InternetDraft  PQC in OpenPGP  March 2023 
Kousidis, et al.  Expires 26 September 2023  [Page] 
 Workgroup:
 Network Working Group
 InternetDraft:
 draftwussleropenpgppqc01
 Published:
 Intended Status:
 Informational
 Expires:
PostQuantum Cryptography in OpenPGP
Abstract
This document defines a postquantum publickey algorithm extension for the OpenPGP protocol. Given the generally assumed threat of a cryptographically relevant quantum computer, this extension provides a basis for longterm secure OpenPGP signatures and ciphertexts. Specifically, it defines composite publickey encryption based on CRYSTALSKyber, composite publickey signatures based on CRYSTALSDilithium, both in combination with elliptic curve cryptography, and SPHINCS+ as a standalone public key signature scheme.¶
About This Document
This note is to be removed before publishing as an RFC.¶
Status information for this document may be found at https://datatracker.ietf.org/doc/draftwussleropenpgppqc/.¶
Discussion of this document takes place on the WG Working Group mailing list (mailto:openpgp@ietf.org), which is archived at https://mailarchive.ietf.org/arch/browse/openpgp/. Subscribe at https://www.ietf.org/mailman/listinfo/openpgp/.¶
Source for this draft and an issue tracker can be found at https://github.com/openpgppqc/draftopenpgppqc.¶
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Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶
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1. Introduction
The OpenPGP protocol supports various traditional publickey algorithms based on the factoring or discrete logarithm problem. As the security of algorithms based on these mathematical problems is endangered by the advent of quantum computers, there is a need to extend OpenPGP by algorithms that remain secure in the presence of quantum computers.¶
Such cryptographic algorithms are referred to as postquantum cryptography. The algorithms defined in this extension were chosen for standardization by the National Institute of Standards and Technology (NIST) in mid 2022 [NISTIR8413] as the result of the NIST PostQuantum Cryptography Standardization process initiated in 2016 [NISTPQC]. Namely, these are CRYSTALSKyber as a Key Encapsulation Mechanism (KEM), a KEM being a modern building block for publickey encryption, and CRYSTALSDilithium as well as SPHINCS+ as signature schemes.¶
For the two CRYSTALS* schemes, this document follows the conservative strategy to deploy postquantum in combination with traditional schemes such that the security is retained even if all schemes but one in the combination are broken. In contrast, the hashedbased signature scheme SPHINCS+ is considered to be sufficiently well understood with respect to its security assumptions in order to be used standalone. To this end, this document specifies the following new set: SPHINCS+ standalone and CRYSTALS* as composite with ECCbased KEM and digital signature schemes. Here, the term "composite" indicates that any data structure or algorithm pertaining to the combination of the two components appears as single data structure or algorithm from the protocol perspective.¶
The document specifies the conventions for interoperability between compliant OpenPGP implementations that make use of this extension and the newly defined algorithms or algorithm combinations.¶
1.1. Conventions used in this Document
1.1.1. Terminology for MultiAlgorithm Schemes
The terminology in this document is oriented towards the definitions in [draftdriscollpqthybridterminology]. Specifically, the terms "multialgorithm", "composite" and "noncomposite" are used in correspondence with the definitions therein. The abbreviation "PQ" is used for postquantum schemes. To denote the combination of postquantum and traditional schemes, the abbreviation "PQ/T" is used. The short form "PQ(/T)" stands for PQ or PQ/T.¶
1.2. PostQuantum Cryptography
This section describes the individual postquantum cryptographic schemes. All schemes listed here are believed to provide security in the presence of a cryptographically relevant quantum computer. However, the mathematical problems on which the two CRYSTALS* schemes and SPHINCS+ are based, are fundamentally different, and accordingly the level of trust commonly placed in them as well as their performance characteristics vary.¶
[Note to the reader: This specification refers to the latest NIST submission papers of each scheme as if it were a specification. This is a temporary solution that is owed to the fact that currently no other specification is available. The goal is to provide a sufficiently precise specification of the algorithms already at the draft stage of this specification, so that it is possible for implementers to create interoperable implementations. As soon as standards by NIST or the IETF for the PQC schemes employed in this specification are available, these will replace the references to the NIST submission papers. Furthermore, we want to point out that, depending on possible changes to the schemes standardized by NIST, this specification may be updated substantially as soon as corresponding information becomes available.]¶
1.2.1. CRYSTALSKyber
CRYSTALSKyber [KYBERSubm] is based on the hardness of solving the learningwitherrors problem in module lattices (MLWE). The scheme is believed to provide security against cryptanalytic attacks by classical as well as quantum computers. This specification defines CRYSTALSKyber only in composite combination with ECCbased encryption schemes in order to provide a prequantum security fallback.¶
1.2.2. CRYSTALSDilithium
CRYSTALSDilithium, defined in [DILITHIUMSubm], is a signature scheme that, like CRYSTALSKyber, is based on the hardness of solving lattice problems in module lattices. Accordingly, this specification only defines CRYSTALSDilithium in composite combination with ECCbased signature schemes.¶
1.2.3. SPHINCS+
SPHINCS+ [SPHINCSSubm] is a stateless hashbased signature scheme. Its security relies on the hardness of finding preimages for cryptographic hash functions. This feature is generally considered to be a high security guarantee. Therefore, this specification defines SPHINCS+ as a standalone signature scheme.¶
In deployments the performance characteristics of SPHINCS+ should be taken into account. We refer to Section 10.1 for a discussion of the performance characteristics of this scheme.¶
1.3. Elliptic Curve Cryptography
The ECCbased encryption is defined here as a KEM. This is in contrast to [ID.ietfopenpgpcryptorefresh] where the ECCbased encryption is defined as a publickey encryption scheme.¶
All elliptic curves for the use in the composite combinations are taken from [ID.ietfopenpgpcryptorefresh]. However, as explained in the following, in the case of Curve25519 encoding changes are applied to the new composite schemes.¶
1.3.1. Curve25519 and Curve448
Curve25519 and Curve448 are defined in [RFC7748] for use in a DiffieHellman key agreement scheme and defined in [RFC8032] for use in a digital signature scheme. For Curve25519 this specification adapts the encoding of objects as defined in [RFC7748] in contrast to [ID.ietfopenpgpcryptorefresh].¶
1.3.2. Generic Prime Curves
For interoperability this extension offers CRYSTALS* in composite combinations with the NIST curves P256, P384 defined in [SP800186] and the Brainpool curves brainpoolP256r1, brainpoolP384r1 defined in [RFC5639].¶
1.4. Standalone and MultiAlgorithm Schemes
This section provides a categorization of the new algorithms and their combinations.¶
1.4.1. Standalone and Composite MultiAlgorithm Schemes
This specification introduces new cryptographic schemes, which can be categorized as follows:¶
 PQ/T multialgorithm publickey encryption, namely a composite combination of CRYSTALSKyber with an ECCbased KEM,¶
 PQ/T multialgorithm digital signature, namely composite combinations of CRYSTALSDilithium with ECCbased signature schemes,¶
 PQ digital signature, namely SPHINCS+ as a standalone cryptographic algorithm.¶
For each of the composite schemes, this specifications mandates that the recipient has to successfully perform the cryptographic algorithms for each of the component schemes used in a cryptrographic message, in order for the message to be deciphered and considered as valid. This means that all component signatures must be verified successfully in order to achieve a successful verification of the composite signature. In the case of the composite publickey decryption, each of the component KEM decapsulation operations must succeed.¶
1.4.2. NonComposite Algorithm Combinations
As the OpenPGP protocol [ID.ietfopenpgpcryptorefresh] allows for multiple signatures to be applied to a single message, it is also possible to realize noncomposite combinations of signatures. Furthermore, multiple OpenPGP signatures may be combined on the application layer. These latter two cases realize noncomposite combinations of signatures. Section 4.4 specifies how implementations should handle the verification of such combinations of signatures.¶
Furthermore, the OpenPGP protocol also allows for parallel encryption to different keys held by the same recipient. Accordingly, if the sender makes use of this feature and sends an encrypted message with multiple PKESK packages for different encryption keys held by the same recipient, a noncomposite multialgorithm publickey encryption is realized where the recipient has to decrypt only one of the PKESK packages in order to decrypt the message. See Section 4.2 for restrictions on parallel encryption mandated by this specification.¶
2. Preliminaries
This section provides some preliminaries for the definitions in the subsequent sections.¶
2.1. Elliptic curves
2.1.1. SEC1 EC Point Wire Format
Elliptic curve points of the generic prime curves are encoded using the SEC1 (uncompressed) format as the following octet string:¶
B = 04  X  Y¶
where X
and Y
are coordinates of the elliptic curve point P = (X, Y)
, and
each coordinate is encoded in the bigendian format and zeropadded to the
adjusted underlying field size. The adjusted underlying field size is the
underlying field size rounded up to the nearest 8bit boundary, as noted in the
"Field size" column in Table 6,
Table 7, or Table 11. This encoding is
compatible with the definition given in [SEC1].¶
2.1.2. Measures to Ensure Secure Implementations
The following paragraphs describe measures that ensure secure implementations according to existing best practices and standards defining the operations of Elliptic Curve Cryptography.¶
Even though the zero point, also called the point at infinity, may occur as a result of arithmetic operations on points of an elliptic curve, it MUST NOT appear in any ECC data structure defined in this document.¶
Furthermore, when performing the explicitly listed operations in Section 5.1.1.1, Section 5.1.1.2 or Section 5.1.1.3 it is REQUIRED to follow the specification and security advisory mandated from the relative elliptic curve specification.¶
3. Supported Public Key Algorithms
This section specifies the composite Kyber + ECC and Dilithium + ECC schemes as well as the standalone SPHINCS+ signature scheme. The composite schemes are fully specified via their algorithm ID. The SPHINCS+ signature schemes are fully specified by their algorithm ID and an additional parameter ID.¶
3.1. Algorithm Specifications
For encryption, the following composite KEM schemes are specified:¶
ID  Algorithm  Requirement  Definition 

29  Kyber768 + X25519  MUST  Section 5.2 
30  Kyber1024 + X448  SHOULD  Section 5.2 
31  Kyber768 + ECDHNISTP256  MAY  Section 5.2 
32  Kyber1024 + ECDHNISTP384  MAY  Section 5.2 
33  Kyber768 + ECDHbrainpoolP256r1  MAY  Section 5.2 
34  Kyber1024 + ECDHbrainpoolP384r1  MAY  Section 5.2 
For signatures, the following (composite) signature schemes are specified:¶
ID  Algorithm  Requirement  Definition 

35  Dilithium3 + Ed25519  MUST  Section 6.2 
36  Dilithium5 + Ed448  SHOULD  Section 6.2 
37  Dilithium3 + ECDSANISTP256  MAY  Section 6.2 
38  Dilithium5 + ECDSANISTP384  MAY  Section 6.2 
39  Dilithium3 + ECDSAbrainpoolP256r1  MAY  Section 6.2 
40  Dilithium5 + ECDSAbrainpoolP384r1  MAY  Section 6.2 
41  SPHINCS+simpleSHA2  SHOULD  Section 1.2.3 
42  SPHINCS+simpleSHAKE  MAY  Section 1.2.3 
3.2. Parameter Specification
3.2.1. SPHINCS+simpleSHA2
For the SPHINCS+simpleSHA2 signature algorithm from Table 2, the following parameters are specified:¶
Parameter ID  Parameter 

1  SPHINCS+simpleSHA2128s 
2  SPHINCS+simpleSHA2128f 
3  SPHINCS+simpleSHA2192s 
4  SPHINCS+simpleSHA2192f 
5  SPHINCS+simpleSHA2256s 
6  SPHINCS+simpleSHA2256f 
All security parameters inherit the requirement of SPHINCS+simpleSHA2 from
Table 2. That is, implementations SHOULD implement the parameters
specified in Table 3. The values 0x00
and 0xFF
are reserved
for future extensions.¶
3.2.2. SPHINCS+simpleSHAKE
For the SPHINCS+simpleSHAKE signature algorithm from Table 2, the following parameters are specified:¶
Parameter ID  Parameter 

1  SPHINCS+simpleSHAKE128s 
2  SPHINCS+simpleSHAKE128f 
3  SPHINCS+simpleSHAKE192s 
4  SPHINCS+simpleSHAKE192f 
5  SPHINCS+simpleSHAKE256s 
6  SPHINCS+simpleSHAKE256f 
All security parameters inherit the requirement of SPHINCS+simpleSHAKE from
Table 2. That is, implementations MAY implement the parameters
specified in Table 4. The values 0x00
and 0xFF
are reserved
for future extensions.¶
4. Algorithm Combinations
4.1. Composite KEMs
Kyber + ECC publickey encryption is meant to involve both the Kyber KEM and an ECCbased KEM in an a priori nonseparable manner. This is achieved via KEM combination, i.e. both key encapsulations/decapsulations are performed in parallel, and the resulting key shares are fed into a key combiner to produce a single shared secret for message encryption.¶
4.2. Parallel PublicKey Encryption
As explained in Section 1.4.2, the OpenPGP protocol inherently supports parallel encryption to different keys of the same recipient. Implementations MUST NOT encrypt a message to a purely traditional publickey encryption key of a recipient if it is encrypted to a PQ/T key of the same recipient.¶
4.3. Composite Signatures
Dilithium + ECC signatures are meant to contain both the Dilithium and the ECC signature data, and an implementation MUST validate both algorithms to state that a signature is valid.¶
4.4. Multiple Signatures
The OpenPGP message format allows multiple signatures of a message, i.e. the attachment of multiple signature packets.¶
An implementation MAY sign a message with a traditional key and a PQ(/T) key from the same sender. This ensures backwards compatibility due to [ID.ietfopenpgpcryptorefresh] Section 5.2.5, since a legacy implementation without PQ(/T) support can fall back on the traditional signature.¶
Newer implementations with PQ(/T) support MAY ignore the traditional signature(s) during validation.¶
Implementations SHOULD consider the message correctly signed if at least one of the nonignored signatures validates successfully.¶
[Note to the reader: The last requirement, that one valid signature is sufficient to identify a message as correctly signed, is an interpretation of [ID.ietfopenpgpcryptorefresh] Section 5.2.5.]¶
5. Composite KEM schemes
5.1. Building Blocks
5.1.1. ECCBased KEMs
In this section we define the encryption, decryption, and data formats for the ECDH component of the composite algorithms.¶
Table 5, Table 6, and Table 7 describe the ECCKEM parameters and artifact lengths. The artefacts in Table 5 follow the encodings described in [RFC7748].¶
X25519  X448  

Algorithm ID reference  29  30 
Field size  32 octets  56 octets 
ECCKEM  x25519Kem (Section 5.1.1.1)  x448Kem (Section 5.1.1.2) 
ECDH public key  32 octets [RFC7748]  56 octets [RFC7748] 
ECDH secret key  32 octets [RFC7748]  56 octets [RFC7748] 
ECDH ephemeral  32 octets [RFC7748]  56 octets [RFC7748] 
ECDH share  32 octets [RFC7748]  56 octets [RFC7748] 
Key share  32 octets  64 octets 
Hash  SHA3256  SHA3512 
NIST P256  NIST P384  

Algorithm ID reference  31  32 
Field size  32 octets  48 octets 
ECCKEM  ecdhKem (Section 5.1.1.3)  ecdhKem (Section 5.1.1.3) 
ECDH public key  65 octets of SEC1encoded public point  97 octets of SEC1encoded public point 
ECDH secret key  32 octets bigendian encoded secret scalar  48 octets bigendian encoded secret scalar 
ECDH ephemeral  65 octets of SEC1encoded ephemeral point  97 octets of SEC1encoded ephemeral point 
ECDH share  65 octets of SEC1encoded shared point  97 octets of SEC1encoded shared point 
Key share  32 octets  64 octets 
Hash  SHA3256  SHA3512 
brainpoolP256r1  brainpoolP384r1  

Algorithm ID reference  33  34 
Field size  32 octets  48 octets 
ECCKEM  ecdhKem (Section 5.1.1.3)  ecdhKem (Section 5.1.1.3) 
ECDH public key  65 octets of SEC1encoded public point  97 octets of SEC1encoded public point 
ECDH secret key  32 octets bigendian encoded secret scalar  48 octets bigendian encoded secret scalar 
ECDH ephemeral  65 octets of SEC1encoded ephemeral point  97 octets of SEC1encoded ephemeral point 
ECDH share  65 octets of SEC1encoded shared point  97 octets of SEC1encoded shared point 
Key share  32 octets  64 octets 
Hash  SHA3256  SHA3512 
The SEC1 format for point encoding is defined in Section 2.1.1.¶
The various procedures to perform the operations of an ECCbased KEM are defined in the following subsections. Specifically, each of these subsections defines the instances of the following operations:¶
(eccCipherText, eccKeyShare) < eccKem.encap(eccPublicKey)¶
and¶
(eccKeyShare) < eccKem.decap(eccPrivateKey, eccCipherText)¶
The placeholder eccKem
has to be replaced with the specific ECCKEM from the
row "ECCKEM" of Table 5, Table 6, and
Table 7.¶
5.1.1.1. X25519KEM
The encapsulation and decapsulation operations of x25519kem
are described
using the function X25519()
and encodings defined in [RFC7748]. The
eccPrivateKey
is denoted as r
, the eccPublicKey
as R
, they are subject
to the equation R = X25519(r, U(P))
. Here, U(P)
denotes the ucoordinate of
the base point of Curve25519.¶
The operation x25519Kem.encap()
is defined as follows:¶
 Generate an ephemeral key pair {
v
,V
} viaV = X25519(v,U(P))
¶  Compute the shared coordinate
X = X25519(v, R)
whereR
is the public keyeccPublicKey
¶  Set the output
eccCipherText
toV
¶  Set the output
eccKeyShare
toSHA3256(X  eccCipherText)
¶
The operation x25519Kem.decap()
is defined as follows:¶
5.1.1.2. X448KEM
The encapsulation and decapsulation operations of x448kem
are described using
the function X448()
and encodings defined in [RFC7748]. The eccPrivateKey
is denoted as r
, the eccPublicKey
as R
, they are subject to the equation
R = X25519(r, U(P))
. Here, U(P)
denotes the ucoordinate of the base point
of Curve448.¶
The operation x448.encap()
is defined as follows:¶
 Generate an ephemeral key pair {
v
,V
} viaV = X448(v,U(P))
¶  Compute the shared coordinate
X = X448(v, R)
whereR
is the public keyeccPublicKey
¶  Set the output
eccCipherText
toV
¶  Set the output
eccKeyShare
toSHA3512(X  eccCipherText)
¶
The operation x448Kem.decap()
is defined as follows:¶
5.1.1.3. ECDHKEM
The operation ecdhKem.encap()
is defined as follows:¶
 Generate an ephemeral key pair {
v
,V=vG
} as defined in [SP800186] or [RFC5639]¶  Compute the shared point
S = vR
, whereR
is the component public keyeccPublicKey
, according to [SP800186] or [RFC5639]¶  Extract the
X
coordinate from the SEC1 encoded pointS = 04  X  Y
as defined in section Section 2.1.1¶  Set the output
eccCipherText
to the SEC1 encoding ofV
¶  Set the output
eccKeyShare
toHash(X  eccCipherText)
, withHash
chosen according to Table 6 or Table 7¶
The operation ecdhKem.decap()
is defined as follows:¶
 Compute the shared Point
S
asrV
, wherer
is theeccPrivateKey
andV
is theeccCipherText
, according to [SP800186] or [RFC5639]¶  Extract the
X
coordinate from the SEC1 encoded pointS = 04  X  Y
as defined in section Section 2.1.1¶  Set the output
eccKeyShare
toHash(X  eccCipherText)
, withHash
chosen according to Table 6 or Table 7¶
5.1.2. KyberKEM
KyberKEM features the following operations:¶
(kyberCipherText, kyberKeyShare) < kyberKem.encap(kyberPublicKey)¶
and¶
(kyberKeyShare) < kyberKem.decap(kyberCipherText, kyberPrivateKey)¶
The above are the operations Kyber.CCAKEM.Enc() and Kyber.CCAKEM.Dec() defined in [KYBERSubm].¶
KyberKEM has the parameterization with the corresponding artifact lengths in octets as given in Table 8. All artifacts are encoded as defined in [KYBERSubm].¶
Algorithm ID reference  KyberKEM  Public key  Secret key  Ciphertext  Key share 

29, 31, 33  kyberKem768  1184  2400  1088  32 
30, 32, 34  kyberKem1024  1568  3186  1568  32 
The placeholder kyberKem
has to be replaced with the specific KyberKEM from
the column "KyberKEM" of Table 8.¶
The procedure to perform kyberKem.encap()
is as follows:¶
 Extract the component public key
kyberPublicKey
that is part of the recipient's composite public key¶  Invoke
(kyberCipherText, keyShare) < kyberKem.encap(kyberPublicKey)
¶  Set
kyberCipherText
as the Kyber ciphertext¶  Set
keyShare
as the Kyber symmetric key share¶
The procedure to perform kyberKem.decap()
is as follows:¶
5.2. Composite Encryption Schemes with Kyber
Table 1 specifies the following Kyber + ECC composite publickey encryption schemes:¶
Algorithm ID reference  KyberKEM  ECCKEM  ECDHKEM curve 

29  kyberKem768  x25519Kem  X25519 
30  kyberKem1024  x448Kem  X448 
31  kyberKem768  ecdhKem  NIST P256 
32  kyberKem1024  ecdhKem  NIST P384 
33  kyberKem768  ecdhKem  brainpoolP256r1 
34  kyberKem1024  ecdhKem  brainpoolP384r1 
The Kyber + ECC composite publickey encryption schemes are built according to the following principal design:¶
 The KyberKEM encapsulation algorithm is invoked to create a Kyber ciphertext together with a Kyber symmetric key share.¶
 The encapsulation algorithm of an ECCbased KEM, namely one out of X25519KEM, X448KEM, or ECDHKEM is invoked to create an ECC ciphertext together with an ECC symmetric key share.¶
 A KeyEncryptionKey (KEK) is computed as the output of a key combiner that receives as input both of the above created symmetric key shares and the protocol binding information.¶
 The session key for content encryption is then wrapped as described in [RFC3394] using AES256 as algorithm and the KEK as key.¶
 The v6 PKESK package's algorithm specific parts are made up of the Kyber ciphertext, the ECC ciphertext, and the wrapped session key¶
5.2.1. Fixed information
For the composite KEM schemes defined in Table 1 the following procedure, justified in Section 9.3, MUST be used to derive a string to use as binding between the KEK and the communication parties.¶
// Input: // algID  the algorithm ID encoded as octet // publicKey  the recipient's encryption subkey packet // serialized as octet string fixedInfo = algID  SHA3256(publicKey)¶
SHA3256 MUST be used to hash the publicKey
of the recipient.¶
5.2.2. Key combiner
For the composite KEM schemes defined in Table 1 the following procedure MUST be used to compute the KEK that wraps a session key. The construction is a onestep key derivation function compliant to [SP80056C] Section 4, based on KMAC256 [SP800185]. It is given by the following algorithm.¶
// multiKeyCombine(eccKeyShare, eccCipherText, // kyberKeyShare, kyberCipherText, // fixedInfo, oBits) // // Input: // eccKeyShare  the ECC key share encoded as an octet string // eccCipherText  the ECC ciphertext encoded as an octet string // kyberKeyShare  the Kyber key share encoded as an octet string // kyberCipherText  the Kyber ciphertext encoded as an octet string // fixedInfo  the fixed information octet string // oBits  the size of the output keying material in bits // // Constants: // domSeparation  the UTF8 encoding of the string // "OpenPGPCompositeKeyDerivationFunction" // counter  the fixed 4 byte value 0x00000001 // customizationString  the UTF8 encoding of the string "KDF" eccKemData = eccKeyShare  eccCipherText kyberKemData = kyberKeyShare  kyberCipherText encData = counter  eccKemData  kyberKemData  fixedInfo MB = KMAC256(domSeparation, encData, oBits, customizationString)¶
Note that the values eccKeyShare
defined in Section 5.1.1 and kyberKeyShare
defined in Section 5.1.2 already use the relative ciphertext in the
derivation. The ciphertext is by design included again in the key combiner to
provide a robust security proof.¶
The value of domSeparation
is the UTF8 encoding of the string
"OpenPGPCompositeKeyDerivationFunction" and MUST be the following octet sequence:¶
domSeparation := 4F 70 65 6E 50 47 50 43 6F 6D 70 6F 73 69 74 65 4B 65 79 44 65 72 69 76 61 74 69 6F 6E 46 75 6E 63 74 69 6F 6E¶
The value of counter
MUST be set to the following octet sequence:¶
counter := 00 00 00 01¶
The value of fixedInfo
MUST be set according to Section 5.2.1.¶
The value of customizationString
is the UTF8 encoding of the string "KDF"
and MUST be set to the following octet sequence:¶
customizationString := 4B 44 46¶
5.2.3. Key generation procedure
The implementation MUST independently generate the Kyber and the ECC component keys. Kyber key generation follows the specification [KYBERSubm] and the artifacts are encoded as fixedlength octet strings. For ECC this is done following the relative specification in [RFC7748], [SP800186], or [RFC5639], and encoding the outputs as fixedlength octet strings in the format specified in table Table 5, Table 6, or Table 7.¶
5.2.4. Encryption procedure
The procedure to perform publickey encryption with a Kyber + ECC composite scheme is as follows:¶
 Take the recipient's authenticated publickey packet
pkComposite
andsessionKey
as input¶  Parse the algorithm ID from
pkComposite
¶  Extract the
eccPublicKey
andkyberPublicKey
component from the algorithm specific data encoded inpkComposite
with the format specified in Section 5.3.2.¶  Instantiate the ECCKEM
eccKem.encap()
and the KyberKEMkyberKem.encap()
depending on the algorithm ID according to Table 9¶  Compute
(eccCipherText, eccKeyShare) := eccKem.encap(eccPublicKey)
¶  Compute
(kyberCipherText, kyberKeyShare) := kyberKem.encap(kyberPublicKey)
¶  Compute
fixedInfo
as specified in Section 5.2.1¶  Compute
KEK := multiKeyCombine(eccKeyShare, eccCipherText, kyberKeyShare, kyberCipherText, fixedInfo, oBits=256)
as defined in Section 5.2.2¶  Compute
C := AESKeyWrap(KEK, sessionKey)
with AES256 as per [RFC3394] that includes a 64 bit integrity check¶  Output
eccCipherText  kyberCipherText  len(C)  C
as specified in Section 5.3.1¶
5.2.5. Decryption procedure
The procedure to perform publickey decryption with a Kyber + ECC composite scheme is as follows:¶
 Take the matching PKESK and own secret key packet as input¶
 From the PKESK extract the algorithm ID and the
encryptedKey
¶  Check that the own and the extracted algorithm ID match¶
 Parse the
eccSecretKey
andkyberSecretKey
from the algorithm specific data of the own secret key encoded in the format specified in Section 5.3.2¶  Instantiate the ECCKEM
eccKem.decap()
and the KyberKEMkyberKem.decap()
depending on the algorithm ID according to Table 9¶  Parse
eccCipherText
,kyberCipherText
, andC
fromencryptedKey
encoded aseccCipherText  kyberCipherText  len(C)  C
as specified in Section 5.3.1¶  Compute
(eccKeyShare) := eccKem.decap(eccCipherText, eccPrivateKey)
¶  Compute
(kyberKeyShare) := kyberKem.decap(kyberCipherText, kyberPrivateKey)
¶  Compute
fixedInfo
as specified in Section 5.2.1¶  Compute
KEK := multiKeyCombine(eccKeyShare, eccCipherText, kyberKeyShare, kyberCipherText, fixedInfo, oBits=256)
as defined in Section 5.2.2¶  Compute
sessionKey := AESKeyUnwrap(KEK, C)
with AES256 as per [RFC3394], aborting if the 64 bit integrity check fails¶  Output
sessionKey
¶
5.3. Packet specifications
5.3.1. PublicKey Encrypted Session Key Packets (Tag 1)
The composite Kyber algorithms MUST be used only with v6 PKESK, as defined in [ID.ietfopenpgpcryptorefresh] Section 5.1.2.¶
The algorithmspecific v6 PKESK parameters consists of:¶
 A fixedlength octet string representing an ECC ephemeral public key in the format associated with the curve as specified in Section 5.1.1.¶
 A fixedlength octet string of the Kyber ciphertext, whose length depends on the algorithm ID as specified in Table 8.¶

A variablelength field containing the symmetric key:¶
 A oneoctet size of the following field;¶
 Octet string of the wrapped symmetric key as described in Section 5.2.4.¶
5.3.2. Key Material Packets
The algorithmspecific public key is this series of values:¶
 A fixedlength octet string representing an EC point public key, in the point format associated with the curve specified in Section 5.1.1.¶
 A fixedlength octet string containing the Kyber public key, whose length depends on the algorithm ID as specified in Table 8.¶
The algorithmspecific secret key is these two values:¶
 A fixedlength octet string of the encoded secret scalar, whose encoding and length depend on the algorithm ID as specified in Section 5.1.1.¶
 A fixedlength octet string containing the Kyber secret key, whose length depends on the algorithm ID as specified in Table 8.¶
6. Composite Signature Schemes
6.1. Building blocks
6.1.1. EdDSABased signatures
To sign and verify with EdDSA the following operations are defined:¶
(eddsaSignature) < eddsa.sign(eddsaPrivateKey, dataDigest)¶
and¶
(verified) < eddsa.verify(eddsaPublicKey, eddsaSignature, dataDigest)¶
The public and private keys, as well as the signature MUST be encoded according to [RFC8032] as fixedlength octet strings. The following table describes the EdDSA parameters and artifact lengths:¶
Algorithm ID reference  Curve  Field size  Public key  Secret key  Signature 

35  Ed25519  32  32  32  64 
36  Ed448  57  57  57  114 
6.1.2. ECDSABased signatures
To sign and verify with ECDSA the following operations are defined:¶
(ecdsaSignatureR, ecdsaSignatureS) < ecdsa.sign(ecdsaPrivateKey, dataDigest)¶
and¶
(verified) < ecdsa.verify(ecdsaPublicKey, ecdsaSignatureR, ecdsaSignatureS, dataDigest)¶
The public keys MUST be encoded in SEC1 format as defined in section
Section 2.1.1. The secret key, as well as both values R
and S
of the
signature MUST each be encoded as a bigendian integer in a fixedlength octet
string of the specified size.¶
The following table describes the ECDSA parameters and artifact lengths:¶
Algorithm ID reference  Curve  Field size  Public key  Secret key  Signature value R  Signature value S 

37  NIST P256  32  65  32  32  32 
38  NIST P384  48  97  48  48  48 
39  brainpoolP256r1  32  65  32  32  32 
40  brainpoolP384r1  48  97  48  48  48 
6.1.3. Dilithium signatures
The procedure for Dilithium signature generation is the function Sign(sk, M)
given in Figure 4 in [DILITHIUMSubm], where sk
is the Dilithium private key
and M
is the data to be signed. OpenPGP does not use the optional randomized
signing given as a variant in the definition of this function, i.e. rho' :=
H(K  mu)
is used. The signing function returns the Dilithium signature. That
is, to sign with Dilithium the following operation is defined:¶
(dilithiumSignature) < dilithium.sign(dilithiumPrivateKey, dataDigest)¶
The procedure for Dilithium signature verification is the function Verify(pk,
M, sigma)
given in Figure 4 in [DILITHIUMSubm], where pk
is the Dilithium
public key, M
is the data to be signed and sigma
is the Dilithium
signature. That is, to verify with Dilithium the following operation is
defined:¶
(verified) < dilithium.verify(dilithiumPublicKey, dataDigest, dilithiumSignature)¶
Dilithium has the parameterization with the corresponding artifact lengths in octets as given in Table 12. All artifacts are encoded as defined in [DILITHIUMSubm].¶
Algorithm ID reference  Dilithium instance  Public key  Secret key  Signature value 

35, 37, 39  Dilithium3  1952  4000  3293 
36, 38, 40  Dilithium5  2592  4864  4595 
6.2. Composite Signature Schemes with Dilithium
6.2.1. Binding hashes
Composite Dilithium + ECC signatures MUST use SHA3256 (hash algorithm ID 12) or SHA3512 (hash algorithm ID 14) as hashing algorithm. Signatures using other hash algorithms MUST be considered invalid.¶
An implementation MUST support SHA3256 and SHOULD support SHA3512, in order to support the hash binding with Dilithium + ECC signatures.¶
6.2.2. Key generation procedure
The implementation MUST independently generate the Dilithium and the ECC component keys. Dilithium key generation follows the specification in [DILITHIUMSubm] and the artifacts are encoded as fixedlength octet strings as defined in Section 6.1.3. For ECC this is done following the relative specification in [RFC7748], [SP800186], or [RFC5639], and encoding the artifacts as specified in Section 6.1.1 or Section 6.1.2 as fixedlength octet strings.¶
6.2.3. Signature Generation
To sign a message M
with Dilithium + EdDSA the following sequence of
operations has to be performed:¶
 Generate
dataDigest
according to [ID.ietfopenpgpcryptorefresh] Section 5.2.4¶  Create the EdDSA signature over
dataDigest
witheddsa.sign()
from Section 6.1.1¶  Create the Dilithium signature over
dataDigest
withdilithium.sign()
from Section 6.1.3¶  Encode the EdDSA and Dilithium signatures according to the packet structure given in Section 6.3.1.¶
To sign a message M
with Dilithium + ECDSA the following sequence of
operations has to be performed:¶
 Generate
dataDigest
according to [ID.ietfopenpgpcryptorefresh] Section 5.2.4¶  Create the ECDSA signature over
dataDigest
withecdsa.sign()
from Section 6.1.2¶  Create the Dilithium signature over
dataDigest
withdilithium.sign()
from Section 6.1.3¶  Encode the ECDSA and Dilithium signatures according to the packet structure given in Section 6.3.1.¶
6.2.4. Signature Verification
To verify a Dilithium + EdDSA signature the following sequence of operations has to be performed:¶
 Verify the EdDSA signature with
eddsa.verify()
from Section 6.1.1¶  Verify the Dilithium signature with
dilithium.verify()
from Section 6.1.3¶
To verify a Dilithium + ECDSA signature the following sequence of operations has to be performed:¶
 Verify the ECDSA signature with
ecdsa.verify()
from Section 6.1.2¶  Verify the Dilithium signature with
dilithium.verify()
from Section 6.1.3¶
As specified in Section 4.3 an implementation MUST validate both signatures, i.e. EdDSA/ECDSA and Dilithium, to state that a composite Dilithium + ECC signature is valid.¶
6.3. Packet Specifications
6.3.1. Signature Packet (Tag 2)
The composite Dilithium + ECC schemes MUST be used only with v6 signatures, as defined in [ID.ietfopenpgpcryptorefresh] Section 5.2.3.¶
The algorithmspecific v6 signature parameters for Dilithium + EdDSA signatures consists of:¶
 A fixedlength octet string representing the EdDSA signature, whose length depends on the algorithm ID as specified in Table 10.¶
 A fixedlength octet string of the Dilithium signature value, whose length depends on the algorithm ID as specified in Table 12.¶
The algorithmspecific v6 signature parameters for Dilithium + ECDSA signatures consists of:¶
 A fixedlength octet string of the bigendian encoded ECDSA value
R
, whose length depends on the algorithm ID as specified in Table 11.¶  A fixedlength octet string of the bigendian encoded ECDSA value
S
, whose length depends on the algorithm ID as specified in Table 11.¶  A fixedlength octet string of the Dilithium signature value, whose length depends on the algorithm ID as specified in Table 12.¶
6.3.2. Key Material Packets
The composite Dilithium + ECC schemes MUST be used only with v6 keys, as defined in [ID.ietfopenpgpcryptorefresh].¶
The algorithmspecific public key for Dilithium + EdDSA keys is this series of values:¶
 A fixedlength octet string representing the EdDSA public key, whose length depends on the algorithm ID as specified in Table 10.¶
 A fixedlength octet string containing the Dilithium public key, whose length depends on the algorithm ID as specified in Table 12.¶
The algorithmspecific private key for Dilithium + EdDSA keys is this series of values:¶
 A fixedlength octet string representing the EdDSA secret key, whose length depends on the algorithm ID as specified in Table 10.¶
 A fixedlength octet string containing the Dilithium secret key, whose length depends on the algorithm ID as specified in Table 12.¶
The algorithmspecific public key for Dilithium + ECDSA keys is this series of values:¶
 A fixedlength octet string representing the ECDSA public key in SEC1 format, as specified in section Section 2.1.1 and with length specified in Table 11.¶
 A fixedlength octet string containing the Dilithium public key, whose length depends on the algorithm ID as specified in Table 12.¶
The algorithmspecific private key for Dilithium + ECDSA keys is this series of values:¶
7. SPHINCS+
7.1. The SPHINCS+ Algorithms
The following table describes the SPHINCS+ parameters and artifact lengths:¶
Parameter ID reference  Parameter name suffix  SPHINCS+ public key  SPHINCS+ secret key  SPHINCS+ signature 

1  128s  32  64  7856 
2  128f  32  64  17088 
3  192s  48  96  16224 
4  192f  48  96  35664 
5  256s  64  128  29792 
6  256f  64  128  49856 
7.1.1. Binding hashes
SPHINCS+ signature packets MUST use the associated hash as specified in Table 14. Signature packets using other hashes MUST be considered invalid.¶
Algorithm ID reference  Parameter ID reference  Hash function  Hash function ID reference 

41  1, 2  SHA256  8 
41  3, 4, 5, 6  SHA512  10 
42  1, 2  SHA3256  12 
42  3, 4, 5, 6  SHA3512  14 
An implementation supporting a specific SPHINCS+ algorithm and parameter MUST also support the matching hash algorithm.¶
7.1.2. Key generation
The SPHINCS+ key generation is performed according to the function
spx_keygen()
specified in [SPHINCSSubm], Sec. 6.2 as Alg. 19. The private
and public key are encoded as defined in [SPHINCSSubm].¶
7.1.3. Signature Generation
The procedure for SPHINCS+ signature generation is the function spx_sign(M,
SK)
specified in [SPHINCSSubm], Sec. 6.4 as Alg. 20. Here, M
is the
dataDigest
generated according to [ID.ietfopenpgpcryptorefresh] Section
5.2.4 and SK
is the SPHINCS+ private key. The global variable RANDOMIZE
specified in Alg. 20 is to be considered as not set, i.e. the variable opt
shall be initialized with PK.seed
. See also Section 9.4.¶
An implementation MUST set the Parameter ID in the signature equal to the issuing private key Parameter ID.¶
7.1.4. Signature Verification
The procedure for SPHINCS+ signature verification is the function
spx_verify(M, SIG, PK)
specified in [SPHINCSSubm], Sec. 6.5 as Alg. 21.
Here, M
is the dataDigest
generated according to
[ID.ietfopenpgpcryptorefresh] Section 5.2.4, SIG
is the signature, and
PK
is the SPHINCS+ public key.¶
An implementation MUST check that the Parameter ID in the signature and in the key match when verifying.¶
7.2. Packet specifications
7.2.1. Signature Packet (Tag 2)
The SPHINCS+ algorithms MUST be used only with v6 signatures, as defined in [ID.ietfopenpgpcryptorefresh] Section 5.2.3.¶
The algorithmspecific v6 Signature parameters consists of:¶
7.2.2. Key Material Packets
The SPHINCS+ algorithms MUST be used only with v6 keys, as defined in [ID.ietfopenpgpcryptorefresh].¶
The algorithmspecific public key is this series of values:¶
 A oneoctet value specifying the SPHINCS+ parameter ID defined in
Table 3 and Table 4. The values
0x00
and0xFF
are reserved for future extensions.¶  A fixedlength octet string containing the SPHINCS+ public key, whose length depends on the parameter ID as specified in Table 13.¶
The algorithmspecific private key is this value:¶
8. Migration Considerations
The postquantum KEM algorithms defined in Table 1 and the signature algorithms defined in Table 2 are a set of new public key algorithms that extend the algorithm selection of [ID.ietfopenpgpcryptorefresh]. During the transition period, the postquantum algorithms will not be supported by all clients. Therefore various migration considerations must be taken into account, in particular backwards compatibility to existing implementations that have not yet been updated to support the postquantum algorithms.¶
8.1. Key preference
Implementations SHOULD prefer PQ(/T) keys when multiple options are available.¶
For instance, if encrypting for a recipient for which both a valid PQ/T and a valid ECC certificate are available, the implementation SHOULD choose the PQ/T certificate. In case a certificate has both a PQ/T and an ECC encryptioncapable valid subkey, the PQ/T subkey SHOULD be preferred.¶
An implementation MAY sign with both a PQ(/T) and an ECC key using multiple signatures over the same data as described in Section 4.4. Signing only with PQ(/T) key material is not backwards compatible.¶
Note that the confidentiality of a message is not postquantum secure when encrypting to multiple recipients if at least one recipient does not support PQ/T encryption schemes. An implementation SHOULD NOT abort the encryption process in this case to allow for a smooth transition to postquantum cryptography.¶
8.2. Key generation strategies
It is REQUIRED to generate fresh secrets when generating PQ(/T) keys. Reusing key material from existing ECC keys in PQ(/T) keys does not provide backwards compatibility, and the fingerprint will differ.¶
An OpenPGP (v6) certificate is composed of a certificationcapable primary key and one or more subkeys for signature, encryption, and authentication. Two migration strategies are recommended:¶
 Generate two independent certificates, one for PQ(/T)capable implementations, and one for legacy implementations. Implementations not understanding PQ(/T) certificates can use the legacy certificate, while PQ(/T)capable implementations will prefer the newer certificate. This allows having an older v4 or v6 ECC certificate for compatibility and a v6 PQ(/T) certificate, at a greater complexity in key distribution.¶
 Attach PQ(/T) encryption and signature subkeys to an existing v6 ECC certificate. Implementations understanding PQ(/T) will be able to parse and use the subkeys, while PQ(/T)incapable implementations can gracefully ignore them. This simplifies key distribution, as only one certificate needs to be communicated and verified, but leaves the primary key vulnerable to quantum computer attacks.¶
9. Security Considerations
9.1. Hashing in ECCKEM
Our construction of the ECCKEMs, in particular the final hashing step in
encapsulation and decapsulation that produces the eccKeyShare
, is standard
and known as hashed ElGamal key encapsulation, a hashed variant of ElGamal
encryption. It ensures INDCCA2 security in the random oracle model under some
DiffieHellman intractability assumptions [CS03].¶
9.2. Key combiner
For the key combination in Section 5.2.2 this specification limits itself to the use of KMAC. The sponge construction used by KMAC was proven to be indifferentiable from a random oracle [BDPA08]. This means, that in contrast to SHA2, which uses a MerkleDamgard construction, no HMACbased construction is required for key combination. Except for a domain separation it is sufficient to simply process the concatenation of any number of key shares when using a spongebased construction like KMAC. The construction using KMAC ensures a standardized domain separation. In this case, the processed message is then the concatenation of any number of key shares.¶
More precisely, for a given capacity c
the indifferentiability proof shows
that assuming there are no weaknesses found in the Keccak permutation, an
attacker has to make an expected number of 2^(c/2)
calls to the permutation
to tell KMAC from a random oracle. For a random oracle, a difference in only a
single bit gives an unrelated, uniformly random output. Hence, to be able to
distinguish a key K
, derived from shared keys K1
and K2
(and ciphertexts
C1
and C2
) as¶
K = KMAC(domainSeparation, counter  K1  C1  K2  C2  fixedInfo, outputBits, customization)¶
from a random bit string, an adversary has to know (or correctly guess) both
key shares K1
and K2
, entirely.¶
The proposed construction in Section 5.2.2 preserves INDCCA2 of any of its ingredient KEMs, i.e. the newly formed combined KEM is INDCCA2 secure as long as at least one of the ingredient KEMs is. Indeed, the above stated indifferentiability from a random oracle qualifies Keccak as a splitkey pseudorandom function as defined in [GHP18]. That is, Keccak behaves like a random function if at least one input shared secret is picked uniformly at random. Our construction can thus be seen as an instantiation of the INDCCA2 preserving Example 3 in Figure 1 of [GHP18], up to some reordering of input shared secrets and ciphertexts. In the random oracle setting, the reordering does not influence the arguments in [GHP18].¶
9.3. Domain separation and binding
The domSeparation
information defined in Section 5.2.2 provides the
domain separation for the key combiner construction. This ensures that the
input keying material is used to generate a KEK for a specific purpose or
context.¶
The fixedInfo
defined in Section 5.2.1 binds the derived KEK to the
chosen algorithm and communication parties. The algorithm ID identifies
univocally the algorithm, the parameters for its instantiation, and the length
of all artifacts, including the derived key. The hash of the recipient's
public key identifies the subkey used to encrypt the message, binding the KEK
to both the Kyber and the ECC key. Given that both algorithms allow a degree of
ciphertext malleability, this prevents transformations onto the ciphertext
without the final recipient's knowledge.¶
This is in line with the Recommendation for ECC in section 5.5 of [SP80056A]. Other fields included in the recommendation are not relevant for the OpenPGP protocol, since the sender is not required to have a key on their own, there are no preshared secrets, and all the other parameters are univocally defined by the algorithm ID.¶
9.4. SPHINCS+
The original specification of SPHINCS+ [SPHINCSSubm] prescribes an optional randomized hashing. This is not used in this specification, as OpenPGP v6 signatures already provide a salted hash of the appropriate size.¶
9.5. Binding hashes in signatures with signature algorithms
In order not to extend the attack surface, we bind the hash algorithm used for message digestion to the hash algorithm used internally by the signature algorithm. Dilithium internally uses a SHAKE256 digest, therefore we require SHA3 in the Dilithium + ECC signature packet. In the case of SPHINCS+ the internal hash algorithm varies based on the algorithm and parameter ID.¶
10. Additional considerations
10.1. Performance Considerations for SPHINCS+
This specification introduces both Dilithium + ECC as well as SPHINCS+ as PQ(/T) signature schemes.¶
Generally, it can be said that Dilithium + ECC provides a performance in terms of execution time and space requirements that is close to that of traditional ECC signature schemes. Implementers may want to offer SPHINCS+ for applications where a higher degree of trust in the signature scheme is required. However, SPHINCS+ has performance characteristics in terms of execution time of the signature generation as well as space requirements for the signature that can be, depending on the parameter choice, far greater than those of traditional or Dilithium + ECC signature schemes.¶
Pertaining to the execution time, the particularly costly operation in SPHINCS+ is the signature generation. In order to achieve short signature generation times, one of the parameter sets with the name ending in the letter "f" for "fast" should be chosen. This comes at the expense of a larger signature size.¶
In order to minimize the space requirements of a SPHINCS+ signature, a parameter set ending in "s" for "small" should be chosen. This comes at the expense of a larger signature generation time.¶
11. IANA Considerations
IANA will add the following registries to the Pretty Good Privacy (PGP)
registry group at https://www.iana.org/assignments/pgpparameters:¶

Registry name:
SPHINCS+simpleSHA2 parameters
¶ 
Registry name:
SPHINCS+simpleSHAKE parameters
¶
Furthermore IANA will add the algorithm IDs defined in Table 1 and
Table 2 to the registry Public Key Algorithms
.¶
12. Contributors
Stephan Ehlen (BSI)
CarlDaniel Hailfinger (BSI)
Andreas Huelsing (TU Eindhoven)
Johannes Roth (MTG AG)¶
13. References
13.1. Normative References
 [ID.ietfopenpgpcryptorefresh]
 Wouters, P., Huigens, D., Winter, J., and N. Yutaka, "OpenPGP", Work in Progress, InternetDraft, draftietfopenpgpcryptorefresh08, , <https://datatracker.ietf.org/doc/html/draftietfopenpgpcryptorefresh08>.
 [RFC3394]
 Schaad, J. and R. Housley, "Advanced Encryption Standard (AES) Key Wrap Algorithm", RFC 3394, DOI 10.17487/RFC3394, , <https://www.rfceditor.org/rfc/rfc3394>.
 [RFC7748]
 Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, , <https://www.rfceditor.org/rfc/rfc7748>.
 [RFC8032]
 Josefsson, S. and I. Liusvaara, "EdwardsCurve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, , <https://www.rfceditor.org/rfc/rfc8032>.
 [RFC8126]
 Cotton, M., Leiba, B., and T. Narten, "Guidelines for Writing an IANA Considerations Section in RFCs", BCP 26, RFC 8126, DOI 10.17487/RFC8126, , <https://www.rfceditor.org/rfc/rfc8126>.
13.2. Informative References
 [BDPA08]
 Bertoni, G., Daemen, J., Peters, M., and G. Assche, "On the Indifferentiability of the Sponge Construction", , <https://doi.org/10.1007/9783540789673_11>.
 [CS03]
 Cramer, R. and V. Shoup, "Design and Analysis of Practical PublicKey Encryption Schemes Secure against Adaptive Chosen Ciphertext Attack", , <https://doi.org/10.1137/S0097539702403773>.
 [DILITHIUMSubm]
 Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schwabe, P., Seiler, G., and D. Stehle, "CRYSTALSDilithium  Algorithm Specifications and Supporting Documentation (Version 3.1)", .
 [draftdriscollpqthybridterminology]
 Driscoll, F., "Terminology for PostQuantum Traditional Hybrid Schemes", , <https://datatracker.ietf.org/doc/html/draftdriscollpqthybridterminology>.
 [GHP18]
 Giacon, F., Heuer, F., and B. Poettering, "KEM Combiners", , <https://doi.org/10.1007/9783319765785_7>.
 [KYBERSubm]
 Avanzi, R., Bos, J., Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schanck, J. M., Schwabe, P., Seiler, G., and D. Stehle, "CRYSTALSKyber (version 3.02)  Submission to round 3 of the NIST postquantum project", .
 [NISTPQC]
 Chen, L., Moody, D., and Y. Liu, "PostQuantum Cryptography Standardization", , <https://csrc.nist.gov/projects/postquantumcryptography/postquantumcryptographystandardization>.
 [NISTIR8413]
 Alagic, G., Apon, D., Cooper, D., Dang, Q., Dang, T., Kelsey, J., Lichtinger, J., Miller, C., Moody, D., Peralta, R., Perlner, R., Robinson, A., SmithTone, D., and Y. Liu, "Status Report on the Third Round of the NIST PostQuantum Cryptography Standardization Process", NIST IR 8413 , , <https://doi.org/10.6028/NIST.IR.8413upd1>.
 [RFC5639]
 Lochter, M. and J. Merkle, "Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation", RFC 5639, DOI 10.17487/RFC5639, , <https://www.rfceditor.org/rfc/rfc5639>.
 [SEC1]
 Standards for Efficient Cryptography Group, "Standards for Efficient Cryptography 1 (SEC 1)", , <https://secg.org/sec1v2.pdf>.
 [SP800185]
 Kelsey, J., Chang, S., and R. Perlner, "SHA3 Derived Functions: cSHAKE, KMAC, TupleHash, and ParallelHash", NIST Special Publication 800185 , , <https://doi.org/10.6028/NIST.SP.800185>.
 [SP800186]
 Chen, L., Moody, D., Regenscheid, A., and K. Randall, "Recommendations for Discrete LogarithmBased Cryptography: Elliptic Curve Domain Parameters", NIST Special Publication 800186 , , <https://doi.org/10.6028/NIST.SP.800186>.
 [SP80056A]
 Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R. Davis, "Recommendation for PairWise KeyEstablishment Schemes Using Discrete Logarithm Cryptography", NIST Special Publication 80056A Rev. 3 , , <https://doi.org/10.6028/NIST.SP.80056Ar3>.
 [SP80056C]
 Barker, E., Chen, L., and R. Davis, "Recommendation for KeyDerivation Methods in KeyEstablishment Schemes", NIST Special Publication 80056C Rev. 2 , , <https://doi.org/10.6028/NIST.SP.80056Cr2>.
 [SPHINCSSubm]
 Aumasson, J., Bernstein, D. J., Beullens, W., Dobraunig, C., Eichlseder, M., Fluhrer, S., Gazdag, S., Huelsing, A., Kampanakis, P., Koelb, S., Lange, T., Lauridsen, M. M., Mendel, F., Niederhagen, R., Rechberger, C., Rijneveld, J., Schwabe, P., and B. Westerbaan, "SPHINCS+  Submission to the 3rd round of the NIST postquantum project. v3.1", .
Acknowledgments
Thanks to Daniel Huigens and Evangelos Karatsiolis for the early review and feedback on this document.¶