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RFC 9858	Additional HSS/LMS Signatures	October 2025
Fluhrer & Dang	Informational	[Page]

Abstract

This document extends HSS/LMS (RFC 8554) by defining parameter sets that use alternative hash functions. These include hash functions that result in signatures with significantly smaller sizes than the signatures that use the RFC 8554 parameter sets and should have sufficient security.¶

This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internet-related 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.¶

Status of This Memo

This document is not an Internet Standards Track specification; it is published for informational purposes.¶

Information about the current status of this document, any errata, and how to provide feedback on it may be obtained athttps://www.rfc-editor.org/info/rfc9858.¶

Copyright Notice

This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) 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.¶

1.Introduction

Stateful hash-based signatures have small private and public keys,are efficient to compute, and are believed to have excellent security.One disadvantage is that the signatures they produce tend to be somewhatlarge (possibly 1-4 kilobytes).This document defines a set of parameter sets for the HSS/LMS stateful hash-based signature method[RFC8554] that reduce the size of the signaturesignificantly or rely on a hash function other than SHA-256 (to increasecryptodiversity).¶

This document represents the consensus of the Crypto Forum Research Group (CFRG) in the IRTF. It is not an IETF product and is not a standard.¶

According to official definitions and common usage, a Leighton-Micali Signature (LMS) is a stateful hash-based signature scheme that is based on a single-level Merkle tree.The Hierarchical Signature System (HSS) is a way of binding several LMS signatures together in a hierarchical manner to increase the number of signatures available.Strictly speaking, all the signatures discussed in this document are HSS signatures (even if the HSS signature consists of a single LMS signature).However, it is common to refer to these signatures as "LMS signatures".This document uses the term "HSS/LMS" to cover both the pedantic and the common meanings.¶

This document is intended to be compatible with the NIST document[NIST_SP_800-208].¶

2.Additional Hash Function Definitions

This section defines three hash functions that are used with the parameter sets defined in Sections3 and4.These hash functions are used where SHA-256 is used in the original parameter sets from[RFC8554].The hash function used is specified by the parameter set that is selected.¶

2.1.192-Bit Hash Function Based on SHA-256

This document defines a SHA-2-based hash function with a 192-bit output.As such, we define SHA-256/192 as a truncated version of SHA-256[FIPS180].That is, it is the result of performing a SHA-256operation to a message and then omitting the final 64 bits of the output.This procedure for truncating the hash output to 192 bits is described in Section 7 of[FIPS180].¶

The following test vector illustrates this:¶

  SHA-256("abc")     = ba7816bf 8f01cfea 414140de 5dae2223                       b00361a3 96177a9c b410ff61 f20015ad  SHA-256/192("abc") = ba7816bf 8f01cfea 414140de 5dae2223                       b00361a3 96177a9c

We use the same initial hash value (initialization vector) as the untruncated SHA-256, rather than defining a distinct one,so that we can use a standard SHA-256 hash implementation without modification.In addition, the fact that anyone gets partial knowledge of the SHA-256 hashof a message by examining the SHA-256/192 hash of the same message isnot a concern for this application.Each message that is hashed is randomized. Any message being signed includesthe C randomizer (a value that is selected by the signer and is included in the hash),which varies per message.Therefore, signing the same message by SHA-256 and by SHA-256/192 will notresult in the same value being hashed, and so the latter hash value is not a prefix of the former one. In addition, all hashes include the I identifier, which is included as a part of the signature process in[RFC8554].This I identifier is selected randomly for each private key (and hence two keys will have different I values with high probability), andso two intermediate hashes computed as a part of signing with two HSS private keys (one with a SHA-256 parameter set and one with a SHA-256/192 parameter set) will also be distinct with high probability.¶

2.2.256-Bit Hash Function Based on SHAKE256

This document defines a SHAKE-based hash function with a 256-bit output.As such, we define SHAKE256/256 to be the first 256 bits of the SHAKE256 extendable-output function (XOF).That is, it is the result of performing a SHAKE-256 operation to a message and then using the first 256 bits of output.See[FIPS202] for more detail.¶

2.3.192-Bit Hash Function Based on SHAKE256

This document defines a SHAKE-based hash function with a 192-bit output.As such, we define SHAKE256/192 to be the first 192 bits of the SHAKE256 XOF.That is, it is the result of performing a SHAKE-256 operation to a message and then using the first 192 bits of output.See[FIPS202] for more detail.¶

3.Additional LM-OTS Parameter Sets

The table below defines the Leighton-Micali One-Time Signature (LM-OTS) parameters that use the hashes defined in Section 2:¶

Table 1
Parameter Set Name	H	n	w	p	ls	id
LMOTS_SHA256_N24_W1	SHA-256/192	24	1	200	8	0x00000005
LMOTS_SHA256_N24_W2	SHA-256/192	24	2	101	6	0x00000006
LMOTS_SHA256_N24_W4	SHA-256/192	24	4	51	4	0x00000007
LMOTS_SHA256_N24_W8	SHA-256/192	24	8	26	0	0x00000008
LMOTS_SHAKE_N32_W1	SHAKE256/256	32	1	265	7	0x00000009
LMOTS_SHAKE_N32_W2	SHAKE256/256	32	2	133	6	0x0000000A
LMOTS_SHAKE_N32_W4	SHAKE256/256	32	4	67	4	0x0000000B
LMOTS_SHAKE_N32_W8	SHAKE256/256	32	8	34	0	0x0000000C
LMOTS_SHAKE_N24_W1	SHAKE256/192	24	1	200	8	0x0000000D
LMOTS_SHAKE_N24_W2	SHAKE256/192	24	2	101	6	0x0000000E
LMOTS_SHAKE_N24_W4	SHAKE256/192	24	4	51	4	0x0000000F
LMOTS_SHAKE_N24_W8	SHAKE256/192	24	8	26	0	0x00000010

Parameter Set Name:: The human-readable name of the parameter set.¶
H:: The second-preimage-resistant cryptographic hash function used within this parameter set.¶
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 n-byte string elements that make up the LM-OTS signature.¶
ls:: The number of left-shift bits used in the checksum function Cksm (used by algorithm 2 of[RFC8554]).¶
id:: The IANA-defined identifier used to denote this specific parameter set, which appears in both public keys and signatures.¶

These values are additions to the entries in Table 1 of[RFC8554].¶

The SHA256_N24, SHAKE_N32, and SHAKE_N24 in the parameter set names denote theSHA-256/192, SHAKE256/256, and SHAKE256/192 hash functions defined inSection 2.¶

Remember that the C message randomizer (which is included in the signature) has the same size (n bytes) as the hash output,and so it shrinks from 32 bytes to 24 bytes for the parameter sets that use either SHA-256/192 or SHAKE256/192.¶

4.Additional LMS Parameter Sets

The table below defines several many-time signature parameters called Leighton-Micali Signature (LMS) parameters,using the SHA-256/192, SHAKE256/256, and SHAKE256/192 hash functions:¶

Table 2
Parameter Set Name	H	m	h	id
LMS_SHA256_M24_H5	SHA-256/192	24	5	0x0000000A
LMS_SHA256_M24_H10	SHA-256/192	24	10	0x0000000B
LMS_SHA256_M24_H15	SHA-256/192	24	15	0x0000000C
LMS_SHA256_M24_H20	SHA-256/192	24	20	0x0000000D
LMS_SHA256_M24_H25	SHA-256/192	24	25	0x0000000E
LMS_SHAKE_M32_H5	SHAKE256/256	32	5	0x0000000F
LMS_SHAKE_M32_H10	SHAKE256/256	32	10	0x00000010
LMS_SHAKE_M32_H15	SHAKE256/256	32	15	0x00000011
LMS_SHAKE_M32_H20	SHAKE256/256	32	20	0x00000012
LMS_SHAKE_M32_H25	SHAKE256/256	32	25	0x00000013
LMS_SHAKE_M24_H5	SHAKE256/192	24	5	0x00000014
LMS_SHAKE_M24_H10	SHAKE256/192	24	10	0x00000015
LMS_SHAKE_M24_H15	SHAKE256/192	24	15	0x00000016
LMS_SHAKE_M24_H20	SHAKE256/192	24	20	0x00000017
LMS_SHAKE_M24_H25	SHAKE256/192	24	25	0x00000018

Parameter Set Name:: The human-readable name of the parameter set.¶
H:: The second-preimage-resistant cryptographic hash function used within this parameter set.¶
m:: The size in bytes of the hash function output.¶
h:: The height of the Merkle tree.¶
id:: The IANA-defined identifier used to denote this specific parameter set, which appears inboth public keys and signatures.¶

These values are additions to the entries in Table 2 of[RFC8554].¶

The SHA256_M24, SHAKE_M32, and SHAKE_M24 in the parameter set names denote theSHA-256/192, SHAKE256/256, and SHAKE256/192 hash functions defined inSection 2.¶

5.Usage for These Additional Hash Functions within HSS

To use the additional hash functions within HSS, one would use the appropriate LM-OTS id fromTable 1 and the appropriate LMS id fromTable 2 and use that additional hash function when computing the hashes for key generation, signature generation, and signature verification.¶

Note that the size of the I Merkle tree identifier remains 16 bytes, independent of what hash function is used.¶

6.Parameter Set Selection

This document, along with[RFC8554], defines four hash functions for use within HSS/LMS: SHA-256, SHA-256/192, SHAKE256/256, and SHAKE256/192.The main reason one would select a hash with a 192-bit output (either SHA-256/192 or SHAKE256/192) would be to reduce the signature size;this comes at the cost of reducing the security margin. However, the security should be sufficient for most uses.¶

In contrast, there is no security or signature size difference between the SHA-256-based parameter sets (SHA-256 or SHA-256/192) versus theSHAKE-based parameter sets (SHAKE256/256 or SHAKE256/192). The reason for selecting between the two would be based on practical considerations,for example, if the implementation happens to have an existing SHA-256 (or SHAKE) implementation or if one of thetwo happens to give better hashing performance on the platform.¶

7.Comparisons of 192-Bit and 256-Bit Parameter Sets

Switching to a 192-bit hash affects the signature size, the computation time, and the security strength.It significantly reduces the signature size and somewhat reduces the computation time, at the cost of security strength. SeeSection 8 for a discussion of the security strength.¶

The impact on signature size and computation time is based on two effects:¶

Each hash that appears in the signature is shorter.¶
We need fewer Winternitz chains (because LM-OTS signs a shorter value).¶

For signature length, both effects are relevant. The first is relevant because the signature consists of a series of hashes and each hash is shorter. The second is relevant because when we need fewer Winternitz chains, we need fewer hashes in each LM-OTS signature.¶

For computation time (for both signature generation and verification), effect 1 is irrelevant (we still need to perform essentially the same hash computation), but effect 2 still applies. For example, with W=8, SHA-256 requires 34 Winternitz chains per LM-OTS signature, but SHA-256/192 requires only 26. Since the vast majority of time (for both signature generation and verification) is spent computing these Winternitz chains, this reduction in the number of chains gives us some performance improvement.¶

The table below gives the space used by both the 256-bit and 192-bit parameter sets for a range of commonly used Winternitz parameters and tree heights:¶

Table 3
ParmSet	Winternitz	256-bit hash	192-bit hash
15	4	2672	1624
15	8	1616	1024
20	4	2832	1744
20	8	1776	1144
15/10	4	5236	3172
15/10	8	3124	1972
15/15	4	5396	3292
15/15	8	3284	2092
20/10	4	5396	3292
20/10	8	3284	2092
20/15	4	5556	3412
20/15	8	3444	2212

ParmSet:: The height of the Merkle tree(s), which is the parameter "h" fromTable 2. 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 top-level Merkle tree and y being the bottom level.¶
Winternitz:: The Winternitz parameter used, which is the parameter "w" fromTable 1. For the tests that use multiple trees, this applies to all of them.¶
256-bit hash:: The size in bytes of a signature, assuming that a 256-bit hash is used in the signature (either SHA-256 or SHAKE256/256).¶
192-bit hash:: The size in bytes of a signature, assuming that a 192-bit hash is used in the signature (either SHA-256/192 or SHAKE256/192).¶

An examination of the signature sizes shows that the 192-bit parameters consistently givea 35-40% reduction in the size of the signature in comparison with the 256-bit parameters.¶

For SHA-256/192, there is a smaller (circa 20%) reduction in the amount of computation required for a signature operation with a 192-bit hash, because fewer Winternitz chains would need to be computed.The SHAKE256/192 signatures may have either a faster or slower computation, depending on the implementation speed of SHAKE versus SHA-256 hashes.¶

The SHAKE256/256-based parameter sets give no space advantage (or disadvantage) over the existing SHA-256-based parameter sets;any performance delta would depend solely on the implementation and whether they can generate SHAKE hashes faster than SHA-256 ones.¶

8.Security Considerations

The strength of a signature that uses the SHA-256/192, SHAKE256/256, and SHAKE256/192 hash functions is basedon the difficulty in finding preimages or second preimages to those hash functions.As shown in[Katz16], if we assume that the hash function can be modeled as a random oracle, then the security of the system is at least 8N-1 bits (where N is the size of the hash output in bytes); this gives us a security level of 255 bits for SHAKE256/256 and 191 bits for SHA-256/192 and SHAKE256/192).That is, the strength of SHA-256/192 and SHAKE256/192 can be expected to be equivalent to the strength of AES-192, while the strength of SHAKE256/256 is equivalent to the strength of AES-256.If AES-192 and AES-256 are quantum-resistant, then we expect SHA-256/192, SHAKE256/192, and SHAKE256/256 to also be.¶

If we look at this in a different way, we see that the case of SHAKE256/256 is essentially the same as the existing SHA-256-based signatures; the difficultlyof finding preimages and second preimages is essentially the same, and so they have (barring unexpected cryptographical advances)essentially the same level of security.¶

The case of SHA-256/192 and SHAKE256/192 requires closer analysis.¶

For a classical (non-quantum) computer, there is no known attack better than performing hashesof a large number of distinct preimages. Therefore, a successful attack has a high probabilityof requiring nearly 2¹⁹² hash computations (for either SHA-256/192 or SHAKE256/192).These can be taken as the expected work effort and would appear to be completelyinfeasible in practice.¶

In theory, an attacker with a quantum computer could use Grover's algorithm[Grover96] to reducethe expected complexity to circa 2⁹⁶ hash computations (for N=24). On the otherhand, implementing Grover's algorithm with this number of hash computations wouldrequire performing circa 2⁹⁶ hash computations in succession, which will takemore time than is likely to be acceptable to any attacker.To speed this up,the attacker would need to run a number of instances of Grover's algorithm inparallel. This would necessarily increase the total work effort required,and to an extent, that makes it likely infeasible.This is because if we limit the time taken by Grover's algorithm to 2^t steps (for t <= 96), then to attack a hash preimage problem of 192 bits, it requires a total of 2^(192-t) hash computations, rather than the 2^(192/2) hash computations it would require if we did not limit the time taken.In other words, the hash preimage can be found in 2^t steps by using 2^(192-2t) quantum computers (for t <= 96), with one of the quantum computers finding the preimage.For example, if the adversary is willing to wait for 2⁶⁴ times the time taken by a hash computation (which is over 50 years if a quantum computer can compute a hash in 0.1 nanoseconds), this implies that a total of 2^(192-64) = 2¹²⁸ hash computations will need to be performed, on 2⁶⁴ (18 quintillion) separate quantum computers, each of which computes 2⁶⁴ hash evaluations.¶

Hence, we expect that HSS/LMS based on these hash functions is secure against both classical and quantum computers,even though, in both cases, the expected work effort is less (for the N=24 case) than against either SHA-256 or SHAKE256/256.¶

SHA-256 is subject to a length extension attack.In this attack, if the attacker is given the hash value of an unknown message (and the message length), then the attacker can compute the hash of the message appended with certain strings (even though the attacker does not know the contents of the initial part of the modified message).This would appear to be irrelevant to HSS for two reasons:¶

For the initial message hash, the hash is entirely on public data. Hence, this attack is irrelevant, because the attacker could compute the hash of the message with appended data anyways.¶
The rest of the hashes within HSS are fixed length. Hence, there is no opportunity to perform length extension attacks.¶

In addition, to perform a length extension attack on SHA-256/192, the attacker has to guess the 64 omitted bits (because the attack requires all 256 bits of the hash value); hence, that is even less of a concern than it is for the standard SHA-256.¶

There is one corner case for which the security strength is reduced: if we need to assume that the signer will never deliberately generate a signature that is valid for two different messages.HSS uses randomized hashing when signing a message. That is, when a message is being presented to be signed, the signer generates a random value C and includes that in what is prepended to the message. Because the attacker cannot predict this value, it is infeasible for anyone other than the signer to find a generic collision.That is, practically speaking, a signature that is valid for two colliding messages is feasible only if the signer signed both messages.For this to happen, a signer (that is, the one with the private key and who picks the random C value) would have to break the collision resistance of the hash function to generate those two colliding messages.Note that this does not apply to someone who submits the messages for signing; only the signer could perform this.This would result in a signature that would be valid for two different selected messages. This is a nonstandard assumption for signature schemes and is usually not a concern, as we assume that the signer is trusted to generate signatures for any message.However, if the application needs to assume that it is infeasible for the signer to generate such a signature, then the security strength assumptions are reduced (128 bits for SHAKE256/256 and 96 bits for SHA-256/192 and SHAKE256/192).¶

Some cryptographers have raised the possibility of a multi-target attack (where the attacker has signatures from a large number of public keys and succeeds if they can generate a forgery against any one of those public keys).While no such method of attack has been proposed, the possibility cannot be excluded; if there are a large number of public keys, it might be prudent to consider the possibility of some security loss with N=24.If there are 2^K public keys, this security loss cannot be more than K bits of security.¶

8.1.Note on the Version of SHAKE

[FIPS202] defines both SHAKE128 and SHAKE256. This specification selects SHAKE256, even though it is less efficientfor large messages. The reason is that SHAKE128 has a low upper bound on the difficultyof finding preimages (due to the invertibility of its internal permutation), which would limit the strengthof HSS/LMS (whose strength is based on the difficulty of finding preimages). Hence, we specify the use ofSHAKE256, which has a considerably stronger preimage resistance.¶

9.IANA Considerations

IANA has assigned the code points for the parameter sets inSection 3 in the "LM-OTS Signatures" registry and the parameter sets inSection 4 in the "Leighton-Micali Signatures (LMS)" registry. These assignments are included in[NIST_SP_800-208], but the IANA registrations only reference this document.¶

10.References

10.1.Normative References

[FIPS180]: NIST,"Secure Hash Standard",NIST FIPS 180-4,DOI 10.6028/NIST.FIPS.180-4,August 2015,<https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.180-4.pdf>.
[FIPS202]: NIST,"SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions",NIST FIPS 202,DOI 10.6028/NIST.FIPS.202,August 2015,<https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.202.pdf>.
[RFC8554]: McGrew, D.,Curcio, M., andS. Fluhrer,"Leighton-Micali Hash-Based Signatures",RFC 8554,DOI 10.17487/RFC8554,April 2019,<https://www.rfc-editor.org/info/rfc8554>.

10.2.Informative References

[Grover96]: Grover, L.,"A fast quantum mechanical algorithm for database search",Proceedings of the twenty-eighth annual ACM symposium on Theory of Computing (STOC '96), pp. 212-219,DOI 10.1145/237814.237866,July 1996,<https://doi.org/10.1145/237814.237866>.
[Katz16]: Katz, J.,"Analysis of a Proposed Hash-Based Signature Standard",Security Standardisation Research (SSR 2016), Lecture Notes in Computer Science, vol. 10074, pp. 261-273,DOI 10.1007/978-3-319-49100-4_12,2016,<https://doi.org/10.1007/978-3-319-49100-4_12>.
[NIST_SP_800-208]: Cooper, D.,Apon, D.,Dang, Q.,Davidson, M.,Dworkin, M., andC. Miller,"Recommendation for Stateful Hash-Based Signature Schemes",National Institute of Standards and Technology,NIST SP 800-208,DOI 10.6028/NIST.SP.800-208,October 2020,<https://doi.org/10.6028/NIST.SP.800-208>.

Appendix A.Test Cases

This appendix provides four test cases that can be used to verify ordebug an implementation.This data is formatted with the name of theelements on the left and the value of the elements on the right, inhexadecimal. The concatenation of all of the values within a publickey or signature produces that public key or signature, and valuesthat do not fit within a single line are listed across successivelines.¶

A.1.Test Case 1 - SHA-256/192

--------------------------------------------(note: procedure in Appendix A of [RFC8554] is used)SEED        000102030405060708090a0b0c0d0e0f            1011121314151617I           202122232425262728292a2b2c2d2e2f--------------------------------------------

Figure 1:Private Key for SHA-256/192

--------------------------------------------HSS public keylevels      00000001--------------------------------------------LMS type    0000000a                         # LMS_SHA256_M24_H5LM-OTS type 00000008                         # LMOTS_SHA256_N24_W8I           202122232425262728292a2b2c2d2e2fK           2c571450aed99cfb4f4ac285da148827            96618314508b12d2--------------------------------------------

Figure 2:Public Key for SHA-256/192

--------------------------------------------Message     54657374206d65737361676520666f72  |Test message for|            205348413235362d3139320a          | SHA-256/192.|--------------------------------------------

Figure 3:Message for SHA-256/192

--------------------------------------------HSS signatureNspk        00000000sig[0]:--------------------------------------------LMS signatureq           00000005--------------------------------------------LM-OTS signatureLM-OTS type 00000008                         # LMOTS_SHA256_N24_W8C           0b5040a18c1b5cabcbc85b047402ec62            94a30dd8da8fc3day[0]        e13b9f0875f09361dc77fcc4481ea463            c073716249719193y[1]        614b835b4694c059f12d3aedd34f3db9            3f3580fb88743b8by[2]        3d0648c0537b7a50e433d7ea9d6672ff            fc5f42770feab4f9y[3]        8eb3f3b23fd2061e4d0b38f832860ae7            6673ad1a1a52a900y[4]        5dcf1bfb56fe16ff723627612f9a48f7            90f3c47a67f870b8y[5]        1e919d99919c8db48168838cece0abfb            683da48b9209868by[6]        e8ec10c63d8bf80d36498dfc205dc45d            0dd870572d6d8f1dy[7]        90177cf5137b8bbf7bcb67a46f86f26c            fa5a44cbcaa4e18dy[8]        a099a98b0b3f96d5ac8ac375d8da2a7c            248004ba11d7ac77y[9]        5b9218359cddab4cf8ccc6d54cb7e1b3            5a36ddc9265c0870y[10]       63d2fc6742a7177876476a324b03295b            fed99f2eaf1f3897y[11]       0583c1b2b616aad0f31cd7a4b1bb0a51            e477e94a01bbb4d6y[12]       f8866e2528a159df3d6ce244d2b6518d            1f0212285a3c2d4ay[13]       927054a1e1620b5b02aab0c8c10ed48a            e518ea73cba81fcfy[14]       ff88bff461dac51e7ab4ca75f47a6259            d24820b9995792d1y[15]       39f61ae2a8186ae4e3c9bfe0af2cc717            f424f41aa67f03fay[16]       edb0665115f2067a46843a4cbbd297d5            e83bc1aafc18d1d0y[17]       3b3d894e8595a6526073f02ab0f08b99            fd9eb208b59ff631y[18]       7e5545e6f9ad5f9c183abd043d5acd6e            b2dd4da3f02dbc31y[19]       67b468720a4b8b92ddfe7960998bb7a0            ecf2a26a37598299y[20]       413f7b2aecd39a30cec527b4d9710c44            73639022451f50d0y[21]       1c0457125da0fa4429c07dad859c846c            bbd93ab5b91b01bcy[22]       770b089cfede6f651e86dd7c15989c8b            5321dea9ca608c71y[23]       fd862323072b827cee7a7e28e4e2b999            647233c3456944bby[24]       7aef9187c96b3f5b79fb98bc76c3574d            d06f0e95685e5b3ay[25]       ef3a54c4155fe3ad817749629c30adbe            897c4f4454c86c49--------------------------------------------LMS type    0000000a                         # LMS_SHA256_M24_H5path[0]     e9ca10eaa811b22ae07fb195e3590a33            4ea64209942fbae3path[1]     38d19f152182c807d3c40b189d3fcbea            942f44682439b191path[2]     332d33ae0b761a2a8f984b56b2ac2fd4            ab08223a69ed1f77path[3]     19c7aa7e9eee96504b0e60c6bb5c942d            695f0493eb25f80apath[4]     5871cffd131d0e04ffe5065bc7875e82            d34b40b69dd9f3c1

Figure 4:Signature for SHA-256/192

A.2.Test Vector for SHAKE256/192

--------------------------------------------(note: procedure in Appendix A of [RFC8554] is used)SEED        303132333435363738393a3b3c3d3e3f            4041424344454647I           505152535455565758595a5b5c5d5e5f--------------------------------------------

Figure 5:Private Key for SHAKE256/192

---------------------------------------------HSS public keylevels      00000001--------------------------------------------LMS type    00000014                         # LMS_SHAKE_N24_H5LM-OTS type 00000010                         # LMOTS_SHAKE_N24_W8I           505152535455565758595a5b5c5d5e5fK           db54a4509901051c01e26d9990e55034            7986da87924ff0b1--------------------------------------------

Figure 6:Public Key for SHAKE256/192

--------------------------------------------Message     54657374206d65737361676520666f72  |Test message for|            205348414b453235362d3139320a      | SHAKE256/192.|--------------------------------------------

Figure 7:Message for SHAKE256/192

--------------------------------------------HSS signatureNspk        00000000sig[0]:--------------------------------------------LMS signatureq           00000006--------------------------------------------LM-OTS signatureLM-OTS type 00000010                         # LMOTS_SHAKE_N24_W8C           84219da9ce9fffb16edb94527c6d1056            5587db28062deac4y[0]        208e62fc4fbe9d85deb3c6bd2c01640a            ccb387d8a6093d68y[1]        511234a6a1a50108091c034cb1777e02            b5df466149a66969y[2]        a498e4200c0a0c1bf5d100cdb97d2dd4            0efd3cada278acc5y[3]        a570071a043956112c6deebd1eb3a7b5            6f5f6791515a7b5fy[4]        fddb0ec2d9094bfbc889ea15c3c7b9be            a953efb75ed648f5y[5]        35b9acab66a2e9631e426e4e99b733ca            a6c55963929b77fey[6]        c54a7e703d8162e736875cb6a455d4a9            015c7a6d8fd5fe75y[7]        e402b47036dc3770f4a1dd0a559cb478            c7fb1726005321bey[8]        9d1ac2de94d731ee4ca79cff454c811f            46d11980909f047by[9]        2005e84b6e15378446b1ca691efe491e            a98acc9d3c0f785cy[10]       aba5e2eb3c306811c240ba2280292382            7d582639304a1e97y[11]       83ba5bc9d69d999a7db8f749770c3c04            a152856dc726d806y[12]       7921465b61b3f847b13b2635a45379e5            adc6ff58a99b00e6y[13]       0ac767f7f30175f9f7a140257e218be3            07954b1250c9b419y[14]       02c4fa7c90d8a592945c66e86a76defc            b84500b55598a199y[15]       0faaa10077c74c94895731585c8f900d            e1a1c675bd8b0c18y[16]       0ebe2b5eb3ef8019ece3e1ea7223eb79            06a2042b6262b4aay[17]       25c4b8a05f205c8befeef11ceff12825            08d71bc2a8cfa0a9y[18]       9f73f3e3a74bb4b3c0d8ca2abd0e1c2c            17dafe18b4ee2298y[19]       e87bcfb1305b3c069e6d385569a4067e            d547486dd1a50d6fy[20]       4a58aab96e2fa883a9a39e1bd45541ee            e94efc32faa9a94by[21]       e66dc8538b2dab05aee5efa6b3b2efb3            fd020fe789477a93y[22]       afff9a3e636dbba864a5bffa3e28d13d            49bb597d94865bdey[23]       88c4627f206ab2b465084d6b780666e9            52f8710efd748bd0y[24]       f1ae8f1035087f5028f14affcc5fffe3            32121ae4f87ac5f1y[25]       eac9062608c7d87708f1723f38b23237            a4edf4b49a5cd3d7--------------------------------------------LMS type    00000014                         # LMS_SHAKE_N24_H5path[0]     dd4bdc8f928fb526f6fb7cdb944a7eba            a7fb05d995b5721apath[1]     27096a5007d82f79d063acd434a04e97            f61552f7f81a9317path[2]     b4ec7c87a5ed10c881928fc6ebce6dfc            e9daae9cc9dba690path[3]     7ca9a9dd5f9f573704d5e6cf22a43b04            e64c1ffc7e1c442epath[4]     cb495ba265f465c56291a902e62a461f            6dfda232457fad14

Figure 8:Signature for SHAKE256/192

A.3.Test Vector for SHA-256/256

--------------------------------------------(note: procedure in Appendix A of [RFC8554] is used)SEED        606162636465666768696a6b6c6d6e6f            707172737475767778797a7b7c7d7e7fI           808182838485868788898a8b8c8d8e8f--------------------------------------------

Figure 9:Private Key for SHAKE256/256

--------------------------------------------HSS public keylevels      00000001--------------------------------------------LMS type    0000000f                         # LMS_SHAKE_N32_H5LM-OTS type 0000000c                         # LMOTS_SHAKE_N32_W8I           808182838485868788898a8b8c8d8e8fK           9bb7faee411cae806c16a466c3191a8b            65d0ac31932bbf0c2d07c7a4a36379fe--------------------------------------------

Figure 10:Public Key for SHAKE256/256

--------------------------------------------Message     54657374206d657361676520666f7220  |Test message for|            5348414b453235362d3235360a        |SHAKE256/256.|--------------------------------------------

Figure 11:Message for SHAKE256/256

--------------------------------------------HSS signatureNspk        00000000sig[0]:--------------------------------------------LMS signatureq           00000007--------------------------------------------LM-OTS signatureLM-OTS type 0000000c                         # LMOTS_SHAKE_N32_W8C           b82709f0f00e83759190996233d1ee4f            4ec50534473c02ffa145e8ca2874e32by[0]        16b228118c62b96c9c77678b33183730            debaade8fe607f05c6697bc971519a34y[1]        1d69c00129680b67e75b3bd7d8aa5c8b            71f02669d177a2a0eea896dcd1660f16y[2]        864b302ff321f9c4b8354408d0676050            4f768ebd4e545a9b0ac058c575078e6cy[3]        1403160fb45450d61a9c8c81f6bd69bd            fa26a16e12a265baf79e9e233eb71af6y[4]        34ecc66dc88e10c6e0142942d4843f70            a0242727bc5a2aabf7b0ec12a99090d8y[5]        caeef21303f8ac58b9f200371dc9e41a            b956e1a3efed9d4bbb38975b46c28d5fy[6]        5b3ed19d847bd0a737177263cbc1a226            2d40e80815ee149b6cce2714384c9b7fy[7]        ceb3bbcbd25228dda8306536376f8793            ecadd6020265dab9075f64c773ef97d0y[8]        7352919995b74404cc69a6f3b469445c            9286a6b2c9f6dc839be76618f053de76y[9]        3da3571ef70f805c9cc54b8e501a98b9            8c70785eeb61737eced78b0e380ded4fy[10]       769a9d422786def59700eef3278017ba            bbe5f9063b468ae0dd61d94f9f99d5ccy[11]       36fbec4178d2bda3ad31e1644a2bcce2            08d72d50a7637851aa908b94dc437612y[12]       0d5beab0fb805e1945c41834dd6085e6            db1a3aa78fcb59f62bde68236a10618cy[13]       ff123abe64dae8dabb2e84ca705309c2            ab986d4f8326ba0642272cb3904eb96fy[14]       6f5e3bb8813997881b6a33cac0714e4b            5e7a882ad87e141931f97d612b84e903y[15]       e773139ae377f5ba19ac86198d485fca            97742568f6ff758120a89bf19059b8a6y[16]       bfe2d86b12778164436ab2659ba86676            7fcc435584125fb7924201ee67b535day[17]       f72c5cb31f5a0b1d926324c26e67d4c3            836e301aa09bae8fb3f91f1622b1818cy[18]       cf440f52ca9b5b9b99aba8a6754aae2b            967c4954fa85298ad9b1e74f27a46127y[19]       c36131c8991f0cc2ba57a15d35c91cf8            bc48e8e20d625af4e85d8f9402ec44afy[20]       bd4792b924b839332a64788a7701a300            94b9ec4b9f4b648f168bf457fbb3c959y[21]       4fa87920b645e42aa2fecc9e21e000ca            7d3ff914e15c40a8bc533129a7fd3952y[22]       9376430f355aaf96a0a13d13f2419141            b3cc25843e8c90d0e551a355dd90ad77y[23]       0ea7255214ce11238605de2f000d2001            04d0c3a3e35ae64ea10a3eff37ac7e95y[24]       49217cdf52f307172e2f6c7a2a4543e1            4314036525b1ad53eeaddf0e24b1f369y[25]       14ed22483f2889f61e62b6fb78f5645b            dbb02c9e5bf97db7a0004e87c2a55399y[26]       b61958786c97bd52fa199c27f6bb4d68            c4907933562755bfec5d4fb52f06c289y[27]       d6e852cf6bc773ffd4c07ee2d6cc55f5            7edcfbc8e8692a49ad47a121fe3c1b16y[28]       cab1cc285faf6793ffad7a8c341a49c5            d2dce7069e464cb90a00b2903648b23cy[29]       81a68e21d748a7e7b1df8a593f3894b2            477e8316947ca725d141135202a9442ey[30]       1db33bbd390d2c04401c39b253b78ce2            97b0e14755e46ec08a146d279c67af70y[31]       de256890804d83d6ec5ca3286f1fca9c            72abf6ef868e7f6eb0fddda1b040ececy[32]       9bbc69e2fd8618e9db3bdb0af13dda06            c6617e95afa522d6a2552de15324d991y[33]       19f55e9af11ae3d5614b564c642dbfec            6c644198ce80d2433ac8ee738f9d825e--------------------------------------------LMS type    0000000f                         # LMS_SHAKE_N32_H5path[0]     71d585a35c3a908379f4072d070311db            5d65b242b714bc5a756ba5e228abfa0dpath[1]     1329978a05d5e815cf4d74c1e547ec4a            a3ca956ae927df8b29fb9fab3917a7a4path[2]     ae61ba57e5342e9db12caf6f6dbc5253            de5268d4b0c4ce4ebe6852f012b162fcpath[3]     1c12b9ffc3bcb1d3ac8589777655e22c            d9b99ff1e4346fd0efeaa1da044692e7path[4]     ad6bfc337db69849e54411df8920c228            a2b7762c11e4b1c49efb74486d3931ea

Figure 12:Signature for SHAKE256/256

A.4.Test Vector for SHA-256/192, W=4

--------------------------------------------(note: procedure in Appendix A of [RFC8554] is used)SEED        202122232425262728292a2b2c2d2e2f            3031323334353637I           404142434445464748494a4b4c4d4e4f--------------------------------------------

Figure 13:Private Key for SHA256/192 with W=4

--------------------------------------------HSS public keylevels      00000001--------------------------------------------LMS type    0000000d                         # LMS_SHA256_M24_H20LM-OTS type 00000007                         # LMOTS_SHA256_N24_W4I           404142434445464748494a4b4c4d4e4fK           9c08a50d170406869892802ee4142fcd            eac990f110c2460c--------------------------------------------

Figure 14:Public Key for SHA256/192 with W=4

--------------------------------------------Message     54657374206d65737361676520666f72  |Test message for|            205348413235362f31393220773d34    | SHA256/192 w=4|--------------------------------------------

Figure 15:Message for SHA256/192 with W=4

--------------------------------------------HSS signatureNspk        00000000sig[0]:--------------------------------------------LMS signatureq           00000064--------------------------------------------LM-OTS signatureLM-OTS type 00000007                         # LMOTS_SHA256_N24_W4C          853fa6e1a65fef076acd2485505b93be           9aeb2641e3d3805cy[0]       1887f26f4bcdb6ac0337b76fa5d66038           34287e010b20516fy[1]       7c336df2134c0a981f1ec2bb7baee516           e91e67d3bd16c8d9y[2]       45a7f2be4fd84a604ae3743efc609ee0           e69572e9c6d4a682y[3]       50e877b75d3cae63e9d5c15a32bb3cd1           7045f6b3e195284fy[4]       dd1ee3cfbe18f1cbd06ef3e7af34b184           4d42dac453115a45y[5]       07ed525cec120d054b403c61a7e5034f           ac4be6ef5412d194y[6]       d4b6bbc0ae6cd3fe9993d583ee06f403           0bc832efec24d1f7y[7]       13f5088731b91a98491fa3adf1b322bc           e26df24c8415e3a4y[8]       6bdfe07a6fd48e6d951515758cd64349           91098bf6949249fcy[9]       a338ec235871dd564998d07d9b1b1b8d           644e657fee8039day[10]      8fe195d129faddb12d543b86b0ab8cf6           f26c121783f3b828y[11]      d03f793b42909272f688e4ef6d46e82b           dd1a02b1ff86c3b7y[12]      9920b2e6f19faf75c623242f1f2c549f           84fb2f4c3ffead31y[13]      20d97baea507467bb2da79f132bbe15b           596fdfcb70983107y[14]      ebca2597de9d55bd83bcae5c28a85259           dadb354859986e60y[15]      c8afa0b10bd08a8f9ed9b1ede3377075           fe0ae36349f7d2edy[16]      7bfc9ece0d4cd6972059329419feaf3b           9a1045b6cfa4ae89y[17]      b1cea8950aea4af870d1a3a3909ebc5a           3013d6deb927abc0y[18]      f95093e83cb36a9c1d6f13add19268ac           7a0371f8335b0952y[19]      a57fdb0141d55d937dd6ebb08fee8a5c           f426ac97d54ee7aay[20]      17e6c57be5e62a52a6b1b986730d3a3a           ad8a7d327ddf883ey[21]      6bc7b636eb2a5c4f2a635ae5bada5418           d43dfedb69c0a020y[22]      9334fac89d420d6ad5a2e1df95d26a1b           feb99a5e8455061by[23]      fdf2d6e8394caf8a4be699b8afa38e52           4d4053330af478f8y[24]      5bf33d3ca3a35bc96987282bd513a8f6           a52db9ba36aa9088y[25]      2b3bf573fa275449d8d49eb30bed2bb1           7a0ecc7d8a20807fy[26]      2ea3dd37acd46c713cc2ac9d01a20a30           d6832eef86a1e26dy[27]      1cad7761bf4130a6565572766026509d           eeddaf46b605452by[28]      218a4e137a7ce063b546a35c52510f0e           a2cac879192ec443y[29]      e43b37c5ffa23da7a7fc254324a3de70           5c771794f10ea356y[30]      e5a747e5146fd804a47719803c185b38           0e34b8dcc8269c2by[31]      073d86b2307cf90c6c3ef9271f2d53df           2579f0c4cfb632dby[32]      37a9025965f70b4616673228e98644be           6576417b7a97f104y[33]      350259e7f697408cdf8cf81a3e774162           6ccdb87ad8531264y[34]      cb5ceb7c8c097cec505091a3ee3a826c           54f78169abc2e7d0y[35]      a318dac10250ba940e51e79a3f572fb3           2bf442be6fd81267y[36]      946e6387f9a8c705d945c653f2684655           e3fa6b9ee311d8a0y[37]      91bef9898292fa272fb8761f066c23d8           7aa10d67871cc541y[38]      9c843b796855c51ad1272e9264acd203           5a82b12c2ddbc85ay[39]      dfcd7c22366a36495349391dbf000106           4b8f6b28365445d7y[40]      33e48f1b058a6cb3e71bbb8df3e90406           299894f4ca682943y[41]      ceeba410b33b07716ffc18d6eab75f2d           6372f1133605fa3cy[42]      3ed66f2d8f7c5abe59e87d4500965e34           7523d73cb356c144y[43]      827aaa22b1c72a15293c7400e02aaefc           f36f68a8246900e6y[44]      e6228e7ad19d1450c23434f1e45043dc           2b6db57f20d8f5b3y[45]      44d4162aa651333287cd8bf8fac41c78           d61fe2929209bfe2y[46]      dc5a2f80205c043b22e540a29f0ea0a5           ff529e55bf1dfe42y[47]      96fc4bb4ac2e875322ab115db479fe97           9d64f78409af4ec3y[48]      ad3b758fff83af1b9c48e90ca39366f4           26c2fb921df55c72y[49]      786a9217723945a1ac1a66af7def4f8b           367001732cce0e5by[50]      ac91ac9d603807f8bab105b46d315d4c           b88feb1c8686884b--------------------------------------------LMS type    0000000d                         # LMS_SHA256_M24_H20path[0]    13d1a8ef00c5811c15c4d774fdcf7515           5315aff53ebdff8fpath[1]    b6a54f12c165963dd5690cc9842b0e21           90afc5443497584cpath[2]    832155599d00aced84bb3b59170396f7           db4fa84aa8577f76path[3]    cf9367d6e99d3d5be3555d7156b004f2           002f505681b1ad22path[4]    9b9b46a666672aa8ee662c3a0456a9ad           da7a44fbaca46789path[5]    577dcd36dc5cdff34b864d0a32492a0a           cbcaa6c011748f20path[6]    5b91ab2ab84f2333fb3e3b9acaecdac3           8b58aa5f32e718e2path[7]    25631ed6674cccb8c119acbd4992ab31           30a6e912deec5983path[8]    5ab52fbc549430f8b403e4a2a51cc7f4           6fc143d365763aa1path[9]    708fd25bcd657a790e54718d97090624           2a3b8a97dff18e91path[10]   a44c4ba818a8dd2d242251265b023b82           6077eb740f6682e6path[11]   c4ada2b85a67988d406132c2ad899099           e44cfe610c3a5af7path[12]   0b406224411a59597e5dda0f31cd16c9           14b67e96141661f0path[13]   074f43eb02273481bc324ded26c64f23           88559d8c8bd0ef8bpath[14]   34ca4afebfac2a689b4246c264241488           dcf922350dc44f7bpath[15]   c09d57dc1126291b2318810e0f44801c           071e572fd032c780path[16]   f44c9503a4c03c37417dc96422ba0849           c37956f9fd5d33eapath[17]   4fcab84276effec652ca77d7d47ac93c           633d99e0a236f03dpath[18]   5587d1990ffaef737fced1f5cdd8f373           844e9f316aad41a0path[19]   b12302639f83a2d74c9fe30d305a942b           c0c30352a5e44dfb

Figure 16:Signature for SHA256/192 with W=4

Acknowledgements

We would like to thankCarsten Bormann,Russ Housley,Andrey Jivsov,Mallory Knodel,Virendra Kumar,Thomas Pornin, andStanislav Smyshlyaev for their insightful and helpful reviews.¶