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<rfc xmlns:xi="http://www.w3.org/2001/XInclude" ipr="trust200902" docName="draft-ietf-lamps-dilithium-certificates-13" number="9881" updates="" obsoletes="" xml:lang="en" category="std" consensus="true" submissionType="IETF" tocInclude="true" sortRefs="true" symRefs="true" version="3">
  <!-- xml2rfc v2v3 conversion 3.30.2 -->

  <front>
    <title abbrev="ML-DSA in Certificates">Internet X.509 Public Key Infrastructure - -- Algorithm Identifiers for the Module-Lattice-Based Digital Signature Algorithm (ML-DSA)</title>
    <seriesInfo name="Internet-Draft" value="draft-ietf-lamps-dilithium-certificates-13"/> name="RFC" value="9881"/>
    <author initials="J." surname="Massimo" fullname="Jake Massimo">
      <organization>AWS</organization>
      <address>
        <postal>
          <country>US</country>
          <country>United States of America</country>
        </postal>
        <email>jakemas@amazon.com</email>
      </address>
    </author>
    <author initials="P." surname="Kampanakis" fullname="Panos Kampanakis">
      <organization>AWS</organization>
      <address>
        <postal>
          <country>US</country>
          <country>United States of America</country>
        </postal>
        <email>kpanos@amazon.com</email>
      </address>
    </author>
    <author initials="S." surname="Turner" fullname="Sean Turner">
      <organization>sn3rd</organization>
      <address>
        <email>sean@sn3rd.com</email>
      </address>
    </author>
    <author initials="B. E." surname="Westerbaan" fullname="Bas Westerbaan">
      <organization>Cloudflare</organization>
      <address>
        <email>bas@cloudflare.com</email>
      </address>
    </author>
    <date year="2025" month="September" day="30"/> month="October"/>
    <area>SEC</area>
    <workgroup>LAMPS WG</workgroup>
    <keyword>ML-DSA Certificate X.509 PKIX</keyword>
    <workgroup>lamps</workgroup>
    <keyword>ML-DSA</keyword>
    <keyword>Certificate</keyword>
    <keyword>X.509</keyword>
    <keyword>PKIX</keyword>

    <abstract>
      <?line 147?>
<t>Digital signatures are used within X.509 certificates, certificates and Certificate
Revocation Lists (CRLs), and to sign messages. This document specifies
the conventions for using FIPS 204, the Module-Lattice-Based Digital
Signature Algorithm (ML-DSA) in Internet X.509 certificates and
certificate revocation lists.
CRLs.  The conventions for the associated
signatures, subject public keys, and private key are also described.</t>
    </abstract>
    <note removeInRFC="true">
      <name>About This Document</name>
      <t>
        The latest revision of this draft can be found at <eref target="https://lamps-wg.github.io/dilithium-certificates/#go.draft-ietf-lamps-dilithium-certificates.html"/>.
        Status information for this document may be found at <eref target="https://datatracker.ietf.org/doc/draft-ietf-lamps-dilithium-certificates/"/>.
      </t>
      <t>
        Discussion of this document takes place on the
        Limited Additional Mechanisms for PKIX and SMIME (lamps) Working Group mailing list (<eref target="mailto:spasm@ietf.org"/>),
        which is archived at <eref target="https://mailarchive.ietf.org/arch/browse/spasm/"/>.
        Subscribe at <eref target="https://www.ietf.org/mailman/listinfo/spasm/"/>.
      </t>
      <t>Source for this draft and an issue tracker can be found at
        <eref target="https://github.com/lamps-wg/dilithium-certificates"/>.</t>
    </note>

  </front>
  <middle>
    <?line 156?>

    <section anchor="introduction">
      <name>Introduction</name>
      <t>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
<xref target="NIST-PQC"/> in <xref target="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.</t>
      <t><xref target="FIPS204"/> defines two variants of ML-DSA: a pure and a pre-hash variant. pre-hash.
Only the former is specified in this document.
See <xref target="sec-disallow-hash"/> for the rationale.
The pure variant of ML-DSA supports the typical pre-hash flow. Refer to
<xref target="externalmu"/> for more details.</t>
      <t>Prior to standardisation, standardization, ML-DSA was known as Dilithium.  ML-DSA and
Dilithium are not compatible.</t>
      <section anchor="requirements-language">
        <name>Requirements Language</name>
        <t>The
        <t>
    The key words "<bcp14>MUST</bcp14>", "<bcp14>MUST NOT</bcp14>",
    "<bcp14>REQUIRED</bcp14>", "<bcp14>SHALL</bcp14>", "<bcp14>SHALL NOT</bcp14>",
    "<bcp14>SHOULD</bcp14>", "<bcp14>SHOULD NOT</bcp14>",
    "<bcp14>RECOMMENDED</bcp14>", "<bcp14>NOT RECOMMENDED</bcp14>",
    "<bcp14>MAY</bcp14>", and "<bcp14>OPTIONAL</bcp14>" in this document are to be
    interpreted as described in BCP 14 BCP&nbsp;14 <xref target="RFC2119"/> <xref
    target="RFC8174"/> when, and only when, they appear in all capitals, as
    shown here.</t>
        <?line -18?> here.
        </t>
      </section>
    </section>
    <section anchor="oids">
      <name>Identifiers</name>
<!-- [rfced] The AlgorithmIdentifier type defined in [RFC5912] includes an "&" character before "Params".  Please review and let us know if the ampersand should be included in the first block of sourcecode.

In addition, the first line extends 3 characters beyond the margin.  Please let us know how the line can be broken.

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

From [RFC5912]:
    AlgorithmIdentifier{ALGORITHM-TYPE, ALGORITHM-TYPE:AlgorithmSet} ::=
        SEQUENCE {
            algorithm   ALGORITHM-TYPE.&id({AlgorithmSet}),
            parameters  ALGORITHM-TYPE.
                   &Params({AlgorithmSet}{@algorithm}) OPTIONAL
        }
-->
      <t>The <tt>AlgorithmIdentifier</tt> type is defined in <xref target="RFC5912"/> as follows:</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
    AlgorithmIdentifier{ALGORITHM-TYPE, ALGORITHM-TYPE:AlgorithmSet} ::=
      SEQUENCE {
        algorithm   ALGORITHM-TYPE.id({AlgorithmSet}),
        parameters  ALGORITHM-TYPE.
                      Params({AlgorithmSet}{@algorithm}) OPTIONAL
     }
]]></artwork>
     }]]></sourcecode>
      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5912"/> and is compatible with
the 2021 ASN.1 syntax <xref target="X680"/>. See <xref target="RFC5280"/> for the 1988 ASN.1
syntax.</t>
      </aside>
      <t>The fields in AlgorithmIdentifier have the following meanings:</t>
      <ul spacing="normal">
        <li>
          <t><tt>algorithm</tt> identifies the cryptographic algorithm with an object
identifier (OID).</t>
        </li>
        <li>
          <t><tt>parameters</tt>, which are optional, are the associated parameters for the
algorithm identifier in the algorithm field.</t>
        </li>
      </ul>
      <t>The NIST registered NIST-registered OIDs <xref target="CSOR"/> are:</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
   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) }
]]></artwork> }]]></sourcecode>

      <t>The contents of the <tt>parameters</tt> component for each <tt>algorithm</tt> <bcp14>MUST</bcp14> be
absent.</t>
    </section>
    <section anchor="ml-dsa-signatures-in-pkix">
      <name>ML-DSA Signatures in PKIX</name>
      <t>ML-DSA is a digital signature scheme built upon the
Fiat-Shamir-with-aborts framework <xref target="Fiat-Shamir"/>. The security is based
upon the hardness of lattice problems over module lattices <xref target="Dilithium"/>.
ML-DSA provides three parameter sets for the NIST PQC security categories
2, 3 3, and 5.</t>
      <t>Signatures are used in a number of different ASN.1 structures. As shown
in the ASN.1 representation from <xref target="RFC5280"/> below, in an X.509
certificate, a signature is encoded with an algorithm identifier in the
<tt>signatureAlgorithm</tt> attribute and a <tt>signatureValue</tt> attribute that contains
the actual signature.</t>
      <artwork><![CDATA[
<!-- [rfced] The following sentences makes it seem like this ASN.1 is copied from
[RFC5280], but this exact source code doesn't appear to be in
[RFC5280]. Perhaps the intent is to indicate the use "1988
ASN.1 syntax"?

Current:

   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}))
  }
]]></artwork>

Same issue occurs here:

Current:
  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 }

-->
      <sourcecode type="asn.1"><![CDATA[
  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}))
  }]]></sourcecode>
      <t>Signatures are also used in the CRL list ASN.1 representation from
<xref target="RFC5280"/> below. In a an X.509 CRL, a signature is encoded with an
algorithm identifier in the <tt>signatureAlgorithm</tt> attribute and a
<tt>signatureValue</tt> attribute that contains the actual signature.</t>
      <artwork><![CDATA[

      <sourcecode type="asn.1"><![CDATA[
   CertificateList  ::=  SIGNED{ TBSCertList }
]]></artwork> }]]></sourcecode>

      <t>The following <tt>SIGNATURE-ALGORITHM</tt> ASN.1 classes are for ML-DSA-44,
ML-DSA-65, and ML-DSA-87:</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
  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 }
    }
]]></artwork>
    }]]></sourcecode>
      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5912"/> and is compatible with the
  2021 ASN.1 syntax <xref target="X680"/>. See <xref target="RFC5280"/> for the 1988 ASN.1 syntax.</t>
      </aside>
      <t>The identifiers defined in <xref target="oids"/> can be used as the
<tt>AlgorithmIdentifier</tt> in the <tt>signatureAlgorithm</tt> field in the sequence
<tt>Certificate</tt>/<tt>CertificateList</tt> and in the <tt>signature</tt> field in the sequence
<tt>TBSCertificate</tt>/<tt>TBSCertList</tt> in certificates and CRLs, respectively,
<xref target="RFC5280"/>. The <tt>parameters</tt> of these signature algorithms <bcp14>MUST</bcp14> be
absent, as explained in <xref target="oids"/>. That is, the <tt>AlgorithmIdentifier</tt>
        <bcp14>SHALL</bcp14> be a <tt>SEQUENCE</tt> of one component, the OID id-ml-dsa-*, where *
is 44, 65, or 87 - -- see <xref target="oids"/>.</t>
      <t>The <tt>signatureValue</tt> field contains the corresponding ML-DSA signature
computed upon the ASN.1 DER encoded DER-encoded <tt>tbsCertificate</tt>/<tt>tbsCertList</tt>
        <xref target="RFC5280"/>.  The optional context string (ctx) parameter
as defined in Section 5.2 of <xref target="FIPS204"/> is left to its default value:
the empty string.</t>
      <t>Conforming Certification Authority (CA) implementations <bcp14>MUST</bcp14> specify
the algorithms explicitly by using the OIDs specified in <xref target="oids"/> when
encoding ML-DSA signatures in certificates and CRLs. Conforming client
implementations that process certificates and CRLs using ML-DSA <bcp14>MUST</bcp14>
recognize the corresponding OIDs. Encoding rules for ML-DSA signature
values are specified in <xref target="oids"/>.</t>
    </section>
    <section anchor="ML-DSA-PublicKey">
      <name>ML-DSA Public Keys in PKIX</name>
      <t>In the X.509 certificate, the <tt>subjectPublicKeyInfo</tt> field has the
<tt>SubjectPublicKeyInfo</tt> type, which has the following ASN.1 syntax:</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
  SubjectPublicKeyInfo {PUBLIC-KEY: IOSet} ::= SEQUENCE {
      algorithm        AlgorithmIdentifier {PUBLIC-KEY, {IOSet}},
      subjectPublicKey BIT STRING
  }
]]></artwork>
  }]]></sourcecode>

      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5912"/> and is compatible with the
  2021 ASN.1 syntax <xref target="X680"/>. See <xref target="RFC5280"/> for the 1988 ASN.1 syntax.</t>
      </aside>
      <t>The fields in <tt>SubjectPublicKeyInfo</tt> have the following meaning:</t>
      <ul spacing="normal">
        <li>
          <t><tt>algorithm</tt> is the algorithm identifier and parameters for the
public key (see above).</t>
        </li>
        <li>
          <t><tt>subjectPublicKey</tt> contains the public key.</t>
        </li>
      </ul>
      <t>Algorithm 22 in Section 7.2 of <xref target="FIPS204"/> defines the raw byte string
encoding of an ML-DSA public key. When used in a <tt>SubjectPublicKeyInfo</tt> type,
the <tt>subjectPublicKey BIT STRING</tt> contains this raw byte string encoding of the
public key. When an ML-DSA public key appears outside of a <tt>SubjectPublicKeyInfo</tt> type in an
environment that uses ASN.1 encoding, it could be encoded as an <tt>OCTET STRING</tt> by
using the <tt>ML-DSA-44-PublicKey</tt>, <tt>ML-DSA-65-PublicKey</tt>, and <tt>ML-DSA-87-PublicKey</tt> types
corresponding to the correct key size defined below.</t>
      <t>The <tt>PUBLIC-KEY</tt> ASN.1 types for ML-DSA are defined here:</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
  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))

]]></artwork> (2592))]]></sourcecode>

      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5912"/> and is compatible with the
  2021 ASN.1 syntax <xref target="X680"/>. See <xref target="RFC5280"/> for the 1988 ASN.1 syntax.</t>
      </aside>
      <t><xref target="RFC5958"/> specifies the Asymmetric Key Package's <tt>OneAsymmetricKey</tt> type for
encoding asymmetric keypairs. When an ML-DSA private key or keypair is encoded as
a <tt>OneAsymmetricKey</tt>, it follows the description in <xref target="priv-key"/>.</t>
      <t>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
<tt>ML-DSA-PrivateKey CHOICE</tt> formats defined in <xref target="priv-key"/>. The <tt>seed</tt> format
is <bcp14>RECOMMENDED</bcp14> as it efficiently stores both the private and public key.</t>
      <t><xref target="examples"/> contains example ML-DSA public keys encoded using the
textual encoding defined in <xref target="RFC7468"/>.</t>
    </section>
    <section anchor="key-usage-bits">
      <name>Key Usage Bits</name>
      <t>The intended application for the key is indicated in the <tt>keyUsage</tt>
certificate extension; see <xref section="4.2.1.3" sectionFormat="of" target="RFC5280"/>. If the
<tt>keyUsage</tt> extension is present in a certificate that includes <tt>id-ml-dsa-*</tt>
(where * is 44, 65, or 87 - -- see <xref target="oids"/>) in the <tt>SubjectPublicKeyInfo</tt>,
then the subject public key can only be used
for verifying digital signatures on certificates or CRLs, or those used in an
entity authentication service, a data origin authentication service, an
integrity service, and/or a non-repudiation service that protects against
the signing entity falsely denying some action. This means that the
<tt>keyUsage</tt> extention extension <bcp14>MUST</bcp14> have at least one of the following bits set:</t>
      <artwork><![CDATA[
  digitalSignature
  nonRepudiation
  keyCertSign
  cRLSign
]]></artwork>

<ul spacing="normal">
  <li>digitalSignature</li>
  <li>nonRepudiation</li>
  <li>keyCertSign</li>
  <li>cRLSign</li>
</ul>
      <t>ML-DSA subject public keys cannot be used to establish keys or encrypt data, so the
<tt>keyUsage</tt> extention extension <bcp14>MUST NOT</bcp14> have any of the following bits set:</t>
      <artwork><![CDATA[
   keyEncipherment,
   dataEncipherment,
   keyAgreement,
   encipherOnly, and
   decipherOnly.
]]></artwork>

<ul spacing="normal">
   <li>keyEncipherment</li>
   <li>dataEncipherment</li>
   <li>keyAgreement</li>
   <li>encipherOnly</li>
   <li>decipherOnly</li>
 </ul>
      <t>Requirements about the <tt>keyUsage</tt> extension bits defined in <xref target="RFC5280"/>
still apply.</t>
    </section>
    <section anchor="priv-key">
      <name>Private Key Format</name>
      <t><xref target="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 <tt>ML-DSA.KeyGen_internal(xi)</tt> (algorithm 6).</t>
      <t>"Asymmetric Key Packages" <xref target="RFC5958"/> specifies 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
<tt>OneAsymmetricKey</tt> is replicated below.</t>
      <artwork><![CDATA[
      <sourcecode type="asn.1"><![CDATA[
  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 ]],
    ...
  }
]]></artwork>
  }]]></sourcecode>
      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5958"/> and is compatible with the
  2021 ASN.1 syntax <xref target="X680"/>.</t>
      </aside>
      <t>For ML-DSA private keys, the <tt>privateKey</tt> field in <tt>OneAsymmetricKey</tt> contains one of
the following DER-encoded <tt>CHOICE</tt> structures. The <tt>seed</tt> format is a
fixed 32 byte 32-byte <tt>OCTET STRING</tt> (34 bytes total with the <tt>0x8020</tt> tag and
length) for all security levels, while the <tt>expandedKey</tt> and <tt>both</tt> formats
vary in size by security level:</t>
      <artwork><![CDATA[

      <sourcecode type="asn.1"><![CDATA[
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))
      }
  }
]]></artwork>
  }]]></sourcecode>

      <aside>
        <t>NOTE: The above syntax is from <xref target="RFC5912"/> and is compatible with the
  2021 ASN.1 syntax <xref target="X680"/>. See <xref target="RFC5280"/> for the 1988 ASN.1 syntax.</t>
      </aside>
      <t>The <tt>CHOICE</tt> allows three representations of the private key:</t>
      <ol spacing="normal" type="1"><li>
          <t>The <tt>seed</tt> format (tag <tt>[0]</tt>) contains just the 32-byte seed value (xi)
from which both the expanded private key and public key can be derived
using <tt>ML-DSA.KeyGen_internal(xi)</tt>.</t>
        </li>
        <li>
          <t>The <tt>expandedKey</tt> format contains the expanded private key that was
derived from the seed.</t>
        </li>
        <li>
          <t>The <tt>both</tt> 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.</t>
        </li>
      </ol>
      <t>When encoding an ML-DSA private key in a <tt>OneAsymmetricKey</tt> object, any of
these three formats may be used, though the seed format is <bcp14>RECOMMENDED</bcp14>
for storage efficiency.</t>
      <t>The <tt>privateKeyAlgorithm</tt> field uses the <tt>AlgorithmIdentifier</tt> structure with
the appropriate OID as defined in <xref target="oids"/>. If present, the <tt>publicKey</tt>
field will hold the encoded public key as defined in <xref target="ML-DSA-PublicKey"/>.</t>
<aside>
      <t>NOTE: While the private key can be stored in multiple formats, the seed-only
format is <bcp14>RECOMMENDED</bcp14> as it is the most compact representation. Both the
expanded private key and the public key can be deterministically derived
from the seed using <tt>ML-DSA.KeyGen_internal(xi)</tt>. Alternatively, the public
key can be generated from the private key. While the <tt>publicKey</tt> field
and <tt>expandedKey</tt> format are technically redundant when using the seed-only
format, they <bcp14>MAY</bcp14> be included to enable keypair consistency checks during
import operations.</t>
</aside>
      <t>When parsing the private key, the ASN.1 tag explicitly indicates which
variant of <tt>CHOICE</tt> is present. Implementations should use the context-specific tag <tt>IMPLICIT [0]</tt>
(raw value <tt>0x80</tt>) for <tt>seed</tt>, <tt>OCTET STRING</tt> (<tt>0x04</tt>) for <tt>expandedKey</tt>, and
<tt>SEQUENCE</tt> (<tt>0x30</tt>) for <tt>both</tt> to parse the private key, rather than any
other heuristic like length of the enclosing <tt>OCTET STRING</tt>.</t>
      <t><xref target="examples"/> contains example ML-DSA private keys encoded using the
textual encoding defined in <xref target="RFC7468"/>.</t>
    </section>
    <section anchor="iana-considerations">
      <name>IANA Considerations</name>
      <t>For the ASN.1 module in <xref target="asn1"/>, IANA is requested to assign an has assigned the following 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).</t> (1.3.6.1.5.5.7.0): </t>

<table anchor="oid">
  <name></name>
  <thead>
    <tr>
      <th>Decimal</th>
      <th>Description</th>
      <th>Reference</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>119</td>
      <td>id-mod-x509-ml-dsa-2025</td>
      <td>RFC 9881</td>
    </tr>
  </tbody>
</table>
    </section>
    <section anchor="operational-considerations">
      <name>Operational Considerations</name>
      <section anchor="private-key-format">
        <name>Private Key Format</name>
        <t>An <tt>ML-DSA.KeyGen seed (xi)</tt> represents the <bcp14>RECOMMENDED</bcp14> format for storing
and transmitting ML-DSA private keys. This format is explicitly permitted
by <xref target="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 <tt>ML-DSA.KeyGen_internal(xi)</tt> can be accessed for
this purpose.</t>
        <t>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
recreated, even if the full expanded private key is available. For this reason reason,
it is <bcp14>RECOMMENDED</bcp14> 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 seed xi that is randomly generated at line 1 of Algorithm 1
<tt>ML-DSA.KeyGen</tt> in the returned output.</t>
        <t>When encoding an ML-DSA private key in a OneAsymmetricKey object, any
of these three formats may be used, though the seed format is
<bcp14>RECOMMENDED</bcp14> for storage efficiency.</t>
      </section>
      <section anchor="private-key-consistency-testing">
        <name>Private Key Consistency Testing</name>
        <t>When receiving a private key that contains both the seed and the
expandedKey, the recipient <bcp14>SHOULD</bcp14> 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 <tt>seed</tt> and
<tt>expandedKey</tt> match.</t>
        <t>If the check is done and the <tt>seed</tt> and the <tt>expandedKey</tt> are not consistent,
the recipient <bcp14>MUST</bcp14> reject the private key as malformed.</t>
        <t>The seed consistency check consists of regenerating the expanded form from
the seed via <tt>ML-DSA.KeyGen_internal</tt> <tt>ML-DSA.KeyGen_internal</tt>, and ensuring it is bytewise equal to
the value presented in the private key.</t>
        <t><xref target="example-bad"/> includes some examples of inconsistent seeds and expanded private
keys.</t>
      </section>
      <section anchor="sec-disallow-hash">
        <name>Rationale for disallowing Disallowing HashML-DSA</name>
        <t>The HashML-DSA mode defined in Section 5.4 of <xref target="FIPS204"/> <bcp14>MUST NOT</bcp14> be
used; in other words, public keys identified by
<tt>id-hash-ml-dsa-44-with-sha512</tt>, <tt>id-hash-ml-dsa-65-with-sha512</tt>, and
<tt>id-hash-ml-dsa-87-with-sha512</tt> <bcp14>MUST NOT</bcp14> be in X.509 certificates used for
CRLs, OCSP, certificate issuance issuance, and related PKIX protocols. This restriction
is primarily to increase interoperability.</t>
        <t>ML-DSA and HashML-DSA are incompatible algorithms that require
different <tt>Verify()</tt> routines. This introduces the complexity of
informing the verifier whether to use <tt>ML-DSA.Verify()</tt> or
<tt>HashML-DSA.Verify()</tt>. 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 <tt>ML-DSA</tt> or
<tt>HashML-DSA</tt> at the time of certificate creation. This is anticipated
to cause operational issues in contexts where the operator does not
know whether the key will need to produce pure or pre-hashed signatures
at key generation key-generation time. The External μ (mu) mode described in <xref target="externalmu"/>
avoids all of these operational concerns.</t>
        <t>A minor security reason for disallowing HashML-DSA is that the design of the
ML-DSA algorithm provides enhanced resistance against collision attacks,
compared with HashML-DSA or conventional RSA or ECDSA signature algorithms.
Specifically, ML-DSA prepends the SHAKE256 hash of the public key (<tt>tr</tt>)
to the message to-be-signed prior to hashing, as described in
line 6 of Algorithm 7 of <xref target="FIPS204"/>. This means that in the unlikely
discovery of a collision attack against the SHA-3 family, an attacker
would have to perform a public-key-specific collision search in order
to find message pairs such that <tt>H(tr || m1) = H(tr || m2)</tt> since m2)</tt>, because a
direct hash collision <tt>H(m1) = H(m2)</tt> will not suffice.
HashML-DSA removes this enhanced security property.
In spite of its lack of targeted collision protection, the practical
security risk of using HashML-DSA in X.509 signatures would be
immaterial. That This is because a hash of the issuing CA's public key
is already included in the Authority Key Identifier (AKI) extension extension, which
is signed as part of the tbsCertificate structure.
Even when it is a SHA-1 hash, general second pre-images against
the AKI hash of existing issuing CAs would be impractical.</t>
      </section>
    </section>
    <section anchor="security-considerations">
      <name>Security Considerations</name>
      <t>The Security Considerations section of <xref target="RFC5280"/> applies to this
specification as well.</t>
      <t>The ML-DSA signature scheme is strongly unforgeable under chosen message
attacks (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.</t>
      <t>ML-DSA depends on high quality random numbers that are suitable for
use in cryptography.  The use of inadequate pseudo-random number
generators (PRNGs) to generate such values can significantly undermine
various security properties. For instance, using an inadequate PRNG
for key generation, might allow an attacker to efficiently recover
the private key by trying a small set of possibilities, rather than
brute force
brute-force search the whole keyspace.  The generation of random
numbers of a sufficient level of quality for use in cryptography
is difficult; see Section 3.6.1 of <xref target="FIPS204"/> for some additional
information.</t>
<!-- [rfced] May we rephrase this as "deliver a secure implementation with greater ease"?  In addition, to what does "it" refer - perhaps it should be "they"?

Original:
   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.
-->

      <t>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 comes with a performance overhead.</t>
      <t>ML-DSA offers both deterministic and randomized signing. Signatures
generated with either mode are compatible and a verifyer verifier can't tell
them apart. In the deterministic case, a signature only depends
on the private key and the message to be signed. This makes
the implementation easier to test and does not require
a randomness source during signing. In the randomized case,
signing mixes in a 256-bit random string from an approved random bit
generator (RBG). When randomized, ML-DSA is easier to harden
against fault and hardware side-channel attacks.</t>
<!-- [rfced] Should "key pair" be "keypair" here as used elsewhere in the document?

Original:
   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, unless its private key was compromised.
-->
      <t>A security property that is 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,
unless its private key was compromised.
The digital signature scheme ML-DSA possess possesses 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). A full discussion
of these properties in ML-DSA can be found at <xref target="CDFFJ21"/>.</t>
    </section>
  </middle>
  <back>
    <references anchor="sec-combined-references">
      <name>References</name>
      <references anchor="sec-normative-references">
        <name>Normative References</name>
        <reference anchor="X680" target="https://www.itu.int/rec/T-REC-X.680">
          <front>
            <title>Information Technology -- Abstract Syntax Notation One (ASN.1): Specification of basic notation</title>
            <author>
              <organization>ITU-T</organization>
            </author>
            <date year="2021" month="February"/>
          </front>
          <seriesInfo name="ITU-T Recommendation" value="X.680"/>
          <seriesInfo name="ISO/IEC" value="8824-1:2021"/>
        </reference>
        <reference anchor="X690" target="https://www.itu.int/rec/T-REC-X.690">
          <front>
            <title>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)</title>
            <author>
              <organization>ITU-T</organization>
            </author>
            <date year="2021" month="February"/>
          </front>
          <seriesInfo name="ITU-T Recommendation" value="X.690"/>
          <seriesInfo name="ISO/IEC" value="8825-1:2021"/>
        </reference>
        <reference anchor="CSOR" target="https://csrc.nist.gov/projects/computer-security-objects-register/algorithm-registration">
          <front>
            <title>Computer Security Objects Register</title>
            <author initials="" surname="NIST" fullname="National Institute of Standards and Technology">
              <organization/>
            </author>
            <date year="2024" month="August" day="20"/>
          </front>
        </reference>
        <reference anchor="FIPS204">
          <front>
            <title>Module-lattice-based digital signature standard</title>
            <author>
              <organization/>
            </author>
            <date month="August" year="2024"/>
          </front>
          <seriesInfo name="DOI" value="10.6028/nist.fips.204"/>
          <refcontent>National Institute of Standards and Technology (U.S.)</refcontent>
        </reference>
        <reference anchor="RFC2119">
          <front>
            <title>Key words

<!-- [rfced] [FIPS204] FYI: We have updated the date for use in RFCs to Indicate Requirement Levels</title>
            <author fullname="S. Bradner" initials="S." surname="Bradner"/>
            <date month="March" year="1997"/>
            <abstract>
              <t>In many standards track documents several words are used this reference from August 2023 to signify the requirements in August
2024. We also replaced the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for original URL (https://csrc.nist.gov/projects/post-quantum-cryptography) with the Internet Community, and requests discussion and suggestions for improvements.</t>
            </abstract>
          </front>
          <seriesInfo name="BCP" value="14"/>
          <seriesInfo name="RFC" value="2119"/>
          <seriesInfo name="DOI" value="10.17487/RFC2119"/>
        </reference>
        <reference anchor="RFC8174">
          <front>
            <title>Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words</title>
            <author fullname="B. Leiba" initials="B." surname="Leiba"/>
            <date month="May" year="2017"/>
            <abstract>
              <t>RFC 2119 specifies common key words one that may be used in protocol specifications. This document aims points to reduce the ambiguity by clarifying that only UPPERCASE usage most recent version of the key words have the defined special meanings.</t>
            </abstract>
          </front>
          <seriesInfo name="BCP" value="14"/>
          <seriesInfo name="RFC" value="8174"/>
          <seriesInfo name="DOI" value="10.17487/RFC8174"/>
        </reference> FIPS 204 (https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.204.pdf).
-->
        <reference anchor="RFC5912"> anchor="FIPS204" target="https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.204.pdf">
          <front>
            <title>New ASN.1 Modules for the Public Key Infrastructure Using X.509 (PKIX)</title>
            <author fullname="P. Hoffman" initials="P." surname="Hoffman"/>
            <author fullname="J. Schaad" initials="J." surname="Schaad"/>
            <date month="June" year="2010"/>
            <abstract>
              <t>The Public Key Infrastructure using X.509 (PKIX) certificate format, and many associated formats, are expressed using ASN.1. The current ASN.1 modules conform to the 1988 version of ASN.1. This document updates those ASN.1 modules to conform to the 2002 version of ASN.1. There are no bits-on-the-wire changes to any
            <title>Module-Lattice-Based Digital Signature Standard</title>
            <author>
              <organization abbrev="NIST">National Institute of the formats; this is simply a change to the syntax. This document is not an Internet Standards Track specification; it is published for informational purposes.</t>
            </abstract> and Technology</organization>
            </author>
            <date day="13" year="2024" month="August"/>
          </front>
          <seriesInfo name="RFC" value="5912"/> name="NIST FIPS" value="204"/>
          <seriesInfo name="DOI" value="10.17487/RFC5912"/> value="10.6028/NIST.FIPS.204 "/>
        </reference>
        <reference anchor="RFC5280">
          <front>
            <title>Internet X.509 Public Key Infrastructure Certificate and Certificate Revocation List (CRL) Profile</title>
            <author fullname="D. Cooper" initials="D." surname="Cooper"/>
            <author fullname="S. Santesson" initials="S." surname="Santesson"/>
            <author fullname="S. Farrell" initials="S." surname="Farrell"/>
            <author fullname="S. Boeyen" initials="S." surname="Boeyen"/>
            <author fullname="R. Housley" initials="R." surname="Housley"/>
            <author fullname="W. Polk" initials="W." surname="Polk"/>
            <date month="May" year="2008"/>
            <abstract>
              <t>This memo profiles

<!-- [rfced] [CSOR] FYI: We updated the X.509 v3 certificate and X.509 v2 certificate revocation list (CRL) date for use in the Internet. An overview of this approach and model is provided as an introduction. The X.509 v3 certificate format is described in detail, with additional information regarding reference from 20 August 2024 to 13 June
2025 to match the format and semantics of Internet name forms. Standard certificate extensions are described and two Internet-specific extensions are defined. A set of required certificate extensions is specified. The X.509 v2 CRL format is described in detail along with standard and Internet-specific extensions. An algorithm for X.509 certification path validation is described. An ASN.1 module and examples are date provided in at the appendices. [STANDARDS-TRACK]</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="5280"/>
          <seriesInfo name="DOI" value="10.17487/RFC5280"/>
        </reference> URL.
-->
        <reference anchor="RFC5958"> anchor="CSOR" target="https://csrc.nist.gov/projects/computer-security-objects-register/algorithm-registration">
          <front>
            <title>Asymmetric Key Packages</title>
            <author fullname="S. Turner" initials="S." surname="Turner"/>
            <date month="August" year="2010"/>
            <abstract>
              <t>This document defines the syntax for private-key information and a content type for it. Private-key information includes a private key for a specified public-key algorithm and a set
            <title>Computer Security Objects Register (CSOR)</title>
            <author>
              <organization abbrev="NIST">National Institute of attributes. The Cryptographic Message Syntax (CMS), as defined in RFC 5652, can be used to digitally sign, digest, authenticate, or encrypt the asymmetric key format content type. This document obsoletes RFC 5208. [STANDARDS-TRACK]</t>
            </abstract> Standards and Technology</organization>
            </author>
            <date year="2025" month="June" day="13"/>
          </front>
          <seriesInfo name="RFC" value="5958"/>
          <seriesInfo name="DOI" value="10.17487/RFC5958"/>
        </reference>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.2119.xml"/>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.8174.xml"/>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.5912.xml"/>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.5280.xml"/>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.5958.xml"/>
      </references>
      <references anchor="sec-informative-references">
        <name>Informative References</name>
        <reference anchor="Dilithium" target="https://pq-crystals.org/dilithium/data/dilithium-specification-round3-20210208.pdf">
          <front>
            <title>CRYSTALS-Dilithium Algorithm Specifications and Supporting Documentation</title> Documentation (Version 3.1)</title>
            <author initials="S." surname="Bai">
              <organization/>
            </author>
            <author initials="L." surname="Ducas">
              <organization/>
            </author>
            <author initials="E." surname="Kiltz">
              <organization/>
            </author>
            <author initials="T." surname="Lepoint">
              <organization/>
            </author>
            <author initials="V." surname="Lyubashevsky">
              <organization/>
            </author>
            <author initials="P." surname="Schwabe">
              <organization/>
            </author>
            <author initials="G." surname="Seiler">
              <organization/>
            </author>
            <author initials="D." surname="Stehlé">
              <organization/>
            </author>
            <date day="8" month="2" year="2021"/>
          </front>
        </reference>
        <reference anchor="Fiat-Shamir" target="https://www.iacr.org/archive/asiacrypt2009/59120596/59120596.pdf">
          <front>
            <title>Fiat-Shamir with aborts: Applications to lattice and factoring-based signatures</title>
            <author initials="V." surname="Lyubashevsky">
              <organization/>
            </author>
            <date year="2009"/>
          </front>
          <seriesInfo name="International
          <refcontent>International Conference on the Theory and Application of Cryptology and Information Security" value=""/> Security</refcontent>
        </reference>

<!-- [rfced] [CDFFJ21] Please review. The original date for this reference is 2021. The version of this paper in the Cryptology ePrint Archive was updated in October 2023 (see the "Note" at this URL: https://eprint.iacr.org/2020/1525).

The 2021 version is available for free from IEEE - https://ieeexplore.ieee.org/document/9519420

Should this reference point to the updated (2023) version at the Cryptology
ePrint Archive or to the 2021 version of this paper?
-->

<!-- Note: IEEE version of [CDFFJ21] -->
        <reference anchor="CDFFJ21" target="https://eprint.iacr.org/2020/1525.pdf"> target="https://ieeexplore.ieee.org/document/9519420">
          <front>
            <title>BUFFing signature schemes beyond unforgeability and the case of post-quantum signatures</title>
            <author initials="C." surname="Cremers">
              <organization/>
            </author>
            <author initials="S." surname="Düzlü">
              <organization/>
            </author>
            <author initials="R." surname="Fiedler">
              <organization/>
            </author>
            <author initials="M." surname="Fischlin">
              <organization/>
            </author>
            <author initials="C." surname="Janson">
              <organization/>
            </author>
            <date year="2021"/>
          </front>
          <seriesInfo name="In Proceedings of the 42nd name="DOI" value="10.1109/SP40001.2021.00093"/>
          <refcontent>2021 IEEE Symposium on Security and Privacy" value=""/> Privacy (SP), pp. 1696-1714</refcontent>
        </reference>

<!-- Note: Cryptology ePrint Archive version of
     [CDFFJ21]
        <reference anchor="CDFFJ21" target="https://eprint.iacr.org/2020/1525.pdf">
          <front>
            <title>BUFFing signature schemes beyond unforgeability and the case of post-quantum signatures</title>
            <author initials="C." surname="Cremers">
              <organization/>
            </author>
            <author initials="S." surname="Düzlü">
              <organization/>
            </author>
            <author initials="R." surname="Fiedler">
              <organization/>
            </author>
            <author initials="M." surname="Fischlin">
              <organization/>
            </author>
            <author initials="C." surname="Janson">
              <organization/>
            </author>
            <date month="October" year="2023"/>
          </front>
          <refcontent>Cryptology ePrint Archive, Paper 2020/1525</refcontent>
        </reference>

-->

<!-- [rfced] [NIST-PQC] FYI: We have updated the date for this reference from 20 December 2016 to
28 July 2025 to match the date provided at the URL.
-->
        <reference anchor="NIST-PQC" target="https://csrc.nist.gov/Projects/post-quantum-cryptography">
          <front>
            <title>Post-Quantum Cryptography Project</title> (PQC)</title>
            <author>
              <organization>National
              <organization abbrev="NIST">National Institute of Standards and Technology (NIST)</organization> Technology</organization>
            </author>
            <date year="2016" month="December" day="20"/> year="2025" month="July" day="28"/>
          </front>
        </reference>
        <reference anchor="FIPS204-ExternalMuFAQ" target="https://csrc.nist.gov/csrc/media/Projects/post-quantum-cryptography/documents/faq/fips204-sec6-03192025.pdf">
          <front>
            <title>FIPS 204 Section 6 FAQ</title>
            <author>
              <organization>National
              <organization abbrev="NIST">National Institute of Standards and Technology (NIST)</organization> Technology</organization>
            </author>
            <date year="2025"/>
          </front>
        </reference>
        <reference anchor="RFC7468">
          <front>
            <title>Textual Encodings of PKIX, PKCS, and CMS Structures</title>
            <author fullname="S. Josefsson" initials="S." surname="Josefsson"/>
            <author fullname="S. Leonard" initials="S." surname="Leonard"/>
            <date month="April" year="2015"/>
            <abstract>
              <t>This document describes and discusses the textual encodings of the Public-Key Infrastructure X.509 (PKIX), Public-Key Cryptography Standards (PKCS), and Cryptographic Message Syntax (CMS). The textual encodings are well-known, are implemented by several applications and libraries, and are widely deployed. This document articulates the de facto rules by which existing implementations operate and defines them so that future implementations can interoperate.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="7468"/>
          <seriesInfo name="DOI" value="10.17487/RFC7468"/>
        </reference>
        <reference anchor="RFC3647">
          <front>
            <title>Internet X.509 Public Key Infrastructure Certificate Policy and Certification Practices Framework</title>
            <author fullname="S. Chokhani" initials="S." surname="Chokhani"/>
            <author fullname="W. Ford" initials="W." surname="Ford"/>
            <author fullname="R. Sabett" initials="R." surname="Sabett"/>
            <author fullname="C. Merrill" initials="C." surname="Merrill"/>
            <author fullname="S. Wu" initials="S." surname="Wu"/>
            <date month="November" year="2003"/>
            <abstract>
              <t>This document presents a framework to assist the writers of certificate policies or certification practice statements for participants within public key infrastructures, such as certification authorities, policy authorities, and communities of interest that wish to rely on certificates. In particular, the framework provides a comprehensive list of topics that potentially (at the writer's discretion) need to be covered in a certificate policy or a certification practice statement. This document supersedes RFC 2527.</t>
            </abstract>
          </front>
          <seriesInfo name="RFC" value="3647"/>
          <seriesInfo name="DOI" value="10.17487/RFC3647"/>
        </reference>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.7468.xml"/>
        <xi:include href="https://bib.ietf.org/public/rfc/bibxml/reference.RFC.3647.xml"/>
      </references>
    </references>
    <?line 726?>

    <section anchor="asn1">
      <name>ASN.1 Module</name>
      <t>This appendix includes the ASN.1 module <xref target="X680"/> for the ML-DSA.  Note that
as per <xref target="RFC5280"/>, certificates use the Distinguished Encoding Rules; see
<xref target="X690"/>. This module imports objects from <xref target="RFC5912"/>.</t>
      <sourcecode type="asn.1" markers="true"><![CDATA[
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)
  id-mod-x509-ml-dsa-2025(119) }

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
]]></sourcecode>
    </section>
    <section anchor="security-strengths">
      <name>Security Strengths</name>
      <t>Instead
<!-- [rfced] Please consider whether "traditional" has a universal meaning as used here:

   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.
-->

      <t>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 attacks. 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 references, respectively.
Levels 2 and 4 use collision search for SHA-256 and SHA-384
as reference.</t> references.</t>
      <t>The parameter sets defined for NIST security levels 2, 3 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.</t>
      <figure
      <table anchor="ML-DSAParameters">
        <name>ML-DSA Parameters</name>
        <artwork><![CDATA[
|=======+=======+=====+========+========+==========+==========|
| 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   |
|=======+=======+=====+========+========+==========+==========|
]]></artwork>
      </figure>
	<thead>
	  <tr>
	    <th>Level</th>
	    <th>(k,l)</th>
	    <th>eta</th>
	    <th>Sig. (B)</th>
	    <th>Public Key(B)</th>
	    <th>Private Seed(B)</th>
	    <th>Private Expand(B)</th>
	  </tr>
	</thead>
	<tbody>
	  <tr>
	    <td>2</td>
	    <td>(4,4)</td>
	    <td>2</td>
	    <td>2420</td>
	    <td>1312</td>
	    <td>32</td>
	    <td>2560</td>
	  </tr>
	  <tr>
	    <td>3</td>
	    <td>(6,5)</td>
	    <td>4</td>
	    <td>3309</td>
	    <td>1952</td>
	    <td>32</td>
	    <td>4032</td>
	  </tr>
	  <tr>
	    <td>5</td>
	    <td>(8,7)</td>
	    <td>2</td>
	    <td>4627</td>
	    <td>2592</td>
	    <td>32</td>
	    <td>4896</td>
	  </tr>
	</tbody>
      </table>

    </section>
    <section anchor="examples">
      <name>Examples</name>
      <t>This appendix contains examples of ML-DSA private keys, public keys,
certificates, and inconsistent seed and expanded private keys.</t>
      <section anchor="example-private">
        <name>Example Private Keys</name>
        <t>The following examples show ML-DSA private keys in different formats,
all derived from the same seed <tt>000102...1e1f</tt>. For each security level,
we show the seed-only format (using a context-specific <tt>[0]</tt> primitive
tag with an implicit encoding of <tt>OCTET STRING</tt>), the <tt>expanded</tt> format,
and <tt>both</tt> formats together.</t>
<aside>
        <t>NOTE: All examples use the same seed value, showing how the same seed
produces different expanded private keys for each security level.</t>
</aside>
        <section anchor="ml-dsa-44-private-key-examples">
          <name>ML-DSA-44 Private Key Examples</name>
          <t>Each of the examples includes the textual encoding <xref target="RFC7468"/> followed by
the so-called "pretty print"; the private keys are the same.</t>
          <section anchor="seed-format">
            <name>Seed Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMRBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
]]></artwork>
}]]></artwork>
          </section>
          <section anchor="expanded-format">
            <name>Expanded Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  OCTET_STRING {
    OCTET_STRING { `d7b2b47254aae0db45e7930d4a98d2c97d8f1397d178
9dafa17024b316e9bec939ce0f7f77f8db5644dcda366bfe4734bd95f435ff9a
613aa54aa41c2c694c04329a07b1fabb48f52a309f11a1898f848e2322ffe623
ec810db3bee33685854a88269da320d5120bfcfe89a18e30f7114d83aa404a64
6b6c997389860d12522ee0006e2384819186619b260d118664d4a62822184482
402898146148a6614c4248a19208c2382951244808a125c2083108c471201409
14836c18a78084106ec9c07022b56408b0610c07049812445188695900462293
2041062e42b64c01164914284c41a85180460a5116515a0820022244dc9849d1
3251e13065d3c08592a85112a1640039220946621cc70cd9086dd00626524085
80443091062c50c80924c5841a966d4a982c99066da4443220a7645a326e11b5
7020926124138e04852c0a4872c8a051d3082a99208058242024074e59148810
a46460c06de0b28d1b1909203422c024410943710a212061a2015222521b8080
9a340013934dd3322922170a9892691a14512027219cc02062a2814818691a85
4d8344695b2041031242cb184601a90d0c023183b0215a224ac89205d9906904
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5c82280bb46c03428d53100c14010ee1365288842491020a63462620062911c2
28d0204802b36ca236095a8648cbb4618b4662c440821a890910024d24b24520
122524c90588288cc9c04d5948220a276ec134644c90605b4450828649438804
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0da4068dd3947142b26c1a84611ba32842b42808a0711ac531e0a04c01376524
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c53a61d867420b04f092355753bc65a06368fd41295fd09924132c6f91f67964
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bf431afab34efbdf782474d2fd57118f646214934ed99cba3b003e8d67a3836f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` }
  }
}
]]></artwork>
}]]></artwork>

          </section>
          <section anchor="both-format">
            <name>Both Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----
]]></artwork>

            <artwork><![CDATA[
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
0914836c18a78084106ec9c07022b56408b0610c070498124451886959004622
932041062e42b64c01164914284c41a85180460a5116515a0820022244dc9849
d13251e13065d3c08592a85112a1640039220946621cc70cd9086dd006265240
8580443091062c50c80924c5841a966d4a982c99066da4443220a7645a326e11
b57020926124138e04852c0a4872c8a051d3082a99208058242024074e591488
10a46460c06de0b28d1b1909203422c024410943710a212061a2015222521b80
809a340013934dd3322922170a9892691a14512027219cc02062a2814818691a
854d8344695b2041031242cb184601a90d0c023183b0215a224ac89205d99069
04306a4b064ad2b2011c404081423252327254a6405a18100c321292c2805212
625c82280bb46c03428d53100c14010ee1365288842491020a63462620062911
c228d0204802b36ca236095a8648cbb4618b4662c440821a890910024d24b245
20122524c90588288cc9c04d5948220a276ec134644c90605b44508286494388
0443b28c603080a2882d84a46d8ca629d0c68442064689885100a98d01498de4
380da4068dd3947142b26c1a84611ba32842b42808a0711ac531e0a04c013765
242862142890091061d940221b3360090292d02481200408491844a3222d5c88
44149808a446610195640b390a0c9450ca406ad2b220c0380182308e13b90891
8084148829c0189112350da02422e20406d9c2850428121cc989180272d24029
c20812d8062a9994719bb8682384291a2289144511dc82445096450c4484c0b2
049aa60543862c44326e88442120a84c9a3070e3b82d63268803254903438c48
a809ca147253344e1243081ba704593022d99480e234228142129c302a943426
6104452426281346094a326d11280918b82562281113410d41b21190844c8b12
12a2c688c9c030220606d2188e848630904452128831d9207113c52843060e03
3060cca6845826524c88011ef72562c85ffa43acfa49217f2b172d7bbc14620e
6d980a71aabbdf0c45e9a206ecb1423fee15decc17601300149d9223cd6e6c6e
1fa8e41fc7c64938ab68905fd3dcda50d87082e7d0d71d1bc9b2b84c85523ca8
fe6cad294adf83be15b108ff721d0cc87bc3dd3a7590184b0e845663a91fc9e1
c3c53a61d867420b04f092355753bc65a06368fd41295fd09924132c6f91f679
64c142674a725c343914c4cecf58c074bcaf4558c97bf7911e07aa6d0938f2ee
2bb3c1a8c595d635e84342fdea01dc24b211ad2fc281cf77e59110c7abc54bf0
c86d480b9be276471dc9d603cee98cfdab3e9fcfb703793560549ea4450fa7b3
3fb9169c44b4d25fb9c457f49791cd3da03eac96095813c105132ccda4e63e49
228cd23d8a1f37856f142d93b90db09f82af89258c63aab8047a80c036c9357e
a2046f8dc6354f0c5295f342bb417d3cfeb0b1fd33622c29e14cbbd92e1363c6
5ebd4504b7512329b9670e32e1b2c67a54e7f1a55f8b9f9ea04e8ca3a705e62a
3c5e637374afb7aeb6ddea612cde28f01a202d7aa4e34722d27dd3f9b89894d0
19fd5d4d7119efe3723bba104cb8bb0981e074de3afe200daaaead826cc45f24
4dbf431afab34efbdf782474d2fd57118f646214934ed99cba3b003e8d67a383
6f6f19fc41910ce5163ee3ae99eb84d514eb761e63684ea56f9791d2dd4aac6e
6168b948c817f75a222acb0e8cdc03cc4afe8f67157e1a363b7faeff9f172b98
913677c5a1dd085e9ee4c22052c1af58193116673dcd3bfc5f34b855dcc6c778
85649e9e71f43d4aea0f4b72ca7eda0578ba13d31a658d2d060a9a66ff69ed1b
e7997a2fb1d2723d38f9bfabe18f8e7b3cda906e4e9b5e942c8eaeb296070ebf
d364947a940cc978bed66b37749e6d5dcd7be8c494440e2b84cecfefb98c0bed
fb3c41e3359d2cd7197fbe720c48aa6c6b6465c1ee63e3569c2adc744491370b
7f7826fe0b77a1d19d64101d032b918106b42d2ef73747e5601fe4ba50f23ede
521f031a817d15294a43722e8378784b6db0cf1ba9e8ae911d9201b9ce9cc301
9c6f5c27cb98da26144b64225a7c932b30f761e78a2d59a1d8b83ec6344a2f6d
d47e765706d00bf4a79a6a926c3ba91d812c8f2c797ab1796709e5d168567782
93529f0286d015c3b5399619642a333e9e593d6e3f5353994208e9e6a332851d
7f652522a928b917e27e2d6d42137dfe2ebfa6fb1c67b26c0254528685f7ebdb
e315a68eaa2da769e8a9f42d3e60007c71330926b2c0012d83ead4e4fd1ed872
ccd1972201d2b027f3545ac2d30cd78bc1d740feccbc6fc2a0446c6e30eac51f
5a69098aa2d447f2085b4e4e4b92ccc26921d2de478518cd090ce267aea2d27a
da57fd88b4976d89fb843cdccf49a76ca2679e6801bfa7fb031896fb50629704
b9923936bb5dd385311121cadfb11995e59b73034cf67ed03ab813867648d025
828087e949a9afd16b95d72d99b1edca257aac132ffb7a0709aed5a9c0ff05fb
0f2bbf28409eed7b5f5801be964ced019e1cb7851d3851f10290674e19ffb008
b301c4acf641a2bb14216e1d69cabf52b5ef227496b0f30799a855d117fad374
4a6fa33503ea798b52ddd7ee5426609dbfcd3f0c13b164d6c051f7ed4a119719
a712e388d328402081ff1354b554d2c237afed3b151c4ba8e9f4bdeb8499a306
6e26bbc69e8af089dec71731d1dc529eab17ef7374734c0fe475494c83836bdd
34a03b9bc89914716061bfb98ec6e61c3ed4438edcaf25243c647086b9ea7018
b0d9a8a0b00cecb00abde2498d69c2336101a772cbe4f571523f51bd05882cdf
358b849cc140aa1faf22423a12851ce0e33fd48975a4959fa5c5fe418c939081
91ab6e741b77bfe02cbd698ee795c466d615619e6441382c6eac01834ee9ab73
cea80bbe235c78da91bd79b6f82f899785d68700d393e675c2224d6b7a1ad213
20495679adaed70167b50866713a53109db7b6f7d81304ecdfd83b319b1ef248
306b45ad29e7ddcc863dac56048b5d69ea175011f7614c00a86a863cde1872a8
932878b9ac7e1ac5bda4997b72064f0cd75f4c814e034de11acb9013cf7ea926
b4e7eaace070c7ba2188efad2e431e1223d45dd05c4d8403c2e45cee6413ecbe
7527e873e455c4e610a61839aacc0bd56d2483e78f298b66a478eb2f558cbafc
a86be847baeb02c5b216c8cd88fea4df249b09e670a20703abac24b0a91abc4a
5646601442ba10becfd30993880051d07f56a05a9379e7a8e6befee3f22faa10
6398f7706006e42e9be1ef89d25c272f11a95095c587d713732284de9dbd3c72
17b0689e21d8eb0ff69668` }
    }
  }
}
]]></artwork>
}]]></artwork>

          </section>
        </section>
        <section anchor="ml-dsa-65-private-key-examples">
          <name>ML-DSA-65 Private Key Examples</name>
          <t>Each of the examples includes the textual encoding <xref target="RFC7468"/> followed by
the so-called "pretty print"; the private keys are the same.</t>
          <section anchor="seed-format-1">
            <name>Seed Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMSBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
]]></artwork>
}]]></artwork>

          </section>
          <section anchor="expanded-format-1">
            <name>Expanded Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
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  }
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96cb884c3c93994748a56f41283bfc41fb89052153a894588c3cb9017f3d6632
6c985637e575acb812346342654025d602de3ba940c19ac1a633dffda977b529
b8013e19c1d6d0680f4dae62c924450ae66aab82f21473061dab3d62b247f907
e3551939ad3f5465e9d08a82bfea17eea1b6b2b923757477f993000b2f43b70f
28aaab1fe9a26ad1fd3361616c0b0e242fe76604b7033a1f30e97e28f526ca3c
880fe2b8d9d1b0c9ff188b31cb9d97425acab9b216d98a6ae355e583da71e886
4ee3d16b0759796190ef545c1e62bfef92af6ca147b13244d6c892fc8ef223ab
3f43f924c2f466097ee8` }
  }
}
]]></artwork>
}]]></artwork>
          </section>
          <section anchor="both-format-1">
            <name>Both Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
MIIP/gIBADALBglghkgBZQMEAxIEgg/qMIIP5gQgAAECAwQFBgcICQoLDA0ODxAR
EhMUFRYXGBkaGxwdHh8Egg/ASGg9kZeOMes93biwRzSC0riKX2JZSf2PWKVh5pa9
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cyRYbHHf2R2QHWIYWBkP7NAMzRELusWflsuITDyTmUdIpW9BKDv8QfuJBSFTqJRY
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D02uYskkRQrmaquC8hRzBh2rPWKyR/kH41UZOa0/VGXp0IqCv+oX7qG2srkjdXR3
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/xiLMcudl0JayrmyFtmKauNV5YPaceiGTuPRawdZeWGQ71RcHmK/75KvbKFHsTJE
1siS/I7yI6s/Q/kkwvRmCX7o
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
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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28c21c7ab7faed2b21cfade2da55223aaab2af9b41c7332341746341b39aa2f4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` }
    }
  }
}
]]></artwork>
}]]></artwork>
          </section>
        </section>
        <section anchor="ml-dsa-87-private-key-examples">
          <name>ML-DSA-87 Private Key Examples</name>
          <t>Each of the examples includes the textual encoding <xref target="RFC7468"/> followed by
the so-called "pretty print"; the private keys are the same.</t>
          <section anchor="seed-format-2">
            <name>Seed Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
MDQCAQAwCwYJYIZIAWUDBAMTBCKAIAABAgMEBQYHCAkKCwwNDg8QERITFBUWFxgZ
GhscHR4f
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  OCTET_STRING {
    [0 PRIMITIVE] { `000102030405060708090a0b0c0d0e0f10111213141
5161718191a1b1c1d1e1f` }
  }
}
]]></artwork>
}]]></artwork>

          </section>
          <section anchor="expanded-format-2">
            <name>Expanded Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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YqfwGqbWaT3uyQsjxmmFoCMH4KHK5ZimcyTboPUvIkMidekyVwZcO35eHP4d/U0N
8IbfISQ0FKLSfiAjCoKb5OtMgsFtNfeLDl4ZgzLgAHS7ZGEvqxfUyJccto5e2rA2
nxFXs0aavYOE4tlVPxt454bh7p0LmNOfg8zs830evTqdY67HZhZKEBcaT9jGPa8Y
LEISWMX1KapVy3664uFlIxXh9x6KdBMUENAyR+3hHTTbkfbwiqJHj9eJZ5wElJ9x
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rdg39vRAzDYPk/Mv7oqWY3Esa704yEq3tUgj7DY+t+QutZ/B/OYPvVUwez7IX9na
8yBte0s5F/HIt6kuPGfYmID98uR/WgyZRZXbFwr0G6v1oltNwcQt1qnbJx52TeL7
AVpJqFDHkZvkcAajNuLjJf3lOsWZVU0KfeTvRexAw51rr/MRvu512J4CrTH0vkvS
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Al2UU0c2bALz3WNo1OR+hdPSqXBb1XlhhS5aV5+TscUUxTn0nqEWOipJOw78tH9H
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kx3BcpjM73ZkXH0zCgXCzkD4m4VGjzV6IXdR4VRjEwTsTgS7RbNniQnHSvUc43A2
TY9PfrHmHgAodCnJlh3oMiypomKbEwnYAOkrwdxQVdzHl/M4ZusM/Y1JAlDUj/yo
Ai9JKQ4tU3YWL7qpgtFkU8gls19lFWNeqSvqcjZ7qlTeP56uppVCqBpBJ/ccuqJX
8yT+/vFPCPvWWgSc0vs2JZSo4j/xomF9tbFY9vAc9Qqw7ZXG5wmEEWQQiwbhtAqw
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uk+KC8GZ4Qz2vXehT91qBgk1FDSOOol0Q0roo2djaca+LPkOZys0P84ErGsi4M9H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r7n6Uo8oosoqp4C5QDg8CZqmWgB0uD/R8Lxbe15Gwl5Ug4s8vPyV+H8dRxs7qJRD
T6WJUv3Ld/FhNyaTMG26To8hbRyOXK/w/oNgpRxgdjZEFp/caoJn8uP5CaYbKmeL
zmrpBAOoNrGnt+jNi1TDcIep4URG2V5pCNLu2/zGU+Av33cfcBp5ueWibtCpR4Qg
cPO1cBdCIRIZ52F2LDfw0KHRuXUP7ld+EggRXGasB+wJHmo/xKpqJTvLqGjt0xVN
yvUWL2FehUkKbKNC80xDrGGj6mv+79hQ4ZDrHY2k0otezusWeMAkM+zV1IslNkBC
V+jKe+9YVfK4E+0vTECURaMxfJvho1ri+00rh5IbkEvywU21FM7gRSUc/CdjdNsV
yZ3qFazeGXxutSSYjjm2Moe+uGdoZaqjutG0O4yrFcvyekmHWeMgOr82npckLwsB
VBSfFKwjPNtzoit/uPCTJb8qzoO7a124oSGitoIUmmkTHMzlIimECxE/x7C8xYQF
v+h/H5X/wulvxVllZ+lDZN+qbZ1abrma5N30JA==
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
SEQUENCE {
  INTEGER { 0 }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  OCTET_STRING {
    OCTET_STRING { `9792bcec2f2430686a82fccf3c2f5ff665e771d7ab41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6d9d5a6eb99ae4ddf424` }
  }
}
]]></artwork>
}]]></artwork>
          </section>
          <section anchor="both-format-2">
            <name>Both Format</name>
            <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
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WWVn6UNk36ptnVpuuZrk3fQk
-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

            <artwork><![CDATA[
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
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c14db514cee045251cfc276374db15c99dea15acde197c6eb524988e39b63287
beb8676865aaa3bad1b43b8cab15cbf27a498759e3203abf369e97242f0b0154
149f14ac233cdb73a22b7fb8f09325bf2ace83bb6b5db8a121a2b682149a6913
1ccce52229840b113fc7b0bcc58405bfe87f1f95ffc2e96fc5596567e94364df
aa6d9d5a6eb99ae4ddf424` }
    }
  }
}
]]></artwork>
}]]></artwork>
          </section>
        </section>
      </section>
      <section anchor="example-public">
        <name>Example Public Keys</name>
        <t>The following is the ML-DSA-44 public key corresponding to the private
key in the previous section. The textual encoding <xref target="RFC7468"/> is
followed by the so-called "pretty print"; the public keys are the same.</t>
        <artwork><![CDATA[
-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----
]]></artwork> KEY-----]]></artwork>

        <artwork><![CDATA[
SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.17 }
  }
  BIT_STRING { `00` `d7b2b47254aae0db45e7930d4a98d2c97d8f1397d17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` }
}
]]></artwork>
}]]></artwork>

        <t>The following is the ML-DSA-65 public key corresponding to the private
key in the previous section.  The textual encoding <xref target="RFC7468"/> is
followed by the so-called "pretty print"; the public keys are the same.</t>

        <artwork><![CDATA[
-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----
]]></artwork> KEY-----]]></artwork>

        <artwork><![CDATA[
SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.18 }
  }
  BIT_STRING { `00` `48683d91978e31eb3dddb8b0473482d2b88a5f62594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` }
}
]]></artwork>
}]]></artwork>

        <t>The following is the ML-DSA-87 public key corresponding to the private
key in the previous section.  The textual encoding <xref target="RFC7468"/> is
followed by the so-called "pretty print"; the public keys are the same.</t>

        <artwork><![CDATA[
-----BEGIN PUBLIC KEY-----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5VvgYiIvi6kKl0BzMmsxISrs4KNKYA==
-----END PUBLIC KEY-----
]]></artwork> KEY-----]]></artwork>

        <artwork><![CDATA[
SEQUENCE {
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
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}
]]></artwork>
}]]></artwork>
      </section>
      <section anchor="example-certificates">
        <name>Example Certificates</name>
        <aside>
          <t>The example certificates in this section have key usage bits set to
<tt>digitalSignature</tt>, <tt>keyCertSign</tt>, and <tt>cRLSign</tt> to lessen the number of
examples, i.e., brevity. Certificate Policies (CPs) <xref target="RFC3647"/>
for production CAs should consider whether this combination is
appropriate.</t>
        </aside>
        <t>The following is a self-signed certificate for the ML-DSA-44 public key in the
previous section. The textual encoding <xref target="RFC7468"/> is followed by the
so-called "pretty print"; the certificates are the same.</t>
        <artwork><![CDATA[
-----BEGIN CERTIFICATE-----
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RUdidnuDkpSm1hIoLz4/SW2Tm6C2u9La5evu7xAfIy1ul8LE3/P0AAAAAAAAAAAA
AAAAABcmOEM=
-----END CERTIFICATE-----
]]></artwork> CERTIFICATE-----]]></artwork>

        <artwork><![CDATA[
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
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8f315ba5925e8edefa679369a2202766151f16a965f9f81ece76cc070b55869e
4db9784cf05c830b3242c8312` }
    }
    [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 }
  }
  BIT_STRING { `00` `67a8951fc308e12f6867611fdc254bf4bcb514f7f5f
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5e1b306d7ec4be2558caa091e2fcc32c355702e971a3f3ac1da421317f3160e2
d7a27b71ed2342065bbf104070c3942494a5257607798a0afb6c3c6cbd9eaf1f
2ff070d233637454762767b839294a6d612282f3e3f496d939ba0b6bbd2dae5e
beeef101f232d6e97c2c4dff3f40000000000000000000000000017263843` }
}
]]></artwork>
}]]></artwork>

        <t>The following is a self-signed certificate for the ML-DSA-65 public key in the
previous section. The textual encoding <xref target="RFC7468"/> is followed by the
so-called "pretty print"; the certificates are the same.</t>

        <artwork><![CDATA[
-----BEGIN CERTIFICATE-----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IA==
-----END CERTIFICATE-----
]]></artwork> CERTIFICATE-----]]></artwork>

        <artwork><![CDATA[
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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00000000000000000000000000000000000060d0f131c20` }
}
]]></artwork>
}]]></artwork>

        <t>The following is a self-signed certificate for the ML-DSA-87 public key in the
previous section. The textual encoding <xref target="RFC7468"/> is followed by the
so-called "pretty print"; the certificates are the same.</t>

        <artwork><![CDATA[
-----BEGIN CERTIFICATE-----
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AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAQMEhUdHiUs
-----END CERTIFICATE-----
]]></artwork> CERTIFICATE-----]]></artwork>

        <artwork><![CDATA[
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 {
            OCTET_STRING { `89886750b57c24db3fc012e61ede59753337
374f` }
          }
        }
      }
    }
  }
  SEQUENCE {
    OBJECT_IDENTIFIER { 2.16.840.1.101.3.4.3.19 }
  }
  BIT_STRING { `00` `08783d565efc7b97440f8bd9818b466be538dbccedb
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0000000000000000000000000000000000000000000040c12151d1e252c` }
}
]]></artwork>
}]]></artwork>
      </section>

      <section anchor="example-bad">
        <name>Example Inconsistent Seed and Expanded Private Keys</name>
        <aside>
          <t>WARNING: These private keys are purposely bad bad; do not use them in
  production systems.</t>
        </aside>
        <t>The following examples demonstrate inconsistent seed and expanded private keys.</t>
        <t>Three <tt>ML-DSA-44-PrivateKey</tt> examples of inconsistent seed and
expanded private keys follow:</t>
        <ol spacing="normal" type="1"><li> type="1">
	  <li>
            <t>The first <tt>ML-DSA-PrivateKey</tt> example includes the <tt>both CHOICE</tt> , i.e., both <tt>seed</tt> and <tt>expandedKey</tt> are included. The <tt>seed</tt> and <tt>expanded</tt> values can be checked for inconsistencies.</t>
          </li>
          <li>
            <t>The second <tt>ML-DSA-PrivateKey</tt> example includes only <tt>expandedKey</tt>.  The public key fails to match the <tt>tr</tt> hash value in the private key.</t>
          </li>
          <li>
            <t>The third <tt>ML-DSA-PrivateKey</tt> example also includes only <tt>expandedKey</tt>. The private <tt>s_1</tt> and <tt>s_2</tt> vectors imply a <tt>t</tt> vector whose private low bits do not match the <tt>t_0</tt> vector portion of the private key (its high bits <tt>t_1</tt> are the primary content of the public key).</t>
          </li>
        </ol>
<!-- [rfced] We are having trouble understanding "then" in this sentence.  Does it indicate a later step or perhaps it should be deleted altogether?

Original:
   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.

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

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

        <t>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 <tt>tr</tt> or <tt>t_0</tt>.</t>
        <t>The following is the first example:</t>
        <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----
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cWFz1hzdOz1ZaMv89k3iYgajdmNIHeUQdz8wwc1621onspo5YlzuruFSorrzz/Ru
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qPVHh3zqURGGSdu1y29uK6M2vjUp0w8NfyuvzbHIy2hJz3Py9kiZotfF4kOgU25D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-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

        <t>The following is the second example:</t>

        <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

        <t>The following is the third example:</t>

        <artwork><![CDATA[
-----BEGIN PRIVATE KEY-----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-----END PRIVATE KEY-----
]]></artwork> KEY-----]]></artwork>

      </section>
    </section>
    <section anchor="externalmu">
      <name>Pre-hashing
      <name>Pre-Hashing (Externalμ-ML-DSA)</name>
      <t>Some applications require pre-hashing that ease operational
requirements around large or inconsistently-sized payloads.
When signing with pre-hashing, the signature generation
process can be separated into a pre-hash step requiring only the message
and other public information, and a core signature
step which that uses the public key.</t>
      <t>In the context of ML-DSA, pre-hashing can be performed with
the HashML-DSA algorithm defined in Section 5.4 of <xref target="FIPS204"/>.
ML-DSA itself supports a an External μ pre-hashing mode mode, which
externalizes the message pre-hashing originally performed inside
the signing operation. This mode is also laid out in
<xref target="FIPS204-ExternalMuFAQ"/>. This document specifies
only the use of ML-DSA's External μ mode, and not HashML-DSA,
in PKIX for reasons laid out in <xref target="sec-disallow-hash"/>.</t>
      <t>Implementations of ML-DSA using the External μ pre-hashing mode requires the following
algorithms, which are modified versions of the algorithms presented in <xref target="FIPS204"/>.
The nomenclature used here has been modified from the NIST FAQ <xref target="FIPS204-ExternalMuFAQ"/>
for clarity.</t>
<!-- [rfced] Should "pre-hash" and "prehash" be consistent throughout?  For example:

Original:
   Pre-hash operation:
...
Figure 2: Computeμ prehash operation
-->

      <t>Pre-hash operation:</t>
      <figure anchor="fig-externalmu-ml-dsa-external">
        <name>Computeμ prehash operation</name> Prehash Operation</name>
<!-- [rfced] The following lines extended beyond 72 characters.  We folded the lines as follows.  Please let us know if any corrections are needed.

Original:
  # The functions `BytesToBits` and `IntegerToBytes` are defined in FIPS 204.

Current:
  # The functions `BytesToBits` and `IntegerToBytes` are defined
  # in FIPS 204.

Original:
    # ... identical to FIPS 204 Algorithm 7, but with Line 6 replaced with

Current:
    # ... identical to FIPS 204 Algorithm 7, but with Line 6 replaced
    # with
-->
        <artwork><![CDATA[
Computeμ(pk, M, ctx):

  # Referred to as 'Externalμ-ML-DSA.Prehash(pk, M, ctx)'
  # in the FIPS 204 FAQ.
  # M is the message, a bit-string
  # μ and ctx are byte-strings.
  # ctx is the context string, which defaults to the empy empty string.

  μ = H(BytesToBits(H(pk, 64) || IntegerToBytes(0, 1) ||
                IntegerToBytes(|ctx|, 1) || ctx) || M, 64)
  # The functions `BytesToBits` and `IntegerToBytes` are defined
  # in FIPS 204.
  return μ
]]></artwork> μ]]></artwork>
      </figure>
      <t>Sign operations:</t>
      <figure anchor="fig-externalmu-ml-dsa-internal">
        <name>The operations Operations for signing Signing μ</name>
<!-- [rfced] We are having trouble parsing "defaulting is Externalμ to false".  Please review.

Original:
ML-DSA.Signμ_internal(sk, M', rnd, isExternalμ=false):
    # μ can be passed as an argument instead of M'
    # defaulting is Externalμ 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.
-->

        <artwork><![CDATA[
Signμ(sk, μ):

  # Referred to as 'Externalμ-ML-DSA.Sign(sk, μ)'
  # in the FIPS 204 FAQ.

  if |μ| != 64 then
    return error  # return an error indication if the input μ 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 = Signμ_internal(sk, μ, rnd, isExternalμ=true)
  return sigma

ML-DSA.Signμ_internal(sk, M', rnd, isExternalμ=false):
    # μ can be passed as an argument instead of M'
    # defaulting is Externalμ to false means that
    # this modified version of Sign_internal can be used
    # in place of the original without interfering with
    # the functioning of pure ML-DSA mode.
    # ... identical to FIPS 204 Algorithm 7, but with Line 6 replaced
    # with
  6: if (isExternalμ):
       μ = M'
     else:
       μ = H(BytesToBits(tr) || M', 64)
]]></artwork> 64)]]></artwork>
      </figure>
      <t>There is no need to specify an External μ <tt>Verify()</tt> routine because
this is identical to the original <tt>ML-DSA.Verify()</tt>. This makes External
μ mode simply an internal optimization of the signer, and
allows an ML-DSA key to sometimes be used with the "one-shot" <tt>Sign()</tt>
API and to sometimes be used with the External μ API without any interoperability concens.</t> concerns.</t>
      <t>The External μ mode requires the <tt>Computeμ</tt> routine to have access to the
hash of the signer's public key key, which may not be available in some architectures,
or require fetching it. That may allow for mismatches between <tt>tr</tt> and <tt>sk</tt>.
At worst, this will produce a signature which that will fail to verify under the
intended public key since a compliant <tt>Verify()</tt> routine will
independently compute <tt>tr</tt> from the public key. That This
is not believed to be a security concern since <tt>μ</tt> is never used as-is
within <tt>ML-DSA.Sign_internal()</tt> (Algorithm 7 in <xref target="FIPS204"/>). Rather,
it is hashed with values unknown to an attacker on lines 7 and 15.
Thus, a signing oracle exposing <tt>Signμ()</tt> does not leak any bits of the secret
key. The External μ mode also requires SHAKE256 to be available to the
<tt>Computeμ</tt> routine.</t>
    </section>
    <section numbered="false" anchor="acknowledgments">
      <name>Acknowledgments</name>
      <t>The authors wish to thank the following people for their
      contributions to this document: Corey Bonnell, Dierdre Connolly, Viktor Dukhovni, Russ Housley,
Alicja Kario, Mike Ounsworth, <contact fullname="Corey Bonnell"/>,
      <contact fullname="Dierdre Connolly"/>, <contact fullname="Viktor
      Dukhovni"/>, <contact fullname="Russ Housley"/>, <contact
      fullname="Alicja Kario"/>, <contact fullname="Mike Ounsworth"/>, and Daniel
      <contact fullname="Daniel Van Geest.</t> Geest"/>.</t>
      <t>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 <contact fullname="Tony Arcieri"/>,
      <contact fullname="Bob Beck"/>, <contact fullname="Dmitry Belyavskiy"/>,
      <contact fullname="David Benjamin"/>, <contact fullname="Daniel
      Bernstein"/>, <contact fullname="Uri Blumenthal"/>, <contact
      fullname="Theo Buehler"/>, <contact fullname="Stephen Farrell"/>,
      <contact fullname="Jean-Pierre Fiset"/>, <contact fullname="Scott
      Fluhrer"/>, <contact fullname="Alex Gaynor"/>, <contact fullname="John
      Gray"/>, <contact fullname="Peter Gutmann"/>, <contact fullname="David
      Hook"/>, <contact fullname="Tim Hudson"/>, <contact fullname="Paul
      Kehrer"/>, <contact fullname="John Kemp"/>, <contact fullname="Watson
      Ladd"/>, <contact fullname="Adam Langley"/>, <contact fullname="John
      Mattsson"/>, <contact fullname="Damien Miller"/>, <contact
      fullname="Robert Relyea"/>, <contact fullname="Michael Richardson"/>,
      <contact fullname="Markku-Juhani O. Saarinen, Rich Salz, Roland Shoemaker, Sophie Schmieg,
Simo Sorce, Michael Saarinen"/>, <contact fullname="Rich
      Salz"/>, <contact fullname="Roland Shoemaker"/>, <contact
      fullname="Sophie Schmieg"/>, <contact fullname="Simo Sorce"/>, <contact
      fullname="Michael St. Johns, Falko Strenzke, Filippo Valsorda, Loganaden
Velvindron, Carl Wallace, and Wei-Jun Wang.</t> Johns"/>, <contact fullname="Falko Strenzke"/>,
      <contact fullname="Filippo Valsorda"/>, <contact fullname="Loganaden
      Velvindron"/>, <contact fullname="Carl Wallace"/>, and <contact
      fullname="Wei-Jun Wang"/>.</t>
    </section>
  </back>

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BRdhpfUf5PwJQAJLAwA= [rfced] Please review <tt> throughout the document for correctness and consistent use.  Please see the list of <tt> use at the following location:
  https://www.rfc-editor.org/authors/rfc9881tt2.txt

The list has been alphabetized and entries are listed only once (that is, if <tt> is used multiple times for the same term, it is only listed once).

Note that we did not review each item on the list.  These are our notes and questions on the items we checked:

a) <tt>expanded</tt> - Should instances of <tt>expanded</tt> be <tt>expandedKey</tt>?

For example:
   For each security level,
   we show the seed-only format (using a context-specific <tt>[0]</tt> primitive
   tag with an implicit encoding of <tt>OCTET STRING</tt>), the <tt>expanded</tt>
   format, and <tt>both</tt> formats together.</t>

b) <tt>seed</tt> - Should occurrences of "seed-only format" be "<tt>seed</tt> format"?

For example:
   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.

b) <tt>OCTET STRING</tt> - seems to be used consistently in <tt> outside of code

c) <tt>OneAsymmetricKey</tt> - Below are examples where "OneAsymmetricKey" does not appear in <tt>:

        <t>When encoding an ML-DSA private key in a OneAsymmetricKey object, any
of these three formats may be used, though the seed format is
<bcp14>RECOMMENDED</bcp14> for storage efficiency.</t>

Note that these are the sizes of the raw keys, not including
ASN.1 encoding overhead from OneAsymmetricKey and SubjectPublicKeyInfo
wrappers.

d) <tt>PUBLIC-KEY</tt> - seems to be used consistently in <tt> outside of code

-->

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