research (SSR) pp. 244260, Lecture Notes in Computer Science Vol. 10074, DOI 10.1007/9783319491004_11, 2016. [XMSS] Buchmann, J., Dahmen, E., and , "XMSS  A Practical Forward Secure Signature Scheme Based on Minimal Security Assumptions.", in PQCrypto 2011: PostQuantum Cryptography pp. 117129, Lecture Notes in Computer Science Vol. 7071, DOI 10.1007/9783642254055_8, 2011. McGrew, et al. Informational [Page 44] RFC 8554 LMS HashBased Signatures April 2019 Appendix A. Pseudorandom Key Generation An implementation MAY use the following pseudorandom process for generating an LMS private key. SEED is an mbyte value that is generated uniformly at random at the start of the process, I is the LMS key pair identifier, q denotes the LMS leaf number of an LMOTS private key, x_q denotes the x array of private elements in the LMOTS private key with leaf number q, i is the index of the private key element, and H is the hash function used in LMOTS. The elements of the LMOTS private keys are computed as: x_q[i] = H(I  u32str(q)  u16str(i)  u8str(0xff)  SEED). This process stretches the mbyte random value SEED into a (much larger) set of pseudorandom values, using a unique counter in each invocation of H. The format of the inputs to H are chosen so that they are distinct from all other uses of H in LMS and LMOTS. A careful reader will note that this is similar to the hash we perform when iterating through the Winternitz chain; however, in that chain, the iteration index will vary between 0 and 254 maximum (for W=8), while the corresponding value in this formula is 255. This algorithm is included in the proof of security in [Fluhrer17] and hence this method is safe when used within the LMS system; however, any other cryptographically secure method of generating private keys would also be safe. Appendix B. LMOTS Parameter Options The LMOTS onetime signature method uses several internal parameters, which are a function of the selected parameter set. These internal parameters include the following: p This is the number of independent Winternitz chains used in the signature; it will be the number of wbit digits needed to hold the nbit hash (u in the below equations), along with the number of digits needed to hold the checksum (v in the below equations) McGrew, et al. Informational [Page 45] RFC 8554 LMS HashBased Signatures April 2019 ls This is the size of the shift needed to move the checksum so that it appears in the checksum digits ls is needed because, while we express the checksum internally as a 16bit value, we don't always express all 16 bits in the signature; for example, if w=4, we might use only the top 12 bits. Because we read the checksum in network order, this means that, without the shift, we'll use the higherorder bits (which may be always 0) and omit the lowerorder bits (where the checksum value actually resides). This shift is here to ensure that the parts of the checksum we need to express (for security) actually contribute to the signature; when multiple such shifts are possible, we take the minimal value. The parameters ls and p are computed as follows: u = ceil(8*n/w) v = ceil((floor(lg((2^w  1) * u)) + 1) / w) ls = 16  (v * w) p = u + v Here, u and v represent the number of wbit fields required to contain the hash of the message and the checksum byte strings, respectively. And as the value of p is the number of wbit 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. The value 16 in the ls computation of ls corresponds to the 16bit value used for the sum variable in Algorithm 2 in Section 4.4 A table illustrating various combinations of n and w with the associated values of u, v, ls, and p is provided in Table 6. McGrew, et al. Informational [Page 46] RFC 8554 LMS HashBased Signatures April 2019 +++++++  Hash  Winternitz  wbit  wbit  Left  Total   Length  Parameter  Elements  Elements  Shift  Number of   in  (w)  in Hash  in  (ls)  wbit   Bytes   (u)  Checksum   Elements   (n)    (v)   (p)  +++++++  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 6 Appendix C. An Iterative Algorithm for Computing an LMS Public Key The LMS public key can be computed using the following algorithm or any equivalent method. The algorithm uses a stack of hashes for data. It also makes use of a hash function with the typical init/update/final interface to hash functions; the result of the invocations hash_init(), hash_update(N[1]), hash_update(N[2]), ... , hash_update(N[n]), v = hash_final(), in that order, is identical to that of the invocation of H(N[1]  N[2]  ...  N[n]). Generating an LMS Public Key from an LMS Private Key for ( i = 0; i < 2^h; i = i + 1 ) { r = i + num_lmots_keys; temp = H(I  u32str(r)  u16str(D_LEAF)  OTS_PUB_HASH[i]) j = i; while (j % 2 == 1) { r = (r  1)/2; j = (j1) / 2; left_side = pop(data stack); temp = H(I  u32str(r)  u16str(D_INTR)  left_side  temp) } push temp onto the data stack } public_key = pop(data stack) McGrew, et al. Informational [Page 47] RFC 8554 LMS HashBased Signatures April 2019 Note that this pseudocode expects that all 2^h leaves of the tree have equal depth  that is, it expects num_lmots_keys to be a power of 2. The maximum depth of the stack will be h1 elements  that is, a total of (h1)*n bytes; for the currently defined parameter sets, this will never be more than 768 bytes of data. Appendix D. Method for Deriving Authentication Path for a Signature The LMS signature consists of u32str(q)  lmots_signature  u32str(type)  path[0]  path[1]  ...  path[h1]. This appendix shows one method of constructing this signature, assuming that the implementation has stored the T[] array that was used to construct the public key. Note that this is not the only possible method; other methods exist that don't assume that you have the entire T[] array in memory. To construct a signature, you perform the following algorithm: Generating an LMS Signature 1. Set type to the typecode of the LMS algorithm. 2. Extract h from the typecode, according to Table 2. 3. Create the LMOTS signature for the message: ots_signature = lmots_sign(message, LMS_PRIV[q]) 4. Compute the array path as follows: i = 0 r = 2^h + q while (i < h) { temp = (r / 2^i) xor 1 path[i] = T[temp] i = i + 1 } 5. S = u32str(q)  ots_signature  u32str(type)  path[0]  path[1]  ...  path[h1] 6. q = q + 1 7. Return S. Here "xor" is the bitwise exclusiveor operation, and / is integer division (that is, rounded down to an integer value). McGrew, et al. Informational [Page 48] RFC 8554 LMS HashBased Signatures April 2019 Appendix E. Example Implementation An example implementation can be found online at <https://github.com/cisco/hashsigs>. Appendix F. Test Cases This section provides test cases that can be used to verify or debug an implementation. This data is formatted with the name of the elements on the left and the hexadecimal value of the elements on the right. The concatenation of all of the values within a public key or signature produces that public key or signature, and values that do not fit within a single line are listed across successive lines. Test Case 1 Public Key  HSS public key levels 00000002  LMS type 00000005 # LM_SHA256_M32_H5 LMOTS type 00000004 # LMOTS_SHA256_N32_W8 I 61a5d57d37f5e46bfb7520806b07a1b8 K 50650e3b31fe4a773ea29a07f09cf2ea 30e579f0df58ef8e298da0434cb2b878   Test Case 1 Message  Message 54686520706f77657273206e6f742064 The powers not d 656c65676174656420746f2074686520 elegated to the  556e6974656420537461746573206279 United States by 2074686520436f6e737469747574696f  the Constitutio 6e2c206e6f722070726f686962697465 n, nor prohibite 6420627920697420746f207468652053 d by it to the S 74617465732c20617265207265736572 tates, are reser 76656420746f20746865205374617465 ved to the State 7320726573706563746976656c792c20 s respectively,  6f7220746f207468652070656f706c65 or to the people 2e0a ..  McGrew, et al. Informational [Page 49] RFC 8554 LMS HashBased Signatures April 2019 Test Case 1 Signature  HSS signature Nspk 00000001 sig[0]:  LMS signature q 00000005  LMOTS signature LMOTS type 00000004 # LMOTS_SHA256_N32_W8 C d32b56671d7eb98833c49b433c272586 bc4a1c8a8970528ffa04b966f9426eb9 y[0] 965a25bfd37f196b9073f3d4a232feb6 9128ec45146f86292f9dff9610a7bf95 y[1] a64c7f60f6261a62043f86c70324b770 7f5b4a8a6e19c114c7be866d488778a0 y[2] e05fd5c6509a6e61d559cf1a77a970de 927d60c70d3de31a7fa0100994e162a2 y[3] 582e8ff1b10cd99d4e8e413ef469559f 7d7ed12c838342f9b9c96b83a4943d16 y[4] 81d84b15357ff48ca579f19f5e71f184 66f2bbef4bf660c2518eb20de2f66e3b y[5] 14784269d7d876f5d35d3fbfc7039a46 2c716bb9f6891a7f41ad133e9e1f6d95 y[6] 60b960e7777c52f060492f2d7c660e14 71e07e72655562035abc9a701b473ecb y[7] c3943c6b9c4f2405a3cb8bf8a691ca51 d3f6ad2f428bab6f3a30f55dd9625563 y[8] f0a75ee390e385e3ae0b906961ecf41a e073a0590c2eb6204f44831c26dd768c y[9] 35b167b28ce8dc988a3748255230cef9 9ebf14e730632f27414489808afab1d1 y[10] e783ed04516de012498682212b078105 79b250365941bcc98142da13609e9768 y[11] aaf65de7620dabec29eb82a17fde35af 15ad238c73f81bdb8dec2fc0e7f93270 y[12] 1099762b37f43c4a3c20010a3d72e2f6 06be108d310e639f09ce7286800d9ef8 y[13] a1a40281cc5a7ea98d2adc7c7400c2fe 5a101552df4e3cccfd0cbf2ddf5dc677 y[14] 9cbbc68fee0c3efe4ec22b83a2caa3e4 8e0809a0a750b73ccdcf3c79e6580c15 y[15] 4f8a58f7f24335eec5c5eb5e0cf01dcf 4439424095fceb077f66ded5bec73b27 y[16] c5b9f64a2a9af2f07c05e99e5cf80f00 252e39db32f6c19674f190c9fbc506d8 McGrew, et al. Informational [Page 50] RFC 8554 LMS HashBased Signatures April 2019 y[17] 26857713afd2ca6bb85cd8c107347552 f30575a5417816ab4db3f603f2df56fb y[18] c413e7d0acd8bdd81352b2471fc1bc4f 1ef296fea1220403466b1afe78b94f7e y[19] cf7cc62fb92be14f18c2192384ebceaf 8801afdf947f698ce9c6ceb696ed70e9 y[20] e87b0144417e8d7baf25eb5f70f09f01 6fc925b4db048ab8d8cb2a661ce3b57a y[21] da67571f5dd546fc22cb1f97e0ebd1a6 5926b1234fd04f171cf469c76b884cf3 y[22] 115cce6f792cc84e36da58960c5f1d76 0f32c12faef477e94c92eb75625b6a37 y[23] 1efc72d60ca5e908b3a7dd69fef02491 50e3eebdfed39cbdc3ce9704882a2072 y[24] c75e13527b7a581a556168783dc1e975 45e31865ddc46b3c957835da252bb732 y[25] 8d3ee2062445dfb85ef8c35f8e1f3371 af34023cef626e0af1e0bc017351aae2 y[26] ab8f5c612ead0b729a1d059d02bfe18e fa971b7300e882360a93b025ff97e9e0 y[27] eec0f3f3f13039a17f88b0cf808f4884 31606cb13f9241f40f44e537d302c64a y[28] 4f1f4ab949b9feefadcb71ab50ef27d6 d6ca8510f150c85fb525bf25703df720 y[29] 9b6066f09c37280d59128d2f0f637c7d 7d7fad4ed1c1ea04e628d221e3d8db77 y[30] b7c878c9411cafc5071a34a00f4cf077 38912753dfce48f07576f0d4f94f42c6 y[31] d76f7ce973e9367095ba7e9a3649b7f4 61d9f9ac1332a4d1044c96aefee67676 y[32] 401b64457c54d65fef6500c59cdfb69a f7b6dddfcb0f086278dd8ad0686078df y[33] b0f3f79cd893d314168648499898fbc0 ced5f95b74e8ff14d735cdea968bee74  LMS type 00000005 # LM_SHA256_M32_H5 path[0] d8b8112f9200a5e50c4a262165bd342c d800b8496810bc716277435ac376728d path[1] 129ac6eda839a6f357b5a04387c5ce97 382a78f2a4372917eefcbf93f63bb591 path[2] 12f5dbe400bd49e4501e859f885bf073 6e90a509b30a26bfac8c17b5991c157e path[3] b5971115aa39efd8d564a6b90282c316 8af2d30ef89d51bf14654510a12b8a14 path[4] 4cca1848cf7da59cc2b3d9d0692dd2a2 0ba3863480e25b1b85ee860c62bf5136  McGrew, et al. Informational [Page 51] RFC 8554 LMS HashBased Signatures April 2019 LMS public key LMS type 00000005 # LM_SHA256_M32_H5 LMOTS type 00000004 # LMOTS_SHA256_N32_W8 I d2f14ff6346af964569f7d6cb880a1b6 K 6c5004917da6eafe4d9ef6c6407b3db0 e5485b122d9ebe15cda93cfec582d7ab  final_signature:  LMS signature q 0000000a  LMOTS signature LMOTS type 00000004 # LMOTS_SHA256_N32_W8 C 0703c491e7558b35011ece3592eaa5da 4d918786771233e8353bc4f62323185c y[0] 95cae05b899e35dffd71705470620998 8ebfdf6e37960bb5c38d7657e8bffeef y[1] 9bc042da4b4525650485c66d0ce19b31 7587c6ba4bffcc428e25d08931e72dfb y[2] 6a120c5612344258b85efdb7db1db9e1 865a73caf96557eb39ed3e3f426933ac y[3] 9eeddb03a1d2374af7bf771855774562 37f9de2d60113c23f846df26fa942008 y[4] a698994c0827d90e86d43e0df7f4bfcd b09b86a373b98288b7094ad81a0185ac y[5] 100e4f2c5fc38c003c1ab6fea479eb2f 5ebe48f584d7159b8ada03586e65ad9c y[6] 969f6aecbfe44cf356888a7b15a3ff07 4f771760b26f9c04884ee1faa329fbf4 y[7] e61af23aee7fa5d4d9a5dfcf43c4c26c e8aea2ce8a2990d7ba7b57108b47dabf y[8] beadb2b25b3cacc1ac0cef346cbb90fb 044beee4fac2603a442bdf7e507243b7 y[9] 319c9944b1586e899d431c7f91bcccc8 690dbf59b28386b2315f3d36ef2eaa3c y[10] f30b2b51f48b71b003dfb08249484201 043f65f5a3ef6bbd61ddfee81aca9ce6 y[11] 0081262a00000480dcbc9a3da6fbef5c 1c0a55e48a0e729f9184fcb1407c3152 y[12] 9db268f6fe50032a363c9801306837fa fabdf957fd97eafc80dbd165e435d0e2 y[13] dfd836a28b354023924b6fb7e48bc0b3 ed95eea64c2d402f4d734c8dc26f3ac5 y[14] 91825daef01eae3c38e3328d00a77dc6 57034f287ccb0f0e1c9a7cbdc828f627 y[15] 205e4737b84b58376551d44c12c3c215 c812a0970789c83de51d6ad787271963 McGrew, et al. Informational [Page 52] RFC 8554 LMS HashBased Signatures April 2019 y[16] 327f0a5fbb6b5907dec02c9a90934af5 a1c63b72c82653605d1dcce51596b3c2 y[17] b45696689f2eb382007497557692caac 4d57b5de9f5569bc2ad0137fd47fb47e y[18] 664fcb6db4971f5b3e07aceda9ac130e 9f38182de994cff192ec0e82fd6d4cb7 y[19] f3fe00812589b7a7ce51544045643301 6b84a59bec6619a1c6c0b37dd1450ed4 y[20] f2d8b584410ceda8025f5d2d8dd0d217 6fc1cf2cc06fa8c82bed4d944e71339e y[21] ce780fd025bd41ec34ebff9d4270a322 4e019fcb444474d482fd2dbe75efb203 y[22] 89cc10cd600abb54c47ede93e08c114e db04117d714dc1d525e11bed8756192f y[23] 929d15462b939ff3f52f2252da2ed64d 8fae88818b1efa2c7b08c8794fb1b214 y[24] aa233db3162833141ea4383f1a6f120b e1db82ce3630b3429114463157a64e91 y[25] 234d475e2f79cbf05e4db6a9407d72c6 bff7d1198b5c4d6aad2831db61274993 y[26] 715a0182c7dc8089e32c8531deed4f74 31c07c02195eba2ef91efb5613c37af7 y[27] ae0c066babc69369700e1dd26eddc0d2 16c781d56e4ce47e3303fa73007ff7b9 y[28] 49ef23be2aa4dbf25206fe45c20dd888 395b2526391a724996a44156beac8082 y[29] 12858792bf8e74cba49dee5e8812e019 da87454bff9e847ed83db07af3137430 y[30] 82f880a278f682c2bd0ad6887cb59f65 2e155987d61bbf6a88d36ee93b6072e6 y[31] 656d9ccbaae3d655852e38deb3a2dcf8 058dc9fb6f2ab3d3b3539eb77b248a66 y[32] 1091d05eb6e2f297774fe6053598457c c61908318de4b826f0fc86d4bb117d33 y[33] e865aa805009cc2918d9c2f840c4da43 a703ad9f5b5806163d7161696b5a0adc  LMS type 00000005 # LM_SHA256_M32_H5 path[0] d5c0d1bebb06048ed6fe2ef2c6cef305 b3ed633941ebc8b3bec9738754cddd60 path[1] e1920ada52f43d055b5031cee6192520 d6a5115514851ce7fd448d4a39fae2ab path[2] 2335b525f484e9b40d6a4a969394843b dcf6d14c48e8015e08ab92662c05c6e9 path[3] f90b65a7a6201689999f32bfd368e5e3 ec9cb70ac7b8399003f175c40885081a path[4] 09ab3034911fe125631051df0408b394 6b0bde790911e8978ba07dd56c73e7ee McGrew, et al. Informational [Page 53] RFC 8554 LMS HashBased Signatures April 2019 Test Case 2 Private Key  (note: procedure in Appendix A is used) Top level LMS tree SEED 558b8966c48ae9cb898b423c83443aae 014a72f1b1ab5cc85cf1d892903b5439 I d08fabd4a2091ff0a8cb4ed834e74534 Second level LMS tree SEED a1c4696e2608035a886100d05cd99945 eb3370731884a8235e2fb3d4d71f2547 I 215f83b7ccb9acbcd08db97b0d04dc2b   Test Case 2 Public Key  HSS public key levels 00000002  LMS type 00000006 # LM_SHA256_M32_H10 LMOTS type 00000003 # LMOTS_SHA256_N32_W4 I d08fabd4a2091ff0a8cb4ed834e74534 K 32a58885cd9ba0431235466bff9651c6 c92124404d45fa53cf161c28f1ad5a8e   Test Case 2 Message  Message 54686520656e756d65726174696f6e20 The enumeration  696e2074686520436f6e737469747574 in the Constitut 696f6e2c206f66206365727461696e20 ion, of certain  7269676874732c207368616c6c206e6f rights, shall no 7420626520636f6e7374727565642074 t be construed t 6f2064656e79206f7220646973706172 o deny or dispar 616765206f7468657273207265746169 age others retai 6e6564206279207468652070656f706c ned by the peopl 652e0a e..  McGrew, et al. Informational [Page 54] RFC 8554 LMS HashBased Signatures April 2019 Test Case 2 Signature  HSS signature Nspk 00000001 sig[0]:  LMS signature q 00000003  LMOTS signature LMOTS type 00000003 # LMOTS_SHA256_N32_W4 C 3d46bee8660f8f215d3f96408a7a64cf 1c4da02b63a55f62c666ef5707a914ce y[0] 0674e8cb7a55f0c48d484f31f3aa4af9 719a74f22cf823b94431d01c926e2a76 y[1] bb71226d279700ec81c9e95fb11a0d10 d065279a5796e265ae17737c44eb8c59 y[2] 4508e126a9a7870bf4360820bdeb9a01 d9693779e416828e75bddd7d8c70d50a y[3] 0ac8ba39810909d445f44cb5bb58de73 7e60cb4345302786ef2c6b14af212ca1 y[4] 9edeaa3bfcfe8baa6621ce88480df237 1dd37add732c9de4ea2ce0dffa53c926 y[5] 49a18d39a50788f4652987f226a1d481 68205df6ae7c58e049a25d4907edc1aa y[6] 90da8aa5e5f7671773e941d805536021 5c6b60dd35463cf2240a9c06d694e9cb y[7] 54e7b1e1bf494d0d1a28c0d31acc7516 1f4f485dfd3cb9578e836ec2dc722f37 y[8] ed30872e07f2b8bd0374eb57d22c614e 09150f6c0d8774a39a6e168211035dc5 y[9] 2988ab46eaca9ec597fb18b4936e66ef 2f0df26e8d1e34da28cbb3af75231372 y[10] 0c7b345434f72d65314328bbb030d0f0 f6d5e47b28ea91008fb11b05017705a8 y[11] be3b2adb83c60a54f9d1d1b2f476f9e3 93eb5695203d2ba6ad815e6a111ea293 y[12] dcc21033f9453d49c8e5a6387f588b1e a4f706217c151e05f55a6eb7997be09d y[13] 56a326a32f9cba1fbe1c07bb49fa04ce cf9df1a1b815483c75d7a27cc88ad1b1 y[14] 238e5ea986b53e087045723ce16187ed a22e33b2c70709e53251025abde89396 y[15] 45fc8c0693e97763928f00b2e3c75af3 942d8ddaee81b59a6f1f67efda0ef81d y[16] 11873b59137f67800b35e81b01563d18 7c4a1575a1acb92d087b517a8833383f McGrew, et al. Informational [Page 55] RFC 8554 LMS HashBased Signatures April 2019 y[17] 05d357ef4678de0c57ff9f1b2da61dfd e5d88318bcdde4d9061cc75c2de3cd47 y[18] 40dd7739ca3ef66f1930026f47d9ebaa 713b07176f76f953e1c2e7f8f271a6ca y[19] 375dbfb83d719b1635a7d8a138919579 44b1c29bb101913e166e11bd5f34186f y[20] a6c0a555c9026b256a6860f4866bd6d0 b5bf90627086c6149133f8282ce6c9b3 y[21] 622442443d5eca959d6c14ca8389d12c 4068b503e4e3c39b635bea245d9d05a2 y[22] 558f249c9661c0427d2e489ca5b5dde2 20a90333f4862aec793223c781997da9 y[23] 8266c12c50ea28b2c438e7a379eb106e ca0c7fd6006e9bf612f3ea0a454ba3bd y[24] b76e8027992e60de01e9094fddeb3349 883914fb17a9621ab929d970d101e45f y[25] 8278c14b032bcab02bd15692d21b6c5c 204abbf077d465553bd6eda645e6c306 y[26] 5d33b10d518a61e15ed0f092c3222628 1a29c8a0f50cde0a8c66236e29c2f310 y[27] a375cebda1dc6bb9a1a01dae6c7aba8e bedc6371a7d52aacb955f83bd6e4f84d y[28] 2949dcc198fb77c7e5cdf6040b0f84fa f82808bf985577f0a2acf2ec7ed7c0b0 y[29] ae8a270e951743ff23e0b2dd12e9c3c8 28fb5598a22461af94d568f29240ba28 y[30] 20c4591f71c088f96e095dd98beae456 579ebbba36f6d9ca2613d1c26eee4d8c y[31] 73217ac5962b5f3147b492e8831597fd 89b64aa7fde82e1974d2f6779504dc21 y[32] 435eb3109350756b9fdabe1c6f368081 bd40b27ebcb9819a75d7df8bb07bb05d y[33] b1bab705a4b7e37125186339464ad8fa aa4f052cc1272919fde3e025bb64aa8e y[34] 0eb1fcbfcc25acb5f718ce4f7c2182fb 393a1814b0e942490e52d3bca817b2b2 y[35] 6e90d4c9b0cc38608a6cef5eb153af08 58acc867c9922aed43bb67d7b33acc51 y[36] 9313d28d41a5c6fe6cf3595dd5ee63f0 a4c4065a083590b275788bee7ad875a7 y[37] f88dd73720708c6c6c0ecf1f43bbaada e6f208557fdc07bd4ed91f88ce4c0de8 y[38] 42761c70c186bfdafafc444834bd3418 be4253a71eaf41d718753ad07754ca3e y[39] ffd5960b0336981795721426803599ed 5b2b7516920efcbe32ada4bcf6c73bd2 y[40] 9e3fa152d9adeca36020fdeeee1b7395 21d3ea8c0da497003df1513897b0f547 McGrew, et al. Informational [Page 56] RFC 8554 LMS HashBased Signatures April 2019 y[41] 94a873670b8d93bcca2ae47e64424b74 23e1f078d9554bb5232cc6de8aae9b83 y[42] fa5b9510beb39ccf4b4e1d9c0f19d5e1 7f58e5b8705d9a6837a7d9bf99cd1338 y[43] 7af256a8491671f1f2f22af253bcff54 b673199bdb7d05d81064ef05f80f0153 y[44] d0be7919684b23da8d42ff3effdb7ca0 985033f389181f47659138003d712b5e y[45] c0a614d31cc7487f52de8664916af79c 98456b2c94a8038083db55391e347586 y[46] 2250274a1de2584fec975fb09536792c fbfcf6192856cc76eb5b13dc4709e2f7 y[47] 301ddff26ec1b23de2d188c999166c74 e1e14bbc15f457cf4e471ae13dcbdd9c y[48] 50f4d646fc6278e8fe7eb6cb5c94100f a870187380b777ed19d7868fd8ca7ceb y[49] 7fa7d5cc861c5bdac98e7495eb0a2cee c1924ae979f44c5390ebedddc65d6ec1 y[50] 1287d978b8df064219bc5679f7d7b264 a76ff272b2ac9f2f7cfc9fdcfb6a5142 y[51] 8240027afd9d52a79b647c90c2709e06 0ed70f87299dd798d68f4fadd3da6c51 y[52] d839f851f98f67840b964ebe73f8cec4 1572538ec6bc131034ca2894eb736b3b y[53] da93d9f5f6fa6f6c0f03ce43362b8414 940355fb54d3dfdd03633ae108f3de3e y[54] bc85a3ff51efeea3bc2cf27e1658f178 9ee612c83d0f5fd56f7cd071930e2946 y[55] beeecaa04dccea9f97786001475e0294 bc2852f62eb5d39bb9fbeef75916efe4 y[56] 4a662ecae37ede27e9d6eadfdeb8f8b2 b2dbccbf96fa6dbaf7321fb0e701f4d4 y[57] 29c2f4dcd153a2742574126e5eaccc77 686acf6e3ee48f423766e0fc466810a9 y[58] 05ff5453ec99897b56bc55dd49b99114 2f65043f2d744eeb935ba7f4ef23cf80 y[59] cc5a8a335d3619d781e7454826df720e ec82e06034c44699b5f0c44a8787752e y[60] 057fa3419b5bb0e25d30981e41cb1361 322dba8f69931cf42fad3f3bce6ded5b y[61] 8bfc3d20a2148861b2afc14562ddd27f 12897abf0685288dcc5c4982f8260268 y[62] 46a24bf77e383c7aacab1ab692b29ed8 c018a65f3dc2b87ff619a633c41b4fad y[63] b1c78725c1f8f922f6009787b1964247 df0136b1bc614ab575c59a16d089917b y[64] d4a8b6f04d95c581279a139be09fcf6e 98a470a0bceca191fce476f9370021cb McGrew, et al. Informational [Page 57] RFC 8554 LMS HashBased Signatures April 2019 y[65] c05518a7efd35d89d8577c990a5e1996 1ba16203c959c91829ba7497cffcbb4b y[66] 294546454fa5388a23a22e805a5ca35f 956598848bda678615fec28afd5da61a  LMS type 00000006 # LM_SHA256_M32_H10 path[0] b326493313053ced3876db9d23714818 1b7173bc7d042cefb4dbe94d2e58cd21 path[1] a769db4657a103279ba8ef3a629ca84e e836172a9c50e51f45581741cf808315 path[2] 0b491cb4ecbbabec128e7c81a46e62a6 7b57640a0a78be1cbf7dd9d419a10cd8 path[3] 686d16621a80816bfdb5bdc56211d72c a70b81f1117d129529a7570cf79cf52a path[4] 7028a48538ecdd3b38d3d5d62d262465 95c4fb73a525a5ed2c30524ebb1d8cc8 path[5] 2e0c19bc4977c6898ff95fd3d310b0ba e71696cef93c6a552456bf96e9d075e3 path[6] 83bb7543c675842bafbfc7cdb88483b3 276c29d4f0a341c2d406e40d4653b7e4 path[7] d045851acf6a0a0ea9c710b805cced46 35ee8c107362f0fc8d80c14d0ac49c51 path[8] 6703d26d14752f34c1c0d2c4247581c1 8c2cf4de48e9ce949be7c888e9caebe4 path[9] a415e291fd107d21dc1f084b11582082 49f28f4f7c7e931ba7b3bd0d824a4570  LMS public key LMS type 00000005 # LM_SHA256_M32_H5 LMOTS type 00000004 # LMOTS_SHA256_N32_W8 I 215f83b7ccb9acbcd08db97b0d04dc2b K a1cd035833e0e90059603f26e07ad2aa d152338e7a5e5984bcd5f7bb4eba40b7  final_signature:  LMS signature q 00000004  LMOTS signature LMOTS type 00000004 # LMOTS_SHA256_N32_W8 C 0eb1ed54a2460d512388cad533138d24 0534e97b1e82d33bd927d201dfc24ebb y[0] 11b3649023696f85150b189e50c00e98 850ac343a77b3638319c347d7310269d y[1] 3b7714fa406b8c35b021d54d4fdada7b 9ce5d4ba5b06719e72aaf58c5aae7aca McGrew, et al. Informational [Page 58] RFC 8554 LMS HashBased Signatures April 2019 y[2] 057aa0e2e74e7dcfd17a0823429db629 65b7d563c57b4cec942cc865e29c1dad y[3] 83cac8b4d61aacc457f336e6a10b6632 3f5887bf3523dfcadee158503bfaa89d y[4] c6bf59daa82afd2b5ebb2a9ca6572a60 67cee7c327e9039b3b6ea6a1edc7fdc3 y[5] df927aade10c1c9f2d5ff446450d2a39 98d0f9f6202b5e07c3f97d2458c69d3c y[6] 8190643978d7a7f4d64e97e3f1c4a08a 7c5bc03fd55682c017e2907eab07e5bb y[7] 2f190143475a6043d5e6d5263471f4ee cf6e2575fbc6ff37edfa249d6cda1a09 y[8] f797fd5a3cd53a066700f45863f04b6c 8a58cfd341241e002d0d2c0217472bf1 y[9] 8b636ae547c1771368d9f317835c9b0e f430b3df4034f6af00d0da44f4af7800 y[10] bc7a5cf8a5abdb12dc718b559b74cab9 090e33cc58a955300981c420c4da8ffd y[11] 67df540890a062fe40dba8b2c1c548ce d22473219c534911d48ccaabfb71bc71 y[12] 862f4a24ebd376d288fd4e6fb06ed870 5787c5fedc813cd2697e5b1aac1ced45 y[13] 767b14ce88409eaebb601a93559aae89 3e143d1c395bc326da821d79a9ed41dc y[14] fbe549147f71c092f4f3ac522b5cc572 90706650487bae9bb5671ecc9ccc2ce5 y[15] 1ead87ac01985268521222fb9057df7e d41810b5ef0d4f7cc67368c90f573b1a y[16] c2ce956c365ed38e893ce7b2fae15d36 85a3df2fa3d4cc098fa57dd60d2c9754 y[17] a8ade980ad0f93f6787075c3f680a2ba 1936a8c61d1af52ab7e21f416be09d2a y[18] 8d64c3d3d8582968c2839902229f85ae e297e717c094c8df4a23bb5db658dd37 y[19] 7bf0f4ff3ffd8fba5e383a48574802ed 545bbe7a6b4753533353d73706067640 y[20] 135a7ce517279cd683039747d218647c 86e097b0daa2872d54b8f3e508598762 y[21] 9547b830d8118161b65079fe7bc59a99 e9c3c7380e3e70b7138fe5d9be255150 y[22] 2b698d09ae193972f27d40f38dea264a 0126e637d74ae4c92a6249fa103436d3 y[23] eb0d4029ac712bfc7a5eacbdd7518d6d 4fe903a5ae65527cd65bb0d4e9925ca2 y[24] 4fd7214dc617c150544e423f450c99ce 51ac8005d33acd74f1bed3b17b7266a4 y[25] a3bb86da7eba80b101e15cb79de9a207 852cf91249ef480619ff2af8cabca831 McGrew, et al. Informational [Page 59] RFC 8554 LMS HashBased Signatures April 2019 y[26] 25d1faa94cbb0a03a906f683b3f47a97 c871fd513e510a7a25f283b196075778 y[27] 496152a91c2bf9da76ebe089f4654877 f2d586ae7149c406e663eadeb2b5c7e8 y[28] 2429b9e8cb4834c83464f079995332e4 b3c8f5a72bb4b8c6f74b0d45dc6c1f79 y[29] 952c0b7420df525e37c15377b5f09843 19c3993921e5ccd97e097592064530d3 y[30] 3de3afad5733cbe7703c5296263f7734 2efbf5a04755b0b3c997c4328463e84c y[31] aa2de3ffdcd297baaaacd7ae646e44b5 c0f16044df38fabd296a47b3a838a913 y[32] 982fb2e370c078edb042c84db34ce36b 46ccb76460a690cc86c302457dd1cde1 y[33] 97ec8075e82b393d542075134e2a17ee 70a5e187075d03ae3c853cff60729ba4  LMS type 00000005 # LM_SHA256_M32_H5 path[0] 4de1f6965bdabc676c5a4dc7c35f97f8 2cb0e31c68d04f1dad96314ff09e6b3d path[1] e96aeee300d1f68bf1bca9fc58e40323 36cd819aaf578744e50d1357a0e42867 path[2] 04d341aa0a337b19fe4bc43c2e79964d 4f351089f2e0e41c7c43ae0d49e7f404 path[3] b0f75be80ea3af098c9752420a8ac0ea 2bbb1f4eeba05238aef0d8ce63f0c6e5 path[4] e4041d95398a6f7f3e0ee97cc1591849 d4ed236338b147abde9f51ef9fd4e1c1 Acknowledgements Thanks are due to Chirag Shroff, Andreas Huelsing, Burt Kaliski, Eric Osterweil, Ahmed Kosba, Russ Housley, Philip Lafrance, Alexander Truskovsky, Mark Peruzel, and Jim Schaad for constructive suggestions and valuable detailed review. We especially acknowledge Jerry Solinas, Laurie Law, and Kevin Igoe, who pointed out the security benefits of the approach of Leighton and Micali [USPTO5432852], Jonathan Katz, who gave us security guidance, and Bruno Couillard and Jim Goodman for an especially thorough review. McGrew, et al. Informational [Page 60] RFC 8554 LMS HashBased Signatures April 2019 Authors' Addresses David McGrew Cisco Systems 13600 Dulles Technology Drive Herndon, VA 20171 United States of America Email: mcgrew@cisco.com Michael Curcio Cisco Systems 70252 Kit Creek Road Research Triangle Park, NC 277094987 United States of America Email: micurcio@cisco.com Scott Fluhrer Cisco Systems 170 West Tasman Drive San Jose, CA United States of America Email: sfluhrer@cisco.com McGrew, et al. Informational [Page 61]