Internet-Draft ML-DSA in Certificates April 2025
Massimo, et al. Expires 12 October 2025 [Page]
Workgroup:
LAMPS WG
Internet-Draft:
draft-ietf-lamps-dilithium-certificates-latest
Published:
Intended Status:
Standards Track
Expires:
Authors:
J. Massimo
AWS
P. Kampanakis
AWS
S. Turner
sn3rd
B. E. Westerbaan
Cloudflare

Internet X.509 Public Key Infrastructure: Algorithm Identifiers for ML-DSA

Abstract

Digital signatures are used within X.509 certificates, Certificate Revocation Lists (CRLs), and to sign messages. This document describes the conventions for using FIPS 204, the Module-Lattice-Based Digital Signature Algorithm (ML-DSA) in Internet X.509 certificates and certificate revocation lists. The conventions for the associated signatures, subject public keys, and private key are also described.

About This Document

This note is to be removed before publishing as an RFC.

The latest revision of this draft can be found at https://lamps-wg.github.io/dilithium-certificates/#go.draft-ietf-lamps-dilithium-certificates.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draft-ietf-lamps-dilithium-certificates/.

Discussion of this document takes place on the Limited Additional Mechanisms for PKIX and SMIME (lamps) Working Group mailing list (mailto:spasm@ietf.org), which is archived at https://mailarchive.ietf.org/arch/browse/spasm/. Subscribe at https://www.ietf.org/mailman/listinfo/spasm/.

Source for this draft and an issue tracker can be found at https://github.com/lamps-wg/dilithium-certificates.

Status of This Memo

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

Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at 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 12 October 2025.

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Table of Contents

1. Introduction

The Module-Lattice-Based Digital Signature Algorithm (ML-DSA) is a quantum-resistant digital signature scheme standardized by the US National Institute of Standards and Technology (NIST) PQC project [NIST-PQC] in [FIPS204]. This document specifies the use of the ML-DSA in Public Key Infrastructure X.509 (PKIX) certificates and Certificate Revocation Lists (CRLs) at three security levels: ML-DSA-44, ML-DSA-65, and ML-DSA-87.

[FIPS204] defines two variants of ML-DSA: a pure and a prehash variant. Only the former is specified in this document. See Section 10.1 for the rationale. The pure variant of ML-DSA supports the typical prehash flow, see Appendix D. In short: one cryptographic module can compute the hash mu on line 6 of algorithm 7 of [FIPS204] and pass it to a second module to finish the signature. The first module only needs access to the full message and the public key, whereas the second module only needs access to hash mu and the private key.

Prior to standardisation, ML-DSA was known as Dilithium. ML-DSA and Dilithium are not compatible.

1.1. Requirements Language

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. Identifiers

The AlgorithmIdentifier type is defined in [RFC5912] as follows:

    AlgorithmIdentifier{ALGORITHM-TYPE, ALGORITHM-TYPE:AlgorithmSet} ::=
      SEQUENCE {
        algorithm   ALGORITHM-TYPE.id({AlgorithmSet}),
        parameters  ALGORITHM-TYPE.
                      Params({AlgorithmSet}{@algorithm}) OPTIONAL
     }

The fields in AlgorithmIdentifier have the following meanings:

The OIDs are:

   id-ml-dsa-44 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2)
            country(16) us(840) organization(1) gov(101) csor(3)
            nistAlgorithm(4) sigAlgs(3) id-ml-dsa-44(17) }

   id-ml-dsa-65 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2)
            country(16) us(840) organization(1) gov(101) csor(3)
            nistAlgorithm(4) sigAlgs(3) id-ml-dsa-65(18) }

   id-ml-dsa-87 OBJECT IDENTIFIER ::= { joint-iso-itu-t(2)
            country(16) us(840) organization(1) gov(101) csor(3)
            nistAlgorithm(4) sigAlgs(3) id-ml-dsa-87(19) }

The contents of the parameters component for each algorithm MUST be absent. The ctx value used in the ML-DSA signing and verification [FIPS204] of ML-DSA signatures defined in this specification (X.509 certificates, CRLs) is the empty string.

3. ML-DSA Signatures in PKIX

ML-DSA is a digital signature scheme built upon the Fiat-Shamir-with-aborts framework [Fiat-Shamir]. The security is based upon the hardness of lattice problems over module lattices [Dilithium]. ML-DSA provides three parameter sets for the NIST PQC security categories 2, 3 and 5.

Signatures are used in a number of different ASN.1 structures. As shown in the ASN.1 representation from [RFC5280] below, in an X.509 certificate, a signature is encoded with an algorithm identifier in the signatureAlgorithm attribute and a signatureValue attribute that contains the actual signature.

  Certificate  ::=  SIGNED{ TBSCertificate }

  SIGNED{ToBeSigned} ::= SEQUENCE {
     toBeSigned           ToBeSigned,
     algorithmIdentifier  SEQUENCE {
         algorithm        SIGNATURE-ALGORITHM.
                            &id({SignatureAlgorithms}),
         parameters       SIGNATURE-ALGORITHM.
                            &Params({SignatureAlgorithms}
                              {@algorithmIdentifier.algorithm})
                                OPTIONAL
     },
     signature BIT STRING (CONTAINING SIGNATURE-ALGORITHM.&Value(
                              {SignatureAlgorithms}
                              {@algorithmIdentifier.algorithm}))
  }

Signatures are also used in the CRL list ASN.1 representation from [RFC5280] below. In a X.509 CRL, a signature is encoded with an algorithm identifier in the signatureAlgorithm attribute and a signatureValue attribute that contains the actual signature.

   CertificateList  ::=  SIGNED{ TBSCertList }

The following SIGNATURE-ALGORITHM ASN.1 classes are for ML-DSA-44, ML-DSA-65, and ML-DSA-87:

  sa-ml-dsa-44 SIGNATURE-ALGORITHM ::= {
    IDENTIFIER id-ml-dsa-44
    PARAMS ARE absent
    PUBLIC-KEYS { pk-ml-dsa-44 }
    SMIME-CAPS { IDENTIFIED BY id-ml-dsa-44 }
    }

  sa-ml-dsa-65 SIGNATURE-ALGORITHM ::= {
    IDENTIFIER id-ml-dsa-65
    PARAMS ARE absent
    PUBLIC-KEYS { pk-ml-dsa-65 }
    SMIME-CAPS { IDENTIFIED BY id-ml-dsa-65 }
    }

  sa-ml-dsa-87 SIGNATURE-ALGORITHM ::= {
    IDENTIFIER id-ml-dsa-87
    PARAMS ARE absent
    PUBLIC-KEYS { pk-ml-dsa-87 }
    SMIME-CAPS { IDENTIFIED BY id-ml-dsa-87 }
    }

The identifiers defined in Section 2 can be used as the AlgorithmIdentifier in the signatureAlgorithm field in the sequence Certificate/CertificateList and the signature field in the sequence TBSCertificate/TBSCertList in certificates and CRLs, respectively, [RFC5280]. The parameters of these signature algorithms MUST be absent, as explained in Section 2. That is, the AlgorithmIdentifier SHALL be a SEQUENCE of one component, the OID id-ml-dsa-*.

The signatureValue field contains the corresponding ML-DSA signature computed upon the ASN.1 DER encoded tbsCertificate/tbsCertList [RFC5280].

Conforming Certification Authority (CA) implementations MUST specify the algorithms explicitly by using the OIDs specified in Section 2 when encoding ML-DSA signatures in certificates and CRLs. Conforming client implementations that process certificates and CRLs using ML-DSA MUST recognize the corresponding OIDs. Encoding rules for ML-DSA signature values are specified Section 2.

4. ML-DSA Public Keys in PKIX

In the X.509 certificate, the subjectPublicKeyInfo field has the SubjectPublicKeyInfo type, which has the following ASN.1 syntax:

  SubjectPublicKeyInfo {PUBLIC-KEY: IOSet} ::= SEQUENCE {
      algorithm        AlgorithmIdentifier {PUBLIC-KEY, {IOSet}},
      subjectPublicKey BIT STRING
  }

The fields in SubjectPublicKeyInfo have the following meaning:

The PUBLIC-KEY ASN.1 types for ML-DSA are defined here:

  pk-ml-dsa-44 PUBLIC-KEY ::= {
    IDENTIFIER id-ml-dsa-44
    -- KEY no ASN.1 wrapping --
    CERT-KEY-USAGE
      { digitalSignature, nonRepudiation, keyCertSign, cRLSign }
    PRIVATE-KEY ML-DSA-44-PrivateKey }  -- defined in Section 6

  pk-ml-dsa-65 PUBLIC-KEY ::= {
    IDENTIFIER id-ml-dsa-65
    -- KEY no ASN.1 wrapping --
    CERT-KEY-USAGE
      { digitalSignature, nonRepudiation, keyCertSign, cRLSign }
    PRIVATE-KEY ML-DSA-65-PrivateKey }  -- defined in Section 6

  pk-ml-dsa-87 PUBLIC-KEY ::= {
    IDENTIFIER id-ml-dsa-87
    -- KEY no ASN.1 wrapping --
    CERT-KEY-USAGE
      { digitalSignature, nonRepudiation, keyCertSign, cRLSign }
    PRIVATE-KEY ML-DSA-87-PrivateKey }  -- defined in Section 6

  ML-DSA-44-PublicKey ::= OCTET STRING (SIZE (1312))

  ML-DSA-65-PublicKey ::= OCTET STRING (SIZE (1952))

  ML-DSA-87-PublicKey ::= OCTET STRING (SIZE (2592))

Algorithm 22 in Section 7.2 of [FIPS204] defines the raw byte string encoding of an ML-DSA public key. When used in a SubjectPublicKeyInfo type, the subjectPublicKey BIT STRING contains the raw byte string encoding of the public key.

When an ML-DSA public key appears outside of a SubjectPublicKeyInfo type in an environment that uses ASN.1 encoding, it can be encoded as an OCTET STRING by using the ML-DSA-44-PublicKey, ML-DSA-65-PublicKey, and ML-DSA-87-PublicKey types corresponding to the correct key size.

[RFC5958] describes the Asymmetric Key Package's OneAsymmetricKey type for encoding asymmetric keypairs. When an ML-DSA private key or keypair is encoded as a OneAsymmetricKey, it follows the description in Section 6.

When the ML-DSA private key appears outside of an Asymmetric Key Package in an environment that uses ASN.1 encoding, it can be encoded using one of the the ML-DSA-PrivateKey CHOICE formats defined in Section 6. The seed format is RECOMMENDED as it efficiently stores both the private and public key.

Appendix C contains example ML-DSA public keys encoded using the textual encoding defined in [RFC7468].

5. Key Usage Bits

The intended application for the key is indicated in the keyUsage certificate extension; see Section 4.2.1.3 of [RFC5280]. If the keyUsage extension is present in a certificate that indicates id-ml-dsa-* in the SubjectPublicKeyInfo, then at least one of following MUST be present:

  digitalSignature; or
  nonRepudiation; or
  keyCertSign; or
  cRLSign.

If the keyUsage extension is present in a certificate that indicates id-ml-dsa-* in the SubjectPublicKeyInfo, then the following MUST NOT be present:

   keyEncipherment; or
   dataEncipherment; or
   keyAgreement; or
   encipherOnly; or
   decipherOnly.

Requirements about the keyUsage extension bits defined in [RFC5280] still apply.

6. Private Key Format

[FIPS204] specifies two formats for an ML-DSA private key: a 32-octet seed (xi) and an (expanded) private key. The expanded private key (and public key) is computed from the seed using ML-DSA.KeyGen_internal(xi) (algorithm 6).

"Asymmetric Key Packages" [RFC5958] describes how to encode a private key in a structure that both identifies what algorithm the private key is for and allows for the public key and additional attributes about the key to be included as well. For illustration, the ASN.1 structure OneAsymmetricKey is replicated below.

  OneAsymmetricKey ::= SEQUENCE {
    version                  Version,
    privateKeyAlgorithm      SEQUENCE {
    algorithm                PUBLIC-KEY.&id({PublicKeySet}),
    parameters               PUBLIC-KEY.&Params({PublicKeySet}
                               {@privateKeyAlgorithm.algorithm})
                                  OPTIONAL}
    privateKey               OCTET STRING (CONTAINING
                               PUBLIC-KEY.&PrivateKey({PublicKeySet}
                                 {@privateKeyAlgorithm.algorithm})),
    attributes           [0] Attributes OPTIONAL,
    ...,
    [[2: publicKey       [1] BIT STRING (CONTAINING
                               PUBLIC-KEY.&Params({PublicKeySet}
                                 {@privateKeyAlgorithm.algorithm})
                                 OPTIONAL,
    ...
  }

For ML-DSA private keys, the privateKey field in OneAsymmetricKey contains one of the following DER-encoded CHOICE structures. The seed format is a fixed 32 byte OCTET STRING (34 bytes total with the 0x8020 tag and length) for all security levels, while the expandedKey and both formats vary in size by security level:

ML-DSA-44-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (2560)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (2560))
      }
  }

ML-DSA-65-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (4032)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (4032))
      }
  }

ML-DSA-87-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (4896)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (4896))
      }
  }

The CHOICE allows three representations of the private key:

  1. The seed format (tag [0]) contains just the 32-byte seed value (xi) from which both the expanded private key and public key can be derived using ML-DSA.KeyGen_internal(xi).

  2. The expandedKey format contains the expanded private key that was derived from the seed.

  3. The both format contains both the seed and expanded private key, allowing for for interoperability; some may want to use and retain the seed and others may only support expanded private keys.

When encoding an ML-DSA private key in a OneAsymmetricKey object, any of these three formats may be used, though the seed format is RECOMMENDED for storage efficiency.

The privateKeyAlgorithm field uses the AlgorithmIdentifier structure with the appropriate OID as defined in Section 2. If present, the publicKey field will hold the encoded public key as defined in Section 4.

NOTE: While the private key can be stored in multiple formats, the seed-only format is RECOMMENDED as it is the most compact representation. Both the expanded private key and the public key can be deterministically derived from the seed using ML-DSA.KeyGen_internal(xi). Alternatively, the public key can be generated from the private key. While the publicKey field and expandedKey format are technically redundant when using the seed-only format, they MAY be included to enable keypair consistency checks during import operations.

When parsing the private key, the ASN.1 tag explicitly indicates which variant of CHOICE is present. Implementations should use tag UNIVERSAL IMPLICIT [0] (raw value 0x80) for seed, OCTET STRING (0x04) for expandedKey, and SEQUENCE (0x30) for both to parse the private key, rather than any other heuristic like length of the enclosing OCTET STRING.

Appendix C contains example ML-DSA private keys encoded using the textual encoding defined in [RFC7468].

7. IANA Considerations

For the ASN.1 module in Appendix A, IANA is requested to assign an object identifier (OID) for the module identifier (TBD1) with a Description of "id-mod-x509-ml-dsa-2025". The OID for the module should be allocated in the "SMI Security for PKIX Module Identifier" registry (1.3.6.1.5.5.7.0).

8. Implementation Considerations

An ML-DSA.KeyGen seed (xi) represents the RECOMMENDED format for storing and transmitting ML-DSA private keys. This format is explicitly permitted by [FIPS204] as an acceptable representation of a keypair. In particular, generating the seed in one cryptographic module and then importing or exporting it into another cryptographic module is allowed. The internal key generation function ML-DSA.KeyGen_internal(xi) can be accessed for this purpose.

Note also that unlike other private key compression methods in other algorithms, expanding a private key from a seed is a one-way function, meaning that once a full key is expanded from seed and the seed discarded, the seed cannot be re-created even if the full expanded private key is available. For this reason it is RECOMMENDED that implementations retain and export the seed, even when also exporting the expanded private key. ML-DSA seed extraction can be implemented by including the random seed xi generated at line 1 of Algorithm 1 ML-DSA.KeyGen in the returned output.

9. Private Key Consistency Testing

When receiving a private key that contains both the seed and the expandedKey, the recipient SHOULD perform a seed consistency check to ensure that the sender properly generated the private key. Recipients that do not perform this seed consistency check avoid keygen and compare operations, but are unable to ensure that the seed and expandedKey match.

If the check is done and the seed and the expandedKey are not consistent, the recipient MUST reject the private key as malformed.

The seed consistency check consists of regenerating the expanded form from the seed via ML-DSA.KeyGen_internal and ensuring it is bytewise equal to the value presented in the private key.

Appendix C.4 includes some examples of inconsistent seeds and expanded private keys.

10. Security Considerations

The Security Considerations section of [RFC5280] applies to this specification as well.

The digital signature scheme defined within this document are modeled under strongly existentially unforgeable under chosen message attack (SUF-CMA). For the purpose of estimating security strength, it has been assumed that the attacker has access to signatures for no more than 2^{64} chosen messages.

ML-DSA offers both deterministic and randomized signing. By default ML-DSA signatures are non-deterministic. The private random seed (rho') for the signature is pseudorandomly derived from the signer's private key, the message, and a 256-bit string, rnd - where rnd should be generated by an approved RBG. In the deterministic version, rng is instead a 256-bit constant string. The source of randomness in the randomized mode has been "hedged" against sources of poor entropy, by including the signers private key and message into the derivation. The primary purpose of rnd is to facilitate countermeasures to side-channel attacks and fault attacks on deterministic signatures.

In the design of ML-DSA, care has been taken to make side-channel resilience easier to achieve. For instance, ML-DSA does not depend on Gaussian sampling. Implementations must still take great care not to leak information via various side channels. While deliberate design decisions such as these can help to deliver a greater ease of secure implementation - particularly against side-channel attacks - it does not necessarily provide resistance to more powerful attacks such as differential power analysis. Some amount of side-channel leakage has been demonstrated in parts of the signing algorithm (specifically the bit-unpacking function), from which a demonstration of key recovery has been made over a large sample of signatures. Masking countermeasures exist for ML-DSA, but come with a performance overhead.

A fundamental security property also associated with digital signatures is non-repudiation. Non-repudiation refers to the assurance that the owner of a signature key pair that was capable of generating an existing signature corresponding to certain data cannot convincingly deny having signed the data. The digital signature scheme ML-DSA possess three security properties beyond unforgeability, that are associated with non-repudiation. These are exclusive ownership, message-bound signatures, and non-resignability. These properties are based tightly on the assumed collision resistance of the hash function used (in this case SHAKE-256).

Exclusive ownership is a property in which a signature sigma uniquely determines the public key and message for which it is valid. Message-bound signatures is the property that a valid signature uniquely determines the message for which it is valid, but not necessarily the public key. Non-resignability is the property in which one cannot produce a valid signature under another key given a signature sigma for some unknown message m. These properties are not provided by classical signature schemes such as DSA or ECDSA, and have led to a variety of attacks such as Duplicate-Signature Key Selection (DSKS) attacks, and attacks on the protocols for secure routing. A full discussion of these properties in ML-DSA can be found at [CDFFJ21].

These properties are dependent, in part, on unambiguous public key serialization. It for this reason the public key structure defined in Section 4 is intentionally encoded as a single OCTET STRING.

10.1. Rationale for disallowing HashML-DSA

The HashML-DSA mode defined in Section 5.4 of [FIPS204] MUST NOT be used; in other words, public keys identified by id-hash-ml-dsa-44-with-sha512, id-hash-ml-dsa-65-with-sha512, and id-hash-ml-dsa-87-with-sha512 MUST NOT be in X.509 certificates used for CRLs, OCSP, certificate issuance and related PKIX protocols.

This restriction is for both implementation and security reasons.

The implementation reason for disallowing HashML-DSA stems from the fact that ML-DSA and HashML-DSA are incompatible algorithms that require different Verify() routines. This forwards to the protocol the complexity of informing the client whether to use ML-DSA.Verify() or HashML-DSA.Verify() along with the hash algorithm to use. Additionally, since the same OIDs are used to identify the ML-DSA public keys and ML-DSA signature algorithms, an implementation would need to commit a given public key to be either of type ML-DSA or HashML-DSA at the time of certificate creation. This is anticipated to cause operational issues in contexts where the operator does not know at key generation time whether the key will need to produce pure or pre-hashed signatures. ExternalMu-ML-DSA avoids all of these operational concerns by virtue of having keys and signatures that are indistinguishable from ML-DSA (i.e., ML-DSA and ExternalMu-ML-DSA are mathematically equivalent algorithms). The difference between ML-DSA and ExternalMu-ML-DSA is merely an internal implementation detail of the signer and has no impact on the verifier or network protocol.

The security reason for disallowing HashML-DSA is that the design of the ML-DSA algorithm provides enhanced resistance against signature collision attacks, compared with conventional RSA or ECDSA signature algorithms. Specifically, ML-DSA binds the hash of the public key tr to the message to-be-signed prior to hashing, as described in line 6 of Algorithm 7 of [FIPS204]. In practice, this provides binding to the intended verification public key, preventing some attacks that would otherwise allow a signature to be successfully verified against a non-intended public key. Also, this unlikely, theoretical binding means that in the unlikely discovery of a collision attack against SHA-3, an attacker would have to perform a public-key-specific collision search in order to find message pairs such that H(tr || m1) = H(tr || m2) since a direct hash collision H(m1) = H(m2) will not suffice. HashML-DSA removes both of these enhanced security properties.

11. References

11.1. Normative References

[FIPS204]
National Institute of Standards and Technology (NIST), "Module-Lattice-based Digital Signature Standard", FIPS PUB 204, , <https://csrc.nist.gov/projects/post-quantum-cryptography>.
[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/rfc/rfc2119>.
[RFC5280]
Cooper, D., Santesson, S., Farrell, S., Boeyen, S., Housley, R., and W. Polk, "Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile", RFC 5280, DOI 10.17487/RFC5280, , <https://www.rfc-editor.org/rfc/rfc5280>.
[RFC5912]
Hoffman, P. and J. Schaad, "New ASN.1 Modules for the Public Key Infrastructure Using X.509 (PKIX)", RFC 5912, DOI 10.17487/RFC5912, , <https://www.rfc-editor.org/rfc/rfc5912>.
[RFC5958]
Turner, S., "Asymmetric Key Packages", RFC 5958, DOI 10.17487/RFC5958, , <https://www.rfc-editor.org/rfc/rfc5958>.
[RFC8174]
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.
[X680]
ITU-T, "Information Technology -- Abstract Syntax Notation One (ASN.1): Specification of basic notation", ITU-T Recommendation X.680, ISO/IEC 8824-1:2021, , <https://www.itu.int/rec/T-REC-X.680>.
[X690]
ITU-T, "Information Technology -- Abstract Syntax Notation One (ASN.1): ASN.1 encoding rules: Specification of Basic Encoding Rules (BER), Canonical Encoding Rules (CER) and Distinguished Encoding Rules (DER)", ITU-T Recommendation X.690, ISO/IEC 8825-1:2021, , <https://www.itu.int/rec/T-REC-X.690>.

11.2. Informative References

[CDFFJ21]
Cremers, C., Düzlü, S., Fiedler, R., Fischlin, M., and C. Janson, "BUFFing signature schemes beyond unforgeability and the case of post-quantum signatures", In Proceedings of the 42nd IEEE Symposium on Security and Privacy , , <https://eprint.iacr.org/2020/1525.pdf>.
[Dilithium]
Bai, S., Ducas, L., Lepoint, T., Lyubashevsky, V., Schwabe, P., Seiler, G., and D. Stehlé, "CRYSTALS-Dilithium Algorithm Specifications and Supporting Documentation", , <https://pq-crystals.org/dilithium/data/dilithium-specification-round3-20210208.pdf>.
[Fiat-Shamir]
Lyubashevsky, V., "Fiat-Shamir with aborts: Applications to lattice and factoring-based signatures", International Conference on the Theory and Application of Cryptology and Information Security , , <https://www.iacr.org/archive/asiacrypt2009/59120596/59120596.pdf>.
[NIST-PQC]
National Institute of Standards and Technology (NIST), "Post-Quantum Cryptography Project", , <https://csrc.nist.gov/Projects/post-quantum-cryptography>.
[RFC3647]
Chokhani, S., Ford, W., Sabett, R., Merrill, C., and S. Wu, "Internet X.509 Public Key Infrastructure Certificate Policy and Certification Practices Framework", RFC 3647, DOI 10.17487/RFC3647, , <https://www.rfc-editor.org/rfc/rfc3647>.
[RFC7468]
Josefsson, S. and S. Leonard, "Textual Encodings of PKIX, PKCS, and CMS Structures", RFC 7468, DOI 10.17487/RFC7468, , <https://www.rfc-editor.org/rfc/rfc7468>.

Appendix A. ASN.1 Module

This appendix includes the ASN.1 module [X680] for the ML-DSA. Note that as per [RFC5280], certificates use the Distinguished Encoding Rules; see [X690]. This module imports objects from [RFC5912].

<CODE BEGINS>
X509-ML-DSA-2025
{ iso(1) identified-organization(3) dod(6)
  internet(1) security(5) mechanisms(5) pkix(7) id-mod(0)
  id-mod-x509-ml-dsa-2025(TBD1) }

DEFINITIONS IMPLICIT TAGS ::= BEGIN

EXPORTS ALL;

IMPORTS

 PUBLIC-KEY, SIGNATURE-ALGORITHM
   FROM AlgorithmInformation-2009  -- [RFC 5912]
     { iso(1) identified-organization(3) dod(6) internet(1)
       security(5) mechanisms(5) pkix(7) id-mod(0)
       id-mod-algorithmInformation-02(58) } ;

--
-- ML-DSA Identifiers
--

nistAlgorithms OBJECT IDENTIFIER ::= { joint-iso-itu-t(2)
  country(16) us(840) organization(1) gov(101) csor(3)
  nistAlgorithms(4) }

sigAlgs OBJECT IDENTIFIER ::= { nistAlgorithms 3 }

id-ml-dsa-44 OBJECT IDENTIFIER ::= { sigAlgs 17 }

id-ml-dsa-65 OBJECT IDENTIFIER ::= { sigAlgs 18 }

id-ml-dsa-87 OBJECT IDENTIFIER ::= { sigAlgs 19 }

--
-- Public Key Algorithms
--

PublicKeys PUBLIC-KEY ::= {
  -- This expands PublicKeys from [RFC 5912]
  pk-ml-dsa-44 |
  pk-ml-dsa-65 |
  pk-ml-dsa-87,
  ...
}

--
-- ML-DSA Public Keys
--

pk-ml-dsa-44 PUBLIC-KEY ::= {
  IDENTIFIER id-ml-dsa-44
  -- KEY no ASN.1 wrapping; 1312 octets --
  PARAMS ARE absent
  CERT-KEY-USAGE { digitalSignature, nonRepudiation,
                   keyCertSign, cRLSign }
  PRIVATE-KEY ML-DSA-44-PrivateKey
  }

ML-DSA-44-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (2560)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (2560))
      }
  }

pk-ml-dsa-65 PUBLIC-KEY ::= {
  IDENTIFIER id-ml-dsa-65
  -- KEY no ASN.1 wrapping; 1952 octets --
  PARAMS ARE absent
  CERT-KEY-USAGE { digitalSignature, nonRepudiation,
                   keyCertSign, cRLSign }
  PRIVATE-KEY ML-DSA-65-PrivateKey
  }

ML-DSA-65-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (4032)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (4032))
      }
  }

pk-ml-dsa-87 PUBLIC-KEY ::= {
  IDENTIFIER id-ml-dsa-87
  -- KEY no ASN.1 wrapping; 2592 octets --
  PARAMS ARE absent
  CERT-KEY-USAGE { digitalSignature, nonRepudiation,
                    keyCertSign, cRLSign }
  PRIVATE-KEY ML-DSA-87-PrivateKey
  }

ML-DSA-87-PrivateKey ::= CHOICE {
  seed [0] OCTET STRING (SIZE (32)),
  expandedKey OCTET STRING (SIZE (4896)),
  both SEQUENCE {
      seed OCTET STRING (SIZE (32)),
      expandedKey OCTET STRING (SIZE (4896))
      }
  }

ML-DSA-44-PublicKey ::= OCTET STRING (SIZE (1312))

ML-DSA-65-PublicKey ::= OCTET STRING (SIZE (1952))

ML-DSA-87-PublicKey ::= OCTET STRING (SIZE (2602))

--
-- Signature Algorithms
--

SignatureAlgorithms SIGNATURE-ALGORITHM ::= {
  -- This expands SignatureAlgorithms from [RFC 5912]
  sa-ml-dsa-44 |
  sa-ml-dsa-65 |
  sa-ml-dsa-87,
  ... }

--
-- ML-DSA Signature Algorithm Identifiers
--

sa-ml-dsa-44 SIGNATURE-ALGORITHM ::= {
  IDENTIFIER id-ml-dsa-44
  PARAMS ARE absent
  PUBLIC-KEYS { pk-ml-dsa-44 }
  SMIME-CAPS { IDENTIFIED BY id-ml-dsa-44 }
  }

sa-ml-dsa-65 SIGNATURE-ALGORITHM ::= {
  IDENTIFIER id-ml-dsa-65
  PARAMS ARE absent
  PUBLIC-KEYS { pk-ml-dsa-65 }
  SMIME-CAPS { IDENTIFIED BY id-ml-dsa-65 }
  }

sa-ml-dsa-87 SIGNATURE-ALGORITHM ::= {
  IDENTIFIER id-ml-dsa-87
  PARAMS ARE absent
  PUBLIC-KEYS { pk-ml-dsa-87 }
  SMIME-CAPS { IDENTIFIED BY id-ml-dsa-87 }
  }

END

<CODE ENDS>

Appendix B. Security Strengths

Instead of defining the strength of a quantum algorithm in a traditional manner using the imprecise notion of bits of security, NIST has instead elected to define security levels by picking a reference scheme, which NIST expects to offer notable levels of resistance to both quantum and classical attack. To wit, an algorithm that achieves NIST PQC security level 1 must require computational resources to break the relevant security property, which are greater than those required for a brute-force key search on AES-128. Levels 3 and 5 use AES-192 and AES-256 as reference respectively. Levels 2 and 4 use collision search for SHA-256 and SHA-384 as reference.

The parameter sets defined for NIST security levels 2, 3 and 5 are listed in the Figure 1, along with the resulting signature size, public key, and private key sizes in bytes. Note that these are the sizes of the raw keys, not including ASN.1 encoding overhead from OneAsymmetricKey and SubjectPublicKeyInfo wrappers. Private key sizes are shown for both the seed format and expanded format.

|=======+=======+=====+========+========+==========+==========|
| Level | (k,l) | eta |  Sig.  | Public | Private  | Private  |
|       |       |     |  (B)   | Key(B) | Seed(B)  | Expand(B)|
|=======+=======+=====+========+========+==========+==========|
|   2   | (4,4) |  2  |  2420  |  1312  |    32    |   2560   |
|   3   | (6,5) |  4  |  3309  |  1952  |    32    |   4032   |
|   5   | (8,7) |  2  |  4627  |  2592  |    32    |   4896   |
|=======+=======+=====+========+========+==========+==========|
Figure 1: ML-DSA Parameters

Appendix C. Examples

This appendix contains examples of ML-DSA private keys, public keys, certificates, and inconsistent seed and expanded private keys.

C.1. Example Private Keys

The following examples show ML-DSA private keys in different formats, all derived from the same seed 000102...1e1f. For each security level, we show the seed-only format (using a context-specific [0] primitive tag with an implicit encoding of OCTET STRING), the expanded format, and both formats together.

NOTE: All examples use the same seed value, showing how the same seed produces different expanded private keys for each security level.

C.1.1. ML-DSA-44 Private Key Examples

Each of the examples includes the textual encoding [RFC7468] followed by the so-called "pretty print"; the private keys are the same.

C.1.1.1. Seed Format 
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMRBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
C.1.1.2. Expanded Format 
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  OCTET_STRING {
    OCTET_STRING { `d7b2b47254aae0db45e7930d4a98d2c97d8f1397d178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` }
  }
}
C.1.1.3. Both Format 
-----BEGIN PRIVATE KEY-----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rq2CbMRfJE2/Qxr6s07733gkdNL9VxGPZGIUk07ZnLo7AD6NZ6ODb28Z/EGRDOUW
PuOumeuE1RTrdh5jaE6lb5eR0t1KrG5haLlIyBf3WiIqyw6M3APMSv6PZxV+GjY7
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jq6ylgcOv9NklHqUDMl4vtZrN3SebV3Ne+jElEQOK4TOz++5jAvt+zxB4zWdLNcZ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-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  OCTET_STRING {
    SEQUENCE {
      OCTET_STRING { `000102030405060708090a0b0c0d0e0f1011121314
15161718191a1b1c1d1e1f` }
      OCTET_STRING { `d7b2b47254aae0db45e7930d4a98d2c97d8f1397d1
789dafa17024b316e9bec939ce0f7f77f8db5644dcda366bfe4734bd95f435ff
9a613aa54aa41c2c694c04329a07b1fabb48f52a309f11a1898f848e2322ffe6
23ec810db3bee33685854a88269da320d5120bfcfe89a18e30f7114d83aa404a
646b6c997389860d12522ee0006e2384819186619b260d118664d4a628221844
82402898146148a6614c4248a19208c2382951244808a125c2083108c4712014
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932041062e42b64c01164914284c41a85180460a5116515a0820022244dc9849
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8580443091062c50c80924c5841a966d4a982c99066da4443220a7645a326e11
b57020926124138e04852c0a4872c8a051d3082a99208058242024074e591488
10a46460c06de0b28d1b1909203422c024410943710a212061a2015222521b80
809a340013934dd3322922170a9892691a14512027219cc02062a2814818691a
854d8344695b2041031242cb184601a90d0c023183b0215a224ac89205d99069
04306a4b064ad2b2011c404081423252327254a6405a18100c321292c2805212
625c82280bb46c03428d53100c14010ee1365288842491020a63462620062911
c228d0204802b36ca236095a8648cbb4618b4662c440821a890910024d24b245
20122524c90588288cc9c04d5948220a276ec134644c90605b44508286494388
0443b28c603080a2882d84a46d8ca629d0c68442064689885100a98d01498de4
380da4068dd3947142b26c1a84611ba32842b42808a0711ac531e0a04c013765
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44149808a446610195640b390a0c9450ca406ad2b220c0380182308e13b90891
8084148829c0189112350da02422e20406d9c2850428121cc989180272d24029
c20812d8062a9994719bb8682384291a2289144511dc82445096450c4484c0b2
049aa60543862c44326e88442120a84c9a3070e3b82d63268803254903438c48
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6d980a71aabbdf0c45e9a206ecb1423fee15decc17601300149d9223cd6e6c6e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e7997a2fb1d2723d38f9bfabe18f8e7b3cda906e4e9b5e942c8eaeb296070ebf
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521f031a817d15294a43722e8378784b6db0cf1ba9e8ae911d9201b9ce9cc301
9c6f5c27cb98da26144b64225a7c932b30f761e78a2d59a1d8b83ec6344a2f6d
d47e765706d00bf4a79a6a926c3ba91d812c8f2c797ab1796709e5d168567782
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7f652522a928b917e27e2d6d42137dfe2ebfa6fb1c67b26c0254528685f7ebdb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` }
    }
  }
}

C.1.2. ML-DSA-65 Private Key Examples

Each of the examples includes the textual encoding [RFC7468] followed by the so-called "pretty print"; the private keys are the same.

C.1.2.1. Seed Format 
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMSBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
C.1.2.2. Expanded Format 
-----BEGIN PRIVATE KEY-----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v++Sr2yhR7EyRNbIkvyO8iOrP0P5JML0Zgl+6A==
-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  OCTET_STRING {
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3f43f924c2f466097ee8` }
  }
}
C.1.2.3. Both Format 
-----BEGIN PRIVATE KEY-----
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1siS/I7yI6s/Q/kkwvRmCX7o
-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  OCTET_STRING {
    SEQUENCE {
      OCTET_STRING { `000102030405060708090a0b0c0d0e0f1011121314
15161718191a1b1c1d1e1f` }
      OCTET_STRING { `48683d91978e31eb3dddb8b0473482d2b88a5f6259
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ca2af75ccce734fd191c50089dad53982fddae02531ff93e1f21ff395fc0a128
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9f96cb884c3c93994748a56f41283bfc41fb89052153a894588c3cb9017f3d66
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29b8013e19c1d6d0680f4dae62c924450ae66aab82f21473061dab3d62b247f9
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0f28aaab1fe9a26ad1fd3361616c0b0e242fe76604b7033a1f30e97e28f526ca
3c880fe2b8d9d1b0c9ff188b31cb9d97425acab9b216d98a6ae355e583da71e8
864ee3d16b0759796190ef545c1e62bfef92af6ca147b13244d6c892fc8ef223
ab3f43f924c2f466097ee8` }
    }
  }
}

C.1.3. ML-DSA-87 Private Key Examples

Each of the examples includes the textual encoding [RFC7468] followed by the so-called "pretty print"; the private keys are the same.

C.1.3.1. Seed Format 
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMTBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
C.1.3.2. Expanded Format 
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
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734521b7b07c6388d3a3b99318bf0131504aa9dfbaf548f9d32a9cd4c6893524
b11330a2d3aad3ed2a58966ebb0134465d543fd7797af549f568eaebe957f64f
ec854674902b97558756986946ea3ab7a251cbbea11a687bd43f5d0bd89cd2ca
ba61d5218374990ee8b92219ed25dca011c68a9757c013bd837b2dd734e3751f
64fcb4b23dcd6bc57ea567f5716e17367244751e2303b22a953e772756956cdc
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f81aea948aedcf217aa8fccd7a0771aa2753e1a823bf41c95377a2ffa61b2265
138153ce86d2c87dd07a4b32d27f5f2872641431ce9a18a502aaefd9afc5b0d1
3cd46c357e38e69e1ee945add1992932a5b1e5c5629c9f48f7661853da00787c
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51634a60b504dde38a2e114f7ae9bf176d4a18ba2895a7bb4b47444a9ba8dbb4
c124cd41bbb32f4bcb1de48c4abb510607a001b5a000bba43618b6c19e43517b
45b42405928b67c713881858bad3a42511c2716ff9cd332034b672b52ff16610
805cdbe7544a8a84b66e1c745a73c1b6bcda5b77b951f36c0f7a5372de9e5d1f
9bbcde8843c6909002dda4875e67571af0bec581856c32c09c240e664e761e57
cd0d8dc8a71cb918a5762d111285cd8b5613ddbd0ca08ac0342b2bdee38f96fa
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656c9be3d590753c366c601406c061bc22033a1fd1f4e1111d039b8813b983cb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` }
  }
}
C.1.3.3. Both Format 
-----BEGIN PRIVATE KEY-----
MIITXgIBADALBglghkgBZQMEAxMEghNKMIITRgQgAAECAwQFBgcICQoLDA0ODxAR
EhMUFRYXGBkaGxwdHh8EghMgl5K87C8kMGhqgvzPPC9f9mXncderQbkCWM+n6Q7J
cSTY6e5OkKFsYC9eybw4UX3DDjKdWrJ2c72F9MmwMA93Y4mIZ1C1fCTbP8AS5h7e
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JDZkIgFIFhEJESyUAokiRSxiuEUARSoIlnCQEm4Uk3DURhCERFFYlpEMqSmCskHJ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oIXAxoRcOAbLtmkUhIRTIoKhpkDMhgDEJiKgqAiYNHLUIEFDxJBIFmgbFlIaNwJQ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8/mVF7fv0qM3dAVstWcWdai0kun18mIOuO+TgdPR3xmTi3tf+qxZvIEQ+oe6jXo9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RCKlclKdTJLx67GfHa1NA28v3zHKkQG9+BrqlIrtzyF6qPzNegdxqidT4agjv0HJ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-----END PRIVATE KEY-----

SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  OCTET_STRING {
    SEQUENCE {
      OCTET_STRING { `000102030405060708090a0b0c0d0e0f1011121314
15161718191a1b1c1d1e1f` }
      OCTET_STRING { `9792bcec2f2430686a82fccf3c2f5ff665e771d7ab
41b90258cfa7e90ec97124d8e9ee4e90a16c602f5ec9bc38517dc30e329d5ab2
7673bd85f4c9b0300f776389886750b57c24db3fc012e61ede59753337374fa7
124991549af243496d0637cb3be05a5948235bf79875f896d8fe0cab30c84948
db4d6315aaaf160ac6243664220148161109112c94028922452c62b84500452a
08967090126e149370d446108444515896910ca92982b241c90871c428680496
894840859b226d1c28645912419cb891840489449005cb3462a0869040269220
99291305695c3468a4328e19269259461009a44923424d123661581065012890
1a334c998631d3a249098225431428c0388103154d5b28860887482331529422
25c3c04da49821984020d14286cb40705bb0719c962cc112065346090c450214
466e91b42154b08ce446429a208c0121251341055a402213c90ca0184052c230
cb342c4bc8681ba4604984846330294aa0695b8004d2380a14264ce2b2448ba2
11244649c414520b427103b210922880012488e308110a052819c481002022dc
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0c9808a1a0010892440413245c987182847184325251b0319c422e1aa8280208
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5fabababbe9075fe5687aa451397bb9cfccd051243e9bf5aef24062d335de5fc
e24e9ddbde1191052d80c36df9f8434872f277ed4f5a1ce8ebd3b960824a4e4f
1001b04cb685f9bee4d0ddb0c571598ac2021a6606fd23345c6fbb84f0ce05fe
52734521b7b07c6388d3a3b99318bf0131504aa9dfbaf548f9d32a9cd4c68935
24b11330a2d3aad3ed2a58966ebb0134465d543fd7797af549f568eaebe957f6
4fec854674902b97558756986946ea3ab7a251cbbea11a687bd43f5d0bd89cd2
caba61d5218374990ee8b92219ed25dca011c68a9757c013bd837b2dd734e375
1f64fcb4b23dcd6bc57ea567f5716e17367244751e2303b22a953e772756956c
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0fdf1e09efe9fb800dc114e46be0879a195cc06870e23d2631dae71c3994481c
8761c40d07c5bfca95e718b7b22585af03ed34175a46d57af3518e32a7fc1aa4
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4ae3810a1c90818d52f93fb203e2d8b1b5fa8f60b2d585d9135d648846f138b8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` }
    }
  }
}

C.2. Example Public Keys

The following is the ML-DSA-44 public key corresponding to the private key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the public keys are the same.

-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  BIT_STRING { `00` `d7b2b47254aae0db45e7930d4a98d2c97d8f1397d17
89dafa17024b316e9bec94fc9946d42f19b79a7413bbaa33e7149cb42ed51156
93ac041facb988adeb5fe0e1d8631184995b592c397d2294e2e14f90aa414ba3
826899ac43f4cccacbc26e9a832b95118d5cb433cbef9660b00138e0817f61e7
62ca274c36ad554eb22aac1162e4ab01acba1e38c4efd8f80b65b333d0f72e55
dfe71ce9c1ebb9889e7c56106c0fd73803a2aecfeafded7aa3cb2ceda54d12bd
8cd36a78cf975943b47abd25e880ac452e5742ed1e8d1a82afa86e590c758c15
ae4d2840d92bca1a5090f40496597fca7d8b9513f1a1bda6e950aaa98de46750
7d4a4f5a4f0599216582c3572f62eda8905ab3581670c4a02777a33e0ca7295f
d8f4ff6d1a0a3a7683d65f5f5f7fc60da023e826c5f92144c02f7d1ba1075987
553ea9367fcd76d990b7fa99cd45afdb8836d43e459f5187df058479709a01ea
6835935fa70460990cd3dc1ba401ba94bab1dde41ac67ab3319dcaca06048d4c
4eef27ee13a9c17d0538f430f2d642dc2415660de78877d8d8abc72523978c04
2e4285f4319846c44126242976844c10e556ba215b5a719e59d0c6b2a96d3985
9071fdcc2cde7524a7bedae54e85b318e854e8fe2b2f3edfac9719128270aafd
1e5044c3a4fdafd9ff31f90784b8e8e4596144a0daf586511d3d9962b9ea95af
197b4e5fc60f2b1ed15de3a5bef5f89bdc79d91051d9b2816e74fa54531efdc1
cbe74d448857f476bcd58f21c0b653b3b76a4e076a6559a302718555cc63f748
59aabab925f023861ca8cd0f7badb2871f67d55326d7451135ad45f4a1ba6911
8fbb2c8a30eec9392ef3f977066c9add5c710cc647b1514d217d958c7017c3e9
0fd20c04e674b90486e9370a31a001d32f473979e4906749e7e477fa0b74508f
8a5f2378312b83c25bd388ca0b0fff7478baf42b71667edaac97c46b129643e5
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f495490a5bd4df248b95c4aae7784b605673166ac4245b5b4b082a09e9323e62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` }
}

The following is the ML-DSA-65 public key corresponding to the private key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the public keys are the same.

-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  BIT_STRING { `00` `48683d91978e31eb3dddb8b0473482d2b88a5f62594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` }
}

The following is the ML-DSA-87 public key corresponding to the private key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the public keys are the same.

-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  BIT_STRING { `00` `9792bcec2f2430686a82fccf3c2f5ff665e771d7ab4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` }
}

C.3. Example Certificates

The following is a self-signed certificate for the ML-DSA-44 public key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the certificates are the same.

-----BEGIN CERTIFICATE-----
MIIPlDCCBgqgAwIBAgIUFZ/+byL9XMQsUk32/V4o0N44804wCwYJYIZIAWUDBAMR
MCIxDTALBgNVBAoTBElFVEYxETAPBgNVBAMTCExBTVBTIFdHMB4XDTIwMDIwMzA0
MzIxMFoXDTQwMDEyOTA0MzIxMFowIjENMAsGA1UEChMESUVURjERMA8GA1UEAxMI
TEFNUFMgV0cwggUyMAsGCWCGSAFlAwQDEQOCBSEA17K0clSq4NtF55MNSpjSyX2P
E5fReJ2voXAksxbpvslPyZRtQvGbeadBO7qjPnFJy0LtURVpOsBB+suYit61/g4d
hjEYSZW1ksOX0ilOLhT5CqQUujgmiZrEP0zMrLwm6agyuVEY1ctDPL75ZgsAE44I
F/YediyidMNq1VTrIqrBFi5KsBrLoeOMTv2PgLZbMz0PcuVd/nHOnB67mInnxWEG
wP1zgDoq7P6v3teqPLLO2lTRK9jNNqeM+XWUO0er0l6ICsRS5XQu0ejRqCr6huWQ
x1jBWuTShA2SvKGlCQ9ASWWX/KfYuVE/GhvabpUKqpjeRnUH1KT1pPBZkhZYLDVy
9i7aiQWrNYFnDEoCd3oz4Mpylf2PT/bRoKOnaD1l9fX3/GDaAj6CbF+SFEwC99G6
EHWYdVPqk2f8122ZC3+pnNRa/biDbUPkWfUYffBYR5cJoB6mg1k1+nBGCZDNPcG6
QBupS6sd3kGsZ6szGdysoGBI1MTu8n7hOpwX0FOPQw8tZC3CQVZg3niHfY2KvHJS
OXjAQuQoX0MZhGxEEmJCl2hEwQ5Va6IVtacZ5Z0MayqW05hZBx/cws3nUkp77a5U
6FsxjoVOj+Ky8+36yXGRKCcKr9HlBEw6T9r9n/MfkHhLjo5FlhRKDa9YZRHT2ZYr
nqla8Ze05fxg8rHtFd46W+9fib3HnZEFHZsoFudPpUUx79wcvnTUSIV/R2vNWPIc
C2U7O3ak4HamVZowJxhVXMY/dIWaq6uSXwI4YcqM0Pe62yhx9n1VMm10URNa1F9K
G6aRGPuyyKMO7JOS7z+XcGbJrdXHEMxkexUU0hfZWMcBfD6Q/SDATmdLkEhuk3Cj
GgAdMvRzl55JBnSefkd/oLdFCPil8jeDErg8Jb04jKCw//dHi69CtxZn7arJfEax
KWQ+WG5bBVoMIRlG1PNuZ1vtWGD6BCoxXZgmFk1qkjfDWl+/SVSQpb1N8ki5XEqu
d4S2BWcxZqxCRbW0sIKgnpMj5i8geMW3Z4NEbe/XNq06NwLUmwiYRJAKYYMzl7xE
GbMNepegs4fBkRR0xNQbU+Mql3rLbw6nXbZbs55Z5wHnaVfe9vLURVnDGncSK1IE
47XCGfFoixTtC8C4AbPm6C3NQ+nA6fQXRM2YFb0byIINi7Ej8E+s0bG2hd1aKxuN
u/PtkzZw8JWhgLTxktCLELj6u9/MKyRRjjLuoKXgyQTKhEeACD87DNLQuLavZ7w1
W5SUAl3HsKePqA46Lb/rUTKIUdYHgZjpSTZRrnh+wCUfkiujDp9R32Km1yeEzz3S
BTkxdt+jJKUSvZSXCjbdNKUUqGeR8Os28BRbCatkZRtKAxOymWEaKhxIiRYnWYdo
oxFAYLpEQ0ht9RUioc6IswmFwhb45u0XjdVnswSg1Mr7qIKig0LxepqiauWNtjAI
PSw1j99WbD9dYqQoVnvJ6ozpXKoPNUdLC/qPM5olCrTfzyCDvo7vvBBV4Y/hU3Du
yyYFZtg/8GshGq7EPKKbVMzQD4gVokZe8LRlFcx+QfMSTwnv/3OTCatYspoUWaAL
zlA46TjJZ49y6w5O5f2q5m2fhXP8l/xCtJWfS/i2HXhDPoawM11ukZHE2L9IezkF
wQjP1qwksM633LfPUfhNDtaHuV6uscUzwG8NlwI9kqcIJYN7Wbpst9TlawqHwgOG
KujzFbpZJejt76Z5NpoiAnZhUfFqll+fgeznbMBwtVhp5NuXhM8FyDCzJCyDEqNC
MEAwDgYDVR0PAQH/BAQDAgGGMA8GA1UdEwEB/wQFMAMBAf8wHQYDVR0OBBYEFDKa
B7H6u0j1KjCfEaGJj4SOIyL/MAsGCWCGSAFlAwQDEQOCCXUAZ6iVH8MI4S9oZ2Ef
3CVL9Ly1FPf18v3rcvqOGgMAYWd7hM0nVZfYMVQZWWaxQWcMsOiBE0YNl4oaejiV
wRykGZV3XAnWTd60e8h8TovxyTJ/xK/Vw3hlU+F9YpsPJxQnZUgUMrXnzNC6YeUc
rT3Y+Vk4wjXr7O6vixauM2bzAMU1jse+nrI6HqGj2lhoZwTwSD+Wim5LH4lnCgE0
s2oY1scn3JsCexJ5R5OkjHq2bt9XrBgRORTADQoRtlplL0d3Eze/dDZm/Klby9OR
Ia4HUL7FWtWoy86Y5TiuUjlH1pKZdjMPyj/JXAHRQDtJ5cuoGBL0NlDdATEJNCee
zQfMqzTCyjCn091QkuFjDhQjzJ+sQ6G02w49lw8Kpm1ASuh7BLTPcuz7Z+rLpNjN
jmW67rR6+hHMK474mSKIZnuO3vVKnidntjLhSYc1soxvYPCLWWnl4m3XyjlrnlzD
4Soec2I2AjKNZKCO9KKa81cRzIcNJjc7sbnrLv/hKXNUTESn4s3yAyRPU7N6bVIy
N9ifBvb1U07WMRPI8A7/f9zVCaLYx87ym9P7GGpMjDYrPUQpOaKQdu4ycWuPrlEA
2BoHIVzbHHm9373BT1LjcxjR5SbbhNFg+42hwG284VlVzcLW/XiipaWN8jnONmxt
kLMui9R/wf0TCehilMDDtRznfm37b2ci5o9MP/LrTDRpMVBudDuwIZmLgPQ/bj08
n+VHd8D2WADpR/kEMpDhSwG2P44mwwE4CUKGbHS0qQLOSRwMlQVEzwxpOOrLMusw
JmzoLE0KNsUR6o/3xAlUmjqCZMqYPYxtXgNfJEJDp3V1iqyZK1iES3EQ0/h8m7oZ
3YqNKrEpTgVV7EmVpUjcVszjWgXcSKynVVsWQd3j0Zf83zXRLwmq8+anJ3XNGCSa
IecO2sZxDbaiHhwFYRkt0BGRM2QM//IPMYeXhRa/1svmbOEHGxJG9LqTffkBs+01
Bp7r3/9lRZ+5t3eukpinpJrCT0AgeV3l3ujbzyCiQbboFDaPS4+kKvi+iS2eHjiu
S/WkfP1Go5jksxhkceJFNPsTmGCyXGPy2/haU9hkiMg9/wmuIKm/gxRfIBh/DoIr
1HWZjTuWcBGWTu2NuXeAVO/MbMtpB0u6mWYktHQcVxA2LenU+N5LEPbbHp+AmPQC
RZPqBziTyx/nuVnFD+/EAbPKzeqMKhcTW6nfkKt/Md4zmi1vhWxx7c+wDlo9cyAf
vsS0p5uXKK1wzaC4mBIVdPYNlZtAjBCK8asKpH3/NyYJ8xhsBjxXLLiQifKiGOpA
LLBy/LyJWmo4R4zkAtUILD4FcsIyLMIJlsqWjaNdey7bwGI75hZQkBIF8QJxFVtT
n4HQBtuNe2ek7e72d+bayceJvlUAFXTu6oeX9/UuS7AhuY4giNzI1pNOgNwWXRxx
REmwvPrzJatZZ7cwfsKTezSSQlv2O4q70+2X2h0VtUg/pkz3GknE07S3ggDR9Qkg
bywQS/42luPIADbbAKXhHaBaX/TaD/uZVn+BOZ5sqWmxEbbHtvzlSea02J1Fk4Hq
kWbpuzByCJ25SuDRr+Xyn84ZDnetumQ0lBkc2ro+rZKXw8YGMyt0aX8ZwJxL4qNB
/WFFEproVsOru8G7iwXgt4QP8WRBSp2kTlQUbNTF3gxOTsslkUErTnvcRQ0GpK06
DRQG8wbjgewpHyw7O8Sfi34EjAzic0gwtIp501/MWmKpRUgAow9LPreiaLq2TBIQ
DXEhUb9fEhY77QKeir8cpue3sShqcz9TLa5REJGqsP/8/URk7lZjiI+YWbRLp2U2
D//0NPEq8fxrzNtacZRxSdx2id/yTWumtj5swjFA4yk0tunadltDMgEYuKgR+Jw9
G3/yFTDnepHK41V6x8eE/4JjUAvIJWADDWxudO7oF/wsY0AnUuWe9DkW09g8IWhk
NukDTdpsl08hCLF06qH3MSHJrdUAzs2GGLMCvtrXK2L3k70PcLqMXhbPSr7d1RGW
gW0BlRfR4l+2LJ952SMv3xzuxgT43aX3FFVBxXk7nFrhWJWIpJpuYXRhTqASkzoZ
KzsIRyW0ZbsaIsy0tgzzyhQvdoOoJn+2sKjcCzpfY6tgRD9sfucOm1sGet/cM5YP
iJYei2qKMeYcvACWiI8GNGY37OzhlikbleO4xXnfJwEOYx66NjTHZqkz1/TiCBGU
a7h+l/fnut6VfkxS1yZ2r5Gsdx7DUfNkEeKyzIMnYRA3zw3047lHqH714rV5VbE3
yYEQWvdtYlHMFM2z9DDta59RRATOemm7AA1fYsfodrV/QPJi5qPmvpHtCvfItbdL
Fg88Zh1zV5nV+0doUTXFVR9poJRE9fASlfU5qCJ9Jx5ISfvIkGz1fmfqXhUN9fE7
C0Evl7IYQLguTXFznRvsXvnliwR9Ut/g85JtXUiku4F2ThCBMHBDbov6p128kP+2
7LBgShM4IG80clxon8sWh6y0RLUz1MTamEYZKCXAPZzJoWhbzdNns/QTsjNP8wlu
vBRtdkb6w4Vrm6GO2BXY6pQUBPcoDuymAhfAF9TxRn860OQeMcT/NRsU9Z/8nRnz
3KbAuMTYsQ6qbjuLTDwfF9B4b4YUDQR22z8wlzCNLzgwFlGSI12xhf3ejRlwjGZJ
J/11Up4pEegRS/c+Li2OUvQr9Jxi8XGIdEJZY1T8oVpzDJf3C29gpARWSDAXrFn0
lgZHnqFyebeC1uDW8r/wGtYmI2EC53+FlOF5AFcH+3LzObZzerqwror4UMOA+B5c
QMU5vDv1LFcWLzvJHMXJfCHL5nVSukXCMawr+DbeKjrkseG0UX0gpUbQy0vHIH1K
2geD2xyl3TJ8jCaKOxb/Hu+KfkvtOCsh07TA+cnTV1WHR77svUcMErzHXWOFm8+U
omIXALO1EiDbpu38gERRLkC84eMhRBQjKcdmlcBFsmilt3cfIofypuhMRiIFjIke
00y2GEdQVsZGA/LX1HILqD4dEFDDQI2LPvCG5qe28HTfWspzsqK94IRESzm+Vmdp
IjNzkTyrPI06yMvxaHGajwUtLWCReJOG/uXhswbX7EviVYyqCR4vzDLDVXAulxo/
OsHaQhMX8xYOLXontx7SNCBlu/EEBww5QklKUldgd5igr7bDxsvZ6vHy/wcNIzY3
RUdidnuDkpSm1hIoLz4/SW2Tm6C2u9La5evu7xAfIy1ul8LE3/P0AAAAAAAAAAAA
AAAAABcmOEM=
-----END CERTIFICATE-----

SEQUENCE {
  SEQUENCE {
    [0] {
      INTEGER { 2 }
    }
    INTEGER { `159ffe6f22fd5cc42c524df6fd5e28d0de38f34e` }
    SEQUENCE {
      OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      UTCTime { "200203043210Z" }
      UTCTime { "400129043210Z" }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      SEQUENCE {
        OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
      }
      BIT_STRING { `00` `d7b2b47254aae0db45e7930d4a98d2c97d8f139
7d1789dafa17024b316e9bec94fc9946d42f19b79a7413bbaa33e7149cb42ed5
115693ac041facb988adeb5fe0e1d8631184995b592c397d2294e2e14f90aa41
4ba3826899ac43f4cccacbc26e9a832b95118d5cb433cbef9660b00138e0817f
61e762ca274c36ad554eb22aac1162e4ab01acba1e38c4efd8f80b65b333d0f7
2e55dfe71ce9c1ebb9889e7c56106c0fd73803a2aecfeafded7aa3cb2ceda54d
12bd8cd36a78cf975943b47abd25e880ac452e5742ed1e8d1a82afa86e590c75
8c15ae4d2840d92bca1a5090f40496597fca7d8b9513f1a1bda6e950aaa98de4
67507d4a4f5a4f0599216582c3572f62eda8905ab3581670c4a02777a33e0ca7
295fd8f4ff6d1a0a3a7683d65f5f5f7fc60da023e826c5f92144c02f7d1ba107
5987553ea9367fcd76d990b7fa99cd45afdb8836d43e459f5187df058479709a
01ea6835935fa70460990cd3dc1ba401ba94bab1dde41ac67ab3319dcaca0604
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c3e90fd20c04e674b90486e9370a31a001d32f473979e4906749e7e477fa0b74
508f8a5f2378312b83c25bd388ca0b0fff7478baf42b71667edaac97c46b1296
43e586e5b055a0c211946d4f36e675bed5860fa042a315d9826164d6a9237c35
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3e62f2078c5b76783446defd736ad3a3702d49b089844900a61833397bc4419b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` }
    }
    [3] {
      SEQUENCE {
        SEQUENCE {
          # keyUsage
          OBJECT_IDENTIFIER { 2.5.29.15 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            BIT_STRING { b`1000011` }
          }
        }
        SEQUENCE {
          # basicConstraints
          OBJECT_IDENTIFIER { 2.5.29.19 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            SEQUENCE {
              BOOLEAN { TRUE }
            }
          }
        }
        SEQUENCE {
          # subjectKeyIdentifier
          OBJECT_IDENTIFIER { 2.5.29.14 }
          OCTET_STRING {
            OCTET_STRING { `329a07b1fabb48f52a309f11a1898f848e23
22ff` }
          }
        }
      }
    }
  }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
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beeef101f232d6e97c2c4dff3f40000000000000000000000000017263843` }
}

The following is a self-signed certificate for the ML-DSA-65 public key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the certificates are the same.

-----BEGIN CERTIFICATE-----
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IA==
-----END CERTIFICATE-----

SEQUENCE {
  SEQUENCE {
    [0] {
      INTEGER { 2 }
    }
    INTEGER { `159ffe6f22fd5cc42c524df6fd5e28d0de38f34e` }
    SEQUENCE {
      OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      UTCTime { "200203043210Z" }
      UTCTime { "400129043210Z" }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      SEQUENCE {
        OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
      }
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b2104bc460a051e02f2e8444f` }
    }
    [3] {
      SEQUENCE {
        SEQUENCE {
          # keyUsage
          OBJECT_IDENTIFIER { 2.5.29.15 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            BIT_STRING { b`1000011` }
          }
        }
        SEQUENCE {
          # basicConstraints
          OBJECT_IDENTIFIER { 2.5.29.19 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            SEQUENCE {
              BOOLEAN { TRUE }
            }
          }
        }
        SEQUENCE {
          # subjectKeyIdentifier
          OBJECT_IDENTIFIER { 2.5.29.14 }
          OCTET_STRING {
            OCTET_STRING { `1b0563e3cd3346149c8c9ebcf23b0a4e5a90
0eea` }
          }
        }
      }
    }
  }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
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f962536c34da1798be2d5ef27d7ac30d4cc151e5b1cb9455b97697c77764c2cc
064ac513de1abb8a435e0a1665dfbdc9b0d7cb55ed85cd4e2b4fb416fee68b22
2485fc1811bdd5e8a9c4dcadc4afeaa15aafbdf8f2b527f8062d170bf235138c
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b5b0984f0c5ae016c99988f040288a5730de9275dd323c6c57412ec373a49b70
88429edb88fa3bd74f32e64789965e87129c7bb164f7ee634ea23e0015d9bbf7
0cc4250865ea928ea380697e3d5eb5315001c53a137c1ec950fd44e8bb5ad481
1f810584539f81f81ad421c3f031dbc0376c6be761223f663910a9edd1143e3b
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411659ecccfe7f811f852798599063b486672787b83bc537374d200000000000
00000000000000000000000000000000000060d0f131c20` }
}

The following is a self-signed certificate for the ML-DSA-87 public key in the previous section. The textual encoding [RFC7468] is followed by the so-called "pretty print"; the certificates are the same.

-----BEGIN CERTIFICATE-----
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-----END CERTIFICATE-----

SEQUENCE {
  SEQUENCE {
    [0] {
      INTEGER { 2 }
    }
    INTEGER { `159ffe6f22fd5cc42c524df6fd5e28d0de38f34e` }
    SEQUENCE {
      OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      UTCTime { "200203043210Z" }
      UTCTime { "400129043210Z" }
    }
    SEQUENCE {
      SET {
        SEQUENCE {
          # organizationName
          OBJECT_IDENTIFIER { 2.5.4.10 }
          PrintableString { "IETF" }
        }
      }
      SET {
        SEQUENCE {
          # commonName
          OBJECT_IDENTIFIER { 2.5.4.3 }
          PrintableString { "LAMPS WG" }
        }
      }
    }
    SEQUENCE {
      SEQUENCE {
        OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
      }
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    }
    [3] {
      SEQUENCE {
        SEQUENCE {
          # keyUsage
          OBJECT_IDENTIFIER { 2.5.29.15 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            BIT_STRING { b`1000011` }
          }
        }
        SEQUENCE {
          # basicConstraints
          OBJECT_IDENTIFIER { 2.5.29.19 }
          BOOLEAN { TRUE }
          OCTET_STRING {
            SEQUENCE {
              BOOLEAN { TRUE }
            }
          }
        }
        SEQUENCE {
          # subjectKeyIdentifier
          OBJECT_IDENTIFIER { 2.5.29.14 }
          OCTET_STRING {
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          }
        }
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662b5368f21aa957c822f0a0303ec9f56fe9030f302d16b2c4bf621ed91ab393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35941a70107925ff095153ec610772c5fd524c80c228d80e781e864e295d20c6
896f502887b18ed2307e8eeaea56e2649639349652f509528e4b80ea477d7a6a
453807b26361b1eca6fcec4e06c504166742c12f8a44a359fef2727d34f47201
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e734b7acba49d308a6151c1352de2e20b1fed060d5e6fb58d6f23b2847dc53b6
7519b92dba604e448f3990570f09bd65f685545a02462cd24c72f2f7989cbe87
b0c52da52eea1770aaecf7d568d0b1bf1aed3ca8f58f332299e391b4413b6c6f
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37abdea5af4b4a3a7f1e8a3e8cff1026ddefc29bae39e75249364a953ec52231
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a15840e92822b7166c8611d464af2e8d1baa5bcaf1c1649589a7eb16b889b54e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` }
}

C.4. Example Inconsistent Private Keys

The following examples demonstrate inconsistent seed and expanded private keys.

C.4.1. ML-DSA Inconsistent Seed and Expanded Private Keys

Three ML-DSA-44-PrivateKey examples of inconsistent seed and expanded private keys follow:

  1. The first ML-DSA-PrivateKey example includes the both CHOICE , i.e., both seed and expandedKey are included. The seed and expanded values can be checked for inconsistencies.

  2. The second ML-DSA-PrivateKey example includes only expandedKey. The public key fails to match the tr hash value in the private key.

  3. The third ML-DSA-PrivateKey example also includes only expandedKey. The private s_1 and s_2 vectors imply a t vector whose private low bits do not match the t_0 vector portion of the private key (its high bits t_1 are the primary content of the public key).

The second and third examples would not be detected by implementations that do not regenerate the public key from the private key, or neglect to then check consistency of tr or t_0.

The following is the first example:

-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----

The following is the second example:

-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----

The following is the third example:

-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----

Appendix D. Pre-hashing (ExternalMu-ML-DSA)

Some applications require pre-hashing, where the signature generation process can be separated into a pre-hash step and a core signature step in order to ease operational requirements around large or inconsistently-sized payloads. Pre-hashing can be performed at the protocol layer, but not all protocols support it. Examples in [RFC5280] are certificates and CRLs; these do not include message digesting before signing. This can make signing large CRLs or a high volume of certificates with large public keys challenging.

As mentioned in the introduction, pure ML-DSA signing itself supports a pre-hashing flow by splitting the operation over two modules. In this section, we make this "ExternalMu-ML-DSA" more explicit.

There are two steps. First an ExternalMu-ML-DSA.Prehash() followed by ExternalMu-ML-DSA.Sign(). Together these are functionally equivalent to ML-DSA.Sign() from [FIPS204] in that used in sequence they create exactly the same signatures as regular pure ML-DSA, which can be verified by the unmodified ML-DSA.Verify().

An ML-DSA key and certificate can be used with either ML-DSA or ExternalMu-ML-DSA interchangeably. Note that ExternalMu-ML-DSA describes a different signature API from ML-DSA and therefore might require explicit support from hardware or software cryptographic modules.

Note that the signing mode defined here is different from HashML-DSA defined in Section 5.4 of [FIPS204]. This specification uses exclusively ExternalMu-ML-DSA for pre-hashed use cases. See Section 10.1 for additional discussion of why HashML-DSA is disallowed in PKIX.

All functions and notation used in Figure 2 and Figure 3 are defined in [FIPS204].

External operations:

ExternalMu-ML-DSA.Prehash(pk, M):

  # Simplified flow for computing ML-DSA's mu based on Algorithm 2 and 7
  # of FIPS 204, with ctx being empty string.
  # M is a bit-string, mu is byte-string.

  mu = H(BytesToBits(H(pk, 64) || IntegerToBytes(0, 2)) || M, 64)
  return mu
Figure 2: External steps of ExternalMu-ML-DSA

Internal operations:

ExternalMu-ML-DSA.Sign(sk, mu):

  if |mu| != 64 then
    return error  # return an error indication if the input mu is not
                  # 64 bytes.
  end if

  rnd = rand(32)  # for the optional deterministic variant,
                  # set rnd to all zeroes
  if rnd = NULL then
    return error  # return an error indication if random bit
                  # generation failed
  end if

  sigma = ExternalMu-ML-DSA.Sign_internal(sk, mu, rnd, isExternalMu=true)
  return sigma

ML-DSA.Sign_internal(sk, M', rnd, isExternalMu=false):
    # mu can be passed as an argument instead of M'
    # defaulting isExternalMu to false means that
    # this modified version of Sign_internal can be used
    # in place of the original without interfering with
    # functioning of pure ML-DSA mode.
    # ... identical to FIPS 204 Algorithm 7, but with Line 6 replaced with
  6: if (isExternalMu):
       mu = M'
     else:
       mu = H(BytesToBits(tr) || M', 64)
Figure 3: Internal steps of ExternalMu-ML-DSA

ExternalMu-ML-DSA requires the public key, or its prehash, as input to the Prehash() function. This assumes the signer generating the pre-hash is in possession of the public key before signing and is different from conventional pre-hashing which only requires the message and the hash function as input.

Security-wise, during the signing operation of pure (or "one-step") ML-DSA, the cryptographic module extracts the public key hash tr from the secret key object, and thus there is no possibility of mismatch between tr and sk. In ExternalMu-ML-DSA, the public key or its hash needs to be provided to the Prehash() routine independently of the secret key, and while the exact mechanism by which it is delivered will be implementation-specific, it does open a window for mismatches between tr and sk. First, this will produce a signature which will fail to verify under the intended public key since a compliant Verify() routine will independently compute tr from the public key. Implementors should pay careful attention to how the public key or its hash is delivered to the ExternalMu-ML-DSA.Prehash() routine, and from where they are sourcing this data.

Acknowledgments

The authors wish to thank the following people for their contributions to this document: Dierdre Connolly, Viktor Dukhovni, Russ Housley, Alicja Kario, Mike Ounsworth, and Daniel Van Geest.

In addition, we would like to thank those who contributed to the private key format discussion: Tony Arcieri, Bob Beck, Dmitry Belyavskiy, David Benjamin, Daniel Bernstein, Uri Blumenthal, Theo Buehler, Stephen Farrell, Jean-Pierre Fiset, Scott Fluhrer, Alex Gaynor, John Gray, Peter Gutmann, David Hook, Tim Hudson, Paul Kehrer, John Kemp, Watson Ladd, Adam Langley, John Mattsson, Damien Miller, Robert Relyea, Michael Richardson, Markku-Juhani O. Saarinen, Rich Salz, Roland Shoemaker, Sophie Schmieg, Simo Sorce, Michael St. Johns, Falko Strenzke, Filippo Valsorda, Loganaden Velvindron, Carl Wallace, and Wei-Jun Wang.

Authors' Addresses

Jake Massimo
AWS
United States of America
Panos Kampanakis
AWS
United States of America
Sean Turner
sn3rd
Bas Westerbaan
Cloudflare