Internet-Draft | Additional HSS/LMS Signatures | September 2023 |

Fluhrer & Dang | Expires 21 March 2024 | [Page] |

- Workgroup:
- Crypto Forum Research Group
- Internet-Draft:
- draft-fluhrer-lms-more-parm-sets-11
- Published:
- Intended Status:
- Informational
- Expires:

This note extends HSS/LMS (RFC 8554) by defining parameter sets by including additional hash functions. These include hash functions that result in signatures with significantly smaller size than the signatures using the current parameter sets, and should have sufficient security.¶

This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.¶

This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶

Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."¶

This Internet-Draft will expire on 21 March 2024.¶

Copyright (c) 2023 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/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. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶

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 somewhat large (possibly 1k - 4kbytes). What this draft explores are a set of parameter sets to the HSS/LMS (RFC8554) stateful hash based signature method that reduce the size of the signature significantly or rely on a hash function other than SHA-256 (to increase cryptodiversity).¶

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

Quick note about the distinction between HSS and LMS. LMS is a stateful hash based signature scheme that a based on a single level Merkle tree. 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 that this document discusses are HSS signatures (even if the HSS signature consists of a single LMS signature). However, it is common to refer to these signatures are LMS signatures. This document uses the term HSS/LMS to cover both the pedantic and the common meanings.¶

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 [RFC3979] and [RFC4879] and at https://datatracker.ietf.org/ipr/about.¶

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].¶

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-256 operation to a message, and then omitting the final 64 bits of the output. This is the procedure found in FIPS 180-4 (section 7) for truncating the hash output to 192 bits.¶

The following test vector may illustrate 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 IV 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 you get partial knowledge of the SHA-256 hash of a message by examining the SHA-256/192 hash of the same message is not a concern for this application. Each message that is hashed is randomized. Any message being signed includes the 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 not result 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 [RFC8554] signature process. This I identifier is selected randomly for each private key (and hence two keys will have different I values with high probability), and so two intermediate hashes computed as a part of signing with two HSS private keys (one with a SHA-256 parameter set and one a SHA-256/192 parameter set) will also be distinct with high probability.¶

This document defines a SHAKE-based hash function with a 256 bit output. As such, we define SHAKE256/256 as a hash where you submit the preimage to the SHAKE256 XOF, with the output being 256 bits, see FIPS 202 [FIPS202] for more detail.¶

This document defines a SHAKE-based hash function with a 192 bit output. As such, we define SHAKE256/192 as a hash where you submit the preimage to the SHAKE256 XOF, with the output being 192 bits, see FIPS 202 [FIPS202] for more detail.¶

Here is a table with the LM-OTS parameters defined that use the above hashes:¶

Parameter Set Name | H | n | w | p | ls | id |
---|---|---|---|---|---|---|

LMOTS_SHA256_N24_W1 | SHA-256/192 | 24 | 1 | 200 | 8 | 0x0005 |

LMOTS_SHA256_N24_W2 | SHA-256/192 | 24 | 2 | 101 | 6 | 0x0006 |

LMOTS_SHA256_N24_W4 | SHA-256/192 | 24 | 4 | 51 | 4 | 0x0007 |

LMOTS_SHA256_N24_W8 | SHA-256/192 | 24 | 8 | 26 | 0 | 0x0008 |

LMOTS_SHAKE_N32_W1 | SHAKE256/256 | 32 | 1 | 265 | 7 | 0x0009 |

LMOTS_SHAKE_N32_W2 | SHAKE256/256 | 32 | 2 | 133 | 6 | 0x000a |

LMOTS_SHAKE_N32_W4 | SHAKE256/256 | 32 | 4 | 67 | 4 | 0x000b |

LMOTS_SHAKE_N32_W8 | SHAKE256/256 | 32 | 8 | 34 | 0 | 0x000c |

LMOTS_SHAKE_N24_W1 | SHAKE256/192 | 24 | 1 | 200 | 8 | 0x000d |

LMOTS_SHAKE_N24_W2 | SHAKE256/192 | 24 | 2 | 101 | 6 | 0x000e |

LMOTS_SHAKE_N24_W4 | SHAKE256/192 | 24 | 4 | 51 | 4 | 0x000f |

LMOTS_SHAKE_N24_W8 | SHAKE256/192 | 24 | 8 | 26 | 0 | 0x0010 |

The id is the IANA-defined identifier used to denote this specific parameter set, and which appears in both public keys and signatures.¶

The SHA256_N24, SHAKE_N32, SHAKE_N24 in the parameter set name denote the SHA-256/192, SHAKE256/256 and SHAKE256/192 hash functions defined in Section 3.¶

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.¶

Here is a table with the LM parameters defined that use SHA-256/192, SHAKE256/256 and SHAKE256/192 hash functions:¶

Parameter Set Name | H | m | h | id |
---|---|---|---|---|

LMS_SHA256_M24_H5 | SHA-256/192 | 24 | 5 | 0x000a |

LMS_SHA256_M24_H10 | SHA-256/192 | 24 | 10 | 0x000b |

LMS_SHA256_M24_H15 | SHA-256/192 | 24 | 15 | 0x000c |

LMS_SHA256_M24_H20 | SHA-256/192 | 24 | 20 | 0x000d |

LMS_SHA256_M24_H25 | SHA-256/192 | 24 | 25 | 0x000e |

LMS_SHAKE_M32_H5 | SHAKE256/256 | 32 | 5 | 0x000f |

LMS_SHAKE_M32_H10 | SHAKE256/256 | 32 | 10 | 0x0010 |

LMS_SHAKE_M32_H15 | SHAKE256/256 | 32 | 15 | 0x0011 |

LMS_SHAKE_M32_H20 | SHAKE256/256 | 32 | 20 | 0x0012 |

LMS_SHAKE_M32_H25 | SHAKE256/256 | 32 | 25 | 0x0013 |

LMS_SHAKE_M24_H5 | SHAKE256/192 | 24 | 5 | 0x0014 |

LMS_SHAKE_M24_H10 | SHAKE256/192 | 24 | 10 | 0x0015 |

LMS_SHAKE_M24_H15 | SHAKE256/192 | 24 | 15 | 0x0016 |

LMS_SHAKE_M24_H20 | SHAKE256/192 | 24 | 20 | 0x0017 |

LMS_SHAKE_M24_H25 | SHAKE256/192 | 24 | 25 | 0x0018 |

The id is the IANA-defined identifier used to denote this specific parameter set, and which appears in both public keys and signatures.¶

The SHA256_M24, SHAKE_M32, SHAKE_M24 in the parameter set name denote the SHA-256/192, SHAKE256/256 and SHAKE256/192 hash functions defined in Section 3.¶

To use the additional hash functions within HSS, you would use the appropriate LMOTS id from Table 1 and the appropriate LMS id from Table 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.¶

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. See Section 9 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 relevent (because the signature consists of a series of hashes and each hash is shorter, and because 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), however 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.¶

Here is a table that gives the space used by both the 256 bit parameter sets and the 192 bit parameter sets, for a range of plausible Winternitz parameters and tree heights:¶

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: this is the height of the Merkle tree(s), which is the parameter "h" from Table 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: this is the Winternitz parameter used, with is the parameter "w" from Table 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 show that the 192 bit parameters consistently give a 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 (for reason 2 listed above). 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.¶

This document, along with [RFC8554], defines four hash functions for use within HSS/LMS; namely 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 does this 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 the SHAKE based parameter sets (SHAKE256/256 or SHAKE256/192); the reason for selecting between the two would be based on practical considerations, for example, if your implementation happens to have an existing SHA-256 (or SHAKE) implementation already present or if one of the two happens to give better hashing performance on your platform.¶

The strength of a signature that uses the SHA-256/192, SHAKE256/256 and SHAKE256/192 hash functions is based on 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-2/192 and SHAKE256/192).¶

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 difficultly of 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 (nonquantum) computer, there is no known attack better than performing hashes of a large number of distinct preimages. Therefore, a successful attack has a high probability of requiring nearly 2**192 hash computations (for either SHA-256/192 or SHAKE256/192). These can be taken as the expected work effort, and would appear to be completely infeasible in practice.¶

With a Quantum Computer, an attacker could in theory use Grover's algorithm [Grover96] to reduce the expected complexity required to circa 2**96 hash computations (for N=24). On the other hand, implementing Grover's algorithm with this number of hash computations would require performing circa 2**96 hash computations in succession, which will take more 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 in parallel. This would necessarily increase the total work effort required, and to an extent that makes it likely to be infeasible.¶

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.¶

There is one corner case for which the security strength is reduced: if we need to assume that the signer will never generate a signature which 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 in 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 need to 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 the application needs to make is reduced; 128 bits for SHAKE256/256 and 96 bits for SHA-2/192 and SHAKE256/192.¶

FIPS 202 [FIPS202] defines both SHAKE128 and SHAKE256. This specification selects SHAKE256, even though it is, for large messages, less efficient. The reason is that SHAKE128 has a low upper bound on the difficulty of finding preimages (due to the invertibility of its internal permutation), which would limit the strength of HSS/LMS (whose strength is based on the difficulty of finding preimages). Hence, we specify the use of SHAKE256, which has a considerably stronger preimage resistance.¶

- [FIPS180]
- National Institute of Standards and Technology, "Secure Hash Standard (SHS)", FIPS 180-4, .
- [FIPS202]
- National Institute of Standards and Technology, "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", FIPS 202, .
- [RFC2119]
- Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/info/rfc2119>.
- [RFC3979]
- Bradner, S., Ed., "Intellectual Property Rights in IETF Technology", RFC 3979, DOI 10.17487/RFC3979, , <https://www.rfc-editor.org/info/rfc3979>.
- [RFC4879]
- Narten, T., "Clarification of the Third Party Disclosure Procedure in RFC 3979", RFC 4879, DOI 10.17487/RFC4879, , <https://www.rfc-editor.org/info/rfc4879>.
- [RFC5226]
- Narten, T. and H. Alvestrand, "Guidelines for Writing an IANA Considerations Section in RFCs", RFC 5226, DOI 10.17487/RFC5226, , <https://www.rfc-editor.org/info/rfc5226>.
- [RFC8554]
- McGrew, D., Curcio, M., and S. Fluhrer, "Leighton-Micali Hash-Based Signatures", RFC 8554, DOI 10.17487/RFC8554, , <https://www.rfc-editor.org/info/rfc8554>.

- [Grover96]
- Grover, L.K., "A fast quantum mechanical algorithm for database search", 28th ACM Symposium on the Theory of Computing p. 212, .
- [Katz16]
- Katz, J., "Analysis of a Proposed Hash-Based Signature Standard", SSR 2016: Security Standardisation Research pp. 261-273, DOI 10.1007/978-3-319-49100-4_12, , <https://doi.org/10.1007/978-3-319-49100-4_12>.
- [NIST_SP_800-208]
- National Institute of Standards and Technology, "Recommendation for Stateful Hash-Based Signature Schemes", NIST SP 800-208, .

This section provides three test cases that can be used to verify or debug an implementation, one for each hash function. This data is formatted with the name of the elements on the left, and the value of the elements on the right, in hexadecimal. 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 Private Key for SHA-256/192¶

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

Test Case 1 Public Key for SHA-256/192¶

-------------------------------------------- HSS public key levels 00000001 -------------------------------------------- LMS type 0000000a # LMS_SHA256_M24_H5 LMOTS type 00000008 # LMOTS_SHA256_N24_W8 I 202122232425262728292a2b2c2d2e2f K 2c571450aed99cfb4f4ac285da148827 96618314508b12d2 -------------------------------------------- --------------------------------------------¶

Test Case 1 Message for SHA-256/192¶

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

Test Case 1 Signature for SHA-256/192¶

-------------------------------------------- HSS signature Nspk 00000000 sig[0]: -------------------------------------------- LMS signature q 00000005 -------------------------------------------- LMOTS signature LMOTS type 00000008 # LMOTS_SHA256_N24_W8 C 0b5040a18c1b5cabcbc85b047402ec62 94a30dd8da8fc3da y[0] e13b9f0875f09361dc77fcc4481ea463 c073716249719193 y[1] 614b835b4694c059f12d3aedd34f3db9 3f3580fb88743b8b y[2] 3d0648c0537b7a50e433d7ea9d6672ff fc5f42770feab4f9 y[3] 8eb3f3b23fd2061e4d0b38f832860ae7 6673ad1a1a52a900 y[4] 5dcf1bfb56fe16ff723627612f9a48f7 90f3c47a67f870b8 y[5] 1e919d99919c8db48168838cece0abfb 683da48b9209868b y[6] e8ec10c63d8bf80d36498dfc205dc45d 0dd870572d6d8f1d y[7] 90177cf5137b8bbf7bcb67a46f86f26c fa5a44cbcaa4e18d y[8] a099a98b0b3f96d5ac8ac375d8da2a7c 248004ba11d7ac77 y[9] 5b9218359cddab4cf8ccc6d54cb7e1b3 5a36ddc9265c0870 y[10] 63d2fc6742a7177876476a324b03295b fed99f2eaf1f3897 y[11] 0583c1b2b616aad0f31cd7a4b1bb0a51 e477e94a01bbb4d6 y[12] f8866e2528a159df3d6ce244d2b6518d 1f0212285a3c2d4a y[13] 927054a1e1620b5b02aab0c8c10ed48a e518ea73cba81fcf y[14] ff88bff461dac51e7ab4ca75f47a6259 d24820b9995792d1 y[15] 39f61ae2a8186ae4e3c9bfe0af2cc717 f424f41aa67f03fa y[16] edb0665115f2067a46843a4cbbd297d5 e83bc1aafc18d1d0 y[17] 3b3d894e8595a6526073f02ab0f08b99 fd9eb208b59ff631 y[18] 7e5545e6f9ad5f9c183abd043d5acd6e b2dd4da3f02dbc31 y[19] 67b468720a4b8b92ddfe7960998bb7a0 ecf2a26a37598299 y[20] 413f7b2aecd39a30cec527b4d9710c44 73639022451f50d0 y[21] 1c0457125da0fa4429c07dad859c846c bbd93ab5b91b01bc y[22] 770b089cfede6f651e86dd7c15989c8b 5321dea9ca608c71 y[23] fd862323072b827cee7a7e28e4e2b999 647233c3456944bb y[24] 7aef9187c96b3f5b79fb98bc76c3574d d06f0e95685e5b3a y[25] ef3a54c4155fe3ad817749629c30adbe 897c4f4454c86c49 -------------------------------------------- LMS type 0000000a # LMS_SHA256_M24_H5 path[0] e9ca10eaa811b22ae07fb195e3590a33 4ea64209942fbae3 path[1] 38d19f152182c807d3c40b189d3fcbea 942f44682439b191 path[2] 332d33ae0b761a2a8f984b56b2ac2fd4 ab08223a69ed1f77 path[3] 19c7aa7e9eee96504b0e60c6bb5c942d 695f0493eb25f80a path[4] 5871cffd131d0e04ffe5065bc7875e82 d34b40b69dd9f3c1¶

Test Case 2 Private Key for SHAKE256/192¶

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

Test Case 2 Public Key for SHAKE256/192¶

--------------------------------------------- HSS public key levels 00000001 -------------------------------------------- LMS type 00000014 # LMS_SHAKE256_N24_H5 LMOTS type 00000010 # LMOTS_SHAKE256_N24_W8 I 505152535455565758595a5b5c5d5e5f K db54a4509901051c01e26d9990e55034 7986da87924ff0b1 -------------------------------------------- --------------------------------------------¶

Test Case 2 Message for SHAKE256/192¶

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

Test Case 2 Signature for SHAKE256/192¶

-------------------------------------------- HSS signature Nspk 00000000 sig[0]: -------------------------------------------- LMS signature q 00000006 -------------------------------------------- LMOTS signature LMOTS type 00000010 # LMOTS_SHAKE256_N24_W8 C 84219da9ce9fffb16edb94527c6d1056 5587db28062deac4 y[0] 208e62fc4fbe9d85deb3c6bd2c01640a ccb387d8a6093d68 y[1] 511234a6a1a50108091c034cb1777e02 b5df466149a66969 y[2] a498e4200c0a0c1bf5d100cdb97d2dd4 0efd3cada278acc5 y[3] a570071a043956112c6deebd1eb3a7b5 6f5f6791515a7b5f y[4] fddb0ec2d9094bfbc889ea15c3c7b9be a953efb75ed648f5 y[5] 35b9acab66a2e9631e426e4e99b733ca a6c55963929b77fe y[6] c54a7e703d8162e736875cb6a455d4a9 015c7a6d8fd5fe75 y[7] e402b47036dc3770f4a1dd0a559cb478 c7fb1726005321be y[8] 9d1ac2de94d731ee4ca79cff454c811f 46d11980909f047b y[9] 2005e84b6e15378446b1ca691efe491e a98acc9d3c0f785c y[10] aba5e2eb3c306811c240ba2280292382 7d582639304a1e97 y[11] 83ba5bc9d69d999a7db8f749770c3c04 a152856dc726d806 y[12] 7921465b61b3f847b13b2635a45379e5 adc6ff58a99b00e6 y[13] 0ac767f7f30175f9f7a140257e218be3 07954b1250c9b419 y[14] 02c4fa7c90d8a592945c66e86a76defc b84500b55598a199 y[15] 0faaa10077c74c94895731585c8f900d e1a1c675bd8b0c18 y[16] 0ebe2b5eb3ef8019ece3e1ea7223eb79 06a2042b6262b4aa y[17] 25c4b8a05f205c8befeef11ceff12825 08d71bc2a8cfa0a9 y[18] 9f73f3e3a74bb4b3c0d8ca2abd0e1c2c 17dafe18b4ee2298 y[19] e87bcfb1305b3c069e6d385569a4067e d547486dd1a50d6f y[20] 4a58aab96e2fa883a9a39e1bd45541ee e94efc32faa9a94b y[21] e66dc8538b2dab05aee5efa6b3b2efb3 fd020fe789477a93 y[22] afff9a3e636dbba864a5bffa3e28d13d 49bb597d94865bde y[23] 88c4627f206ab2b465084d6b780666e9 52f8710efd748bd0 y[24] f1ae8f1035087f5028f14affcc5fffe3 32121ae4f87ac5f1 y[25] eac9062608c7d87708f1723f38b23237 a4edf4b49a5cd3d7 -------------------------------------------- LMS type 00000014 # LMS_SHAKE256_N24_H5 path[0] dd4bdc8f928fb526f6fb7cdb944a7eba a7fb05d995b5721a path[1] 27096a5007d82f79d063acd434a04e97 f61552f7f81a9317 path[2] b4ec7c87a5ed10c881928fc6ebce6dfc e9daae9cc9dba690 path[3] 7ca9a9dd5f9f573704d5e6cf22a43b04 e64c1ffc7e1c442e path[4] cb495ba265f465c56291a902e62a461f 6dfda232457fad14¶

Test Case 3 Private Key for SHAKE256/256¶

-------------------------------------------- (note: procedure in Appendix A of [RFC8554] is used) SEED 606162636465666768696a6b6c6d6e6f 707172737475767778797a7b7c7d7e7f I 808182838485868788898a8b8c8d8e8f -------------------------------------------- --------------------------------------------¶

Test Case 3 Public Key for SHAKE256/256¶

-------------------------------------------- HSS public key levels 00000001 -------------------------------------------- LMS type 0000000f # LMS_SHAKE256_N32_H5 LMOTS type 0000000c # LMOTS_SHAKE256_N32_W8 I 808182838485868788898a8b8c8d8e8f K 9bb7faee411cae806c16a466c3191a8b 65d0ac31932bbf0c2d07c7a4a36379fe -------------------------------------------- --------------------------------------------¶

Test Case 3 Message for SHAKE256/256¶

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

Test Case 3 Signature for SHAKE256/256¶

-------------------------------------------- HSS signature Nspk 00000000 sig[0]: -------------------------------------------- LMS signature q 00000007 -------------------------------------------- LMOTS signature LMOTS type 0000000c # LMOTS_SHAKE256_N32_W8 C b82709f0f00e83759190996233d1ee4f 4ec50534473c02ffa145e8ca2874e32b y[0] 16b228118c62b96c9c77678b33183730 debaade8fe607f05c6697bc971519a34 y[1] 1d69c00129680b67e75b3bd7d8aa5c8b 71f02669d177a2a0eea896dcd1660f16 y[2] 864b302ff321f9c4b8354408d0676050 4f768ebd4e545a9b0ac058c575078e6c y[3] 1403160fb45450d61a9c8c81f6bd69bd fa26a16e12a265baf79e9e233eb71af6 y[4] 34ecc66dc88e10c6e0142942d4843f70 a0242727bc5a2aabf7b0ec12a99090d8 y[5] caeef21303f8ac58b9f200371dc9e41a b956e1a3efed9d4bbb38975b46c28d5f y[6] 5b3ed19d847bd0a737177263cbc1a226 2d40e80815ee149b6cce2714384c9b7f y[7] ceb3bbcbd25228dda8306536376f8793 ecadd6020265dab9075f64c773ef97d0 y[8] 7352919995b74404cc69a6f3b469445c 9286a6b2c9f6dc839be76618f053de76 y[9] 3da3571ef70f805c9cc54b8e501a98b9 8c70785eeb61737eced78b0e380ded4f y[10] 769a9d422786def59700eef3278017ba bbe5f9063b468ae0dd61d94f9f99d5cc y[11] 36fbec4178d2bda3ad31e1644a2bcce2 08d72d50a7637851aa908b94dc437612 y[12] 0d5beab0fb805e1945c41834dd6085e6 db1a3aa78fcb59f62bde68236a10618c y[13] ff123abe64dae8dabb2e84ca705309c2 ab986d4f8326ba0642272cb3904eb96f y[14] 6f5e3bb8813997881b6a33cac0714e4b 5e7a882ad87e141931f97d612b84e903 y[15] e773139ae377f5ba19ac86198d485fca 97742568f6ff758120a89bf19059b8a6 y[16] bfe2d86b12778164436ab2659ba86676 7fcc435584125fb7924201ee67b535da y[17] f72c5cb31f5a0b1d926324c26e67d4c3 836e301aa09bae8fb3f91f1622b1818c y[18] cf440f52ca9b5b9b99aba8a6754aae2b 967c4954fa85298ad9b1e74f27a46127 y[19] c36131c8991f0cc2ba57a15d35c91cf8 bc48e8e20d625af4e85d8f9402ec44af y[20] bd4792b924b839332a64788a7701a300 94b9ec4b9f4b648f168bf457fbb3c959 y[21] 4fa87920b645e42aa2fecc9e21e000ca 7d3ff914e15c40a8bc533129a7fd3952 y[22] 9376430f355aaf96a0a13d13f2419141 b3cc25843e8c90d0e551a355dd90ad77 y[23] 0ea7255214ce11238605de2f000d2001 04d0c3a3e35ae64ea10a3eff37ac7e95 y[24] 49217cdf52f307172e2f6c7a2a4543e1 4314036525b1ad53eeaddf0e24b1f369 y[25] 14ed22483f2889f61e62b6fb78f5645b dbb02c9e5bf97db7a0004e87c2a55399 y[26] b61958786c97bd52fa199c27f6bb4d68 c4907933562755bfec5d4fb52f06c289 y[27] d6e852cf6bc773ffd4c07ee2d6cc55f5 7edcfbc8e8692a49ad47a121fe3c1b16 y[28] cab1cc285faf6793ffad7a8c341a49c5 d2dce7069e464cb90a00b2903648b23c y[29] 81a68e21d748a7e7b1df8a593f3894b2 477e8316947ca725d141135202a9442e y[30] 1db33bbd390d2c04401c39b253b78ce2 97b0e14755e46ec08a146d279c67af70 y[31] de256890804d83d6ec5ca3286f1fca9c 72abf6ef868e7f6eb0fddda1b040ecec y[32] 9bbc69e2fd8618e9db3bdb0af13dda06 c6617e95afa522d6a2552de15324d991 y[33] 19f55e9af11ae3d5614b564c642dbfec 6c644198ce80d2433ac8ee738f9d825e -------------------------------------------- LMS type 0000000f # LMS_SHAKE256_N32_H5 path[0] 71d585a35c3a908379f4072d070311db 5d65b242b714bc5a756ba5e228abfa0d path[1] 1329978a05d5e815cf4d74c1e547ec4a a3ca956ae927df8b29fb9fab3917a7a4 path[2] ae61ba57e5342e9db12caf6f6dbc5253 de5268d4b0c4ce4ebe6852f012b162fc path[3] 1c12b9ffc3bcb1d3ac8589777655e22c d9b99ff1e4346fd0efeaa1da044692e7 path[4] ad6bfc337db69849e54411df8920c228 a2b7762c11e4b1c49efb74486d3931ea¶