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title: "Hybrid PQ/T Key Encapsulation Mechanisms" abbrev: hybrid-kems category: info

docname: draft-irtf-cfrg-hybrid-kems-latest submissiontype: IRTF consensus: false v: 3 workgroup: "Crypto Forum"

author:

fullname: Deirdre Connolly
organization: SandboxAQ
email: [email protected]

normative: FIPS202: DOI.10.6028/NIST.FIPS.202 FIPS203: DOI.10.6028/NIST.FIPS.203

informative: ANSIX9.62: title: "Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)" date: Nov, 2005 seriesinfo: "ANS": X9.62-2005 author: - org: ANS AVIRAM: target: https://mailarchive.ietf.org/arch/msg/tls/F4SVeL2xbGPaPB2GW_GkBbD_a5M/ title: "[TLS] Combining Secrets in Hybrid Key Exchange in TLS 1.3" date: 2021-09-01 author: - ins: Nimrod Aviram - ins: Benjamin Dowling - ins: Ilan Komargodski - ins: Kenny Paterson - ins: Eyal Ronen - ins: Eylon Yogev BDG2020: title: "Separate Your Domains: NIST PQC KEMs, Oracle Cloning and Read-Only Indifferentiability" target: https://eprint.iacr.org/2020/241.pdf date: 2020 CDM23: title: "Keeping Up with the KEMs: Stronger Security Notions for KEMs and automated analysis of KEM-based protocols" target: https://eprint.iacr.org/2023/1933.pdf date: 2023 author: - ins: C. Cremers name: Cas Cremers org: CISPA Helmholtz Center for Information Security - ins: A. Dax name: Alexander Dax org: CISPA Helmholtz Center for Information Security - ins: N. Medinger name: Niklas Medinger org: CISPA Helmholtz Center for Information Security FIPS186: DOI.10.6028/NIST.FIPS.186-5 #https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-5.pdf GHP2018: title: "KEM Combiners" target: https://eprint.iacr.org/2018/024.pdf date: 2018 I-D.driscoll-pqt-hybrid-terminology: KSMW2024: target: https://eprint.iacr.org/2024/1233 title: "Binding Security of Implicitly-Rejecting KEMs and Application to BIKE and HQC" author: - ins: J. Kraemer - ins: P. Struck - ins: M. Weishaupl LUCKY13: target: https://ieeexplore.ieee.org/iel7/6547086/6547088/06547131.pdf title: "Lucky Thirteen: Breaking the TLS and DTLS record protocols" author: - ins: N. J. Al Fardan - ins: K. G. Paterson RACCOON: target: https://raccoon-attack.com/ title: "Raccoon Attack: Finding and Exploiting Most-Significant-Bit-Oracles in TLS-DH(E)" author: - ins: R. Merget - ins: M. Brinkmann - ins: N. Aviram - ins: J. Somorovsky - ins: J. Mittmann - ins: J. Schwenk date: 2020-09 HKDF: RFC5869 SCHMIEG2024: title: "Unbindable Kemmy Schmidt: ML-KEM is neither MAL-BIND-K-CT nor MAL-BIND-K-PK" target: https://eprint.iacr.org/2024/523.pdf date: 2024 author: - ins: S. Schmieg name: Sophie Schmieg SEC1: title: "Elliptic Curve Cryptography, Standards for Efficient Cryptography Group, ver. 2" target: https://secg.org/sec1-v2.pdf date: 2009 XWING: title: "X-Wing: The Hybrid KEM You’ve Been Looking For" target: https://eprint.iacr.org/2024/039.pdf date: 2024

XWING-EC-PROOF: https://github.com/formosa-crypto/formosa-x-wing/

--- abstract

This document defines generic techniques to achive hybrid post-quantum/traditional (PQ/T) key encapsulation mechanisms (KEMs) from post-quantum and traditional component algorithms that meet specified security properties. It then uses those generic techniques to construct several concrete instances of hybrid KEMs.

--- middle

Introduction {#intro}

There are many choices that can be made when specifying a hybrid KEM: the constituent KEMs; their security levels; the combiner; and the hash within, to name but a few. Having too many similar options are a burden to the ecosystem.

The aim of this document is provide a small set of techniques for constructing hybrid KEMs designed to achieve specific security properties given conforming component algorithms, that should be suitable for the vast majority of use cases.

Requirements Notation

{::boilerplate bcp14-tagged}

Notation

This document is consistent with all terminology defined in {{I-D.driscoll-pqt-hybrid-terminology}}.

The following terms are used throughout this document:

  • random(n): return a pseudorandom byte string of length n bytes produced by a cryptographically-secure random number generator.
  • concat(x0, ..., xN): Concatenation of byte strings. concat(0x01, 0x0203, 0x040506) = 0x010203040506.
  • I2OSP(n, w): Convert non-negative integer n to a w-length, big-endian byte string, as described in {{!RFC8017}}.
  • OS2IP(x): Convert byte string x to a non-negative integer, as described in {{!RFC8017}}, assuming big-endian byte order.

Cryptographic Dependencies {#cryptographic-deps}

The generic hybrid PQ/T KEM constructions we define depend on the the following cryptographic primitives:

  • Key Encapsulation Mechanism {{kems}};
  • Extendable Output Function (XOF) {{xof}};
  • Key Derivation Function (KDF) {{kdf}}; and
  • Nominal Diffie-Hellman Group {{group}}.

These dependencies are defined in the following subsections.

Key encapsulation mechanisms {#kems}

Key encapsulation mechanisms (KEMs) are cryptographic schemes that consist of four algorithms:

  • KeyGen() -> (pk, sk): A probabilistic key generation algorithm, which generates a public encapsulation key pk and a secret decapsulation key sk, each of which are byte strings.
  • DeriveKey(seed) -> (pk, sk): A deterministic algorithm, which takes as input a seed seed and generates a public encapsulation key pk and a secret decapsulation key sk, each of which are byte strings.
  • Encaps(pk) -> (ct, shared_secret): A probabilistic encapsulation algorithm, which takes as input a public encapsulation key pk and outputs a ciphertext ct and shared secret shared_secret.
  • Decaps(sk, ct) -> shared_secret: A decapsulation algorithm, which takes as input a secret decapsulation key sk and ciphertext ct and outputs a shared secret shared_secret.

KEMs can also provide a deterministic version of Encaps, denoted EncapsDerand, with the following signature:

  • EncapsDerand(pk, randomness) -> (ct, shared_secret): A deterministic encapsulation algorithm, which takes as input a public encapsulation key pk and randomness randomness, and outputs a ciphertext ct and shared secret shared_secret.

Finally, KEMs are also parameterized with the following constants:

  • Nseed, which denotes the number of bytes for a key seed;
  • Npk, which denotes the number of bytes in a public encapsulation key;
  • Nsk, which denotes the number of bytes in a private decapsulation key; and
  • Nct, which denotes the number of bytes in a ciphertext.

XOF {#xof}

Extendable-output function (XOF). A function on bit strings in which the output can be extended to any desired length. Ought to satisfy the following properties as long as the specified output length is sufficiently long to prevent trivial attacks:

  1. (One-way) It is computationally infeasible to find any input that maps to any new pre-specified output.

  2. (Collision-resistant) It is computationally infeasible to find any two distinct inputs that map to the same output.

MUST provide the bit-security required to source input randomness for PQ/T components from a seed that is expanded to a output length, of which a subset is passed to the component key generation algorithms.

Key Derivation Function KDF {#kdf}

A secure key derivation function (KDF) that is modeled as a secure pseudorandom function (PRF) in the standard model {{GHP2018}} and independent random oracle in the random oracle model (ROM).

Nominal Diffie-Hellman Group {#group}

The traditional DH-KEM construction depends on an abelian group of order order. We represent this group as the object G that additionally defines helper functions described below. The group operation for G is addition + with identity element I. For any elements A and B of the group G, A + B = B + A is also a member of G. Also, for any A in G, there exists an element -A such that A + (-A) = (-A) + A = I. For convenience, we use - to denote subtraction, e.g., A - B = A + (-B). Integers, taken modulo the group order order, are called scalars; arithmetic operations on scalars are implicitly performed modulo order. Scalar multiplication is equivalent to the repeated application of the group operation on an element A with itself r-1 times, denoted as ScalarMult(A, r). We denote the sum, difference, and product of two scalars using the +, -, and * operators, respectively. (Note that this means + may refer to group element addition or scalar addition, depending on the type of the operands.) For any element A, ScalarMult(A, order) = I. We denote B as a fixed generator of the group. Scalar base multiplication is equivalent to the repeated application of the group operation on B with itself r-1 times, this is denoted as ScalarBaseMult(r). The set of scalars corresponds to GF(order), which we refer to as the scalar field. It is assumed that group element addition, negation, and equality comparison can be efficiently computed for arbitrary group elements.

This document uses types Element and Scalar to denote elements of the group G and its set of scalars, respectively. We denote Scalar(x) as the conversion of integer input x to the corresponding Scalar value with the same numeric value. For example, Scalar(1) yields a Scalar representing the value 1. We denote equality comparison of these types as == and assignment of values by =. When comparing Scalar values, e.g., for the purposes of sorting lists of Scalar values, the least nonnegative representation mod order is used.

We now detail a number of member functions that can be invoked on G.

  • Order(): Outputs the order of G (i.e., order).
  • Identity(): Outputs the identity Element of the group (i.e., I).
  • RandomScalar(): Outputs a random Scalar element in GF(order), i.e., a random scalar in [0, order - 1].
  • ScalarMult(A, k): Outputs the scalar multiplication between Element A and Scalar k.
  • ScalarBaseMult(k): Outputs the scalar multiplication between Scalar k and the group generator B.
  • SerializeElement(A): Maps an Element A to a canonical byte array buf of fixed length Ne. This function raises an error if A is the identity element of the group.
  • DeserializeElement(buf): Attempts to map a byte array buf to an Element A, and fails if the input is not the valid canonical byte representation of an element of the group. This function raises an error if deserialization fails or if A is the identity element of the group.
  • SerializeScalar(s): Maps a Scalar s to a canonical byte array buf of fixed length Ns.
  • DeserializeScalar(buf): Attempts to map a byte array buf to a Scalar s. This function raises an error if deserialization fails.

Hybrid KEM Constructions {#constructions}

During encapsulation and decapsulation, a hybrid KEM combines its component KEM shared secrets and other info, such as the KEM ciphertexts and encapsulation keys keys, to yield a shared secret. The interface for this function, often called a 'combiner' in the literature, is the SharedSecret function for the constructions in this document. SharedSecret accepts the following inputs:

  • pq_SS: The PQ KEM shared secret.
  • trad_SS: The traditional KEM shared secret.
  • pq_CT: The PQ KEM ciphertext.
  • pq_PK: The PQ KEM public encapsulation key.
  • trad_CT: The traditional KEM ciphertext.
  • trad_PK: The traditional KEM public encapsulation key.
  • label: A domain-separating label; see {{domain-separation}} for more information on the role of the label.

The output of the SharedSecret function is a 32 byte shared secret that is, ultimately, the output of the KEM.

This section describes two generic constructions for hybrid KEMs: one called the KitchenSink, specified in {{KitchenSink}}, and another called QSF, specified in {{QSF}}. The KitchenSink construction is maximally conservative in design, opting for the least assumptions about the component KEMs. The QSF construction is tailored to specific component KEMs and is not generally reusable; specific requirements for component KEMs to be usable in the QSF combiner are detailed in {{QSF}}.

Both make use of the following requirements:

  1. Both component KEMs have IND-CCA security.
  2. KDF as a secure PRF. A key derivation function (KDF) that is modeled as a secure pseudorandom function (PRF) in the standard model {{GHP2018}} and independent random oracle in the random oracle model (ROM).
  3. Fixed-length values. Every instantiation in concrete parameters of the generic constructions is for fixed parameter sizes, KDF choice, and label, allowing the lengths to not also be encoded into the generic construction. The label/KDF/component algorithm parameter sets MUST be disjoint and non-colliding. Moreover, the length of each each public encapsulation key, ciphertext, and shared secret is fixed once the algorithm is assumed to be fixed.

'Kitchen Sink' construction {#KitchenSink}

As indicated by the name, the KitchenSink puts 'the whole transcript' through the KDF. This relies on the minimum security properties of its component algorithms at the cost of more bytes needing to be processed by the KDF.

def KitchenSink-KEM.SharedSecret(pq_SS, trad_SS, pq_CT, pq_PK, trad_CT,
                                 trad_PK, label):
    input = concat(pq_SS, trad_SS, pq_CT, pq_PK,
                   trad_CT, trad_PK, label)
    return KDF(input)

Security properties

Because the entire hybrid KEM ciphertext and encapsulation key material are included in the KDF preimage, the KitchenSink construction is resilient against implementation errors in the component algorithms.

'QSF' construction {#QSF}

Inspired by the generic QSF (Quantum Superiority Fighter) framework in {{XWING}}, which leverages the security properties of a KEM like ML-KEM and an inlined instance of DH-KEM, to elide other public data like the PQ ciphertext and encapsulation key from the KDF input:

def QSF-KEM.SharedSecret(pq_SS, trad_SS, pq_CT, pq_PK, trad_CT,
                         trad_PK, label):
    return KDF(concat(pq_SS, trad_SS, trad_CT, trad_PK, label))

Note that pq_CT and pq_PK are NOT included in the KDF. This is only possible because the component KEMs adhere to the following requirements. The QSF combiner MUST NOT be used in concrete KEM instances that do not satisfy these requirements.

  1. Nominal Diffie-Hellman Group with strong Diffie-Hellman security

A cryptographic group modelable as a nominal group where the strong Diffie-Hellman assumption holds {XWING}. Specically regarding a nominal group, this means that especially the QSF construction's security is based on a computational-Diffie-Hellman-like problem, but no assumption is made about the format of the generated group element - no assumption is made that the shared group element is indistinguishable from random bytes.

The concrete instantiations in this document use elliptic curve groups that have been modeled as nominal groups in the literature.

  1. Post-quantum IND-CCA KEM with ciphertext second preimage resistance

The QSF relies the post-quantum KEM component having IND-CCA security against a post-quantum attacker, and ciphertext second preimage resistance (C2SPI, also known as chosen ciphertext resistance, CCR). C2SPI/CCR is [equivalent to LEAK-BIND-K,PK-CT security][CDM23]

  1. KDF is a secure (post-quantum) PRF, modelable as a random oracle.

Indistinguishability of the final shared secret from a random key is established by modeling the key-derivation function as a random oracle {{XWING}}.

Concrete Hybrid KEM Instances

This section instantiates three concrete KEMs:

  1. QSF-SHA3-256-ML-KEM-768-P-256 {{qsf-p256}}: A hybrid KEM using the QSF combiner based on ML-KEM-768 and P-256.
  2. KitchenSink-HKDF-SHA-256-ML-KEM-768-X25519 {{ks-x25519}}: A hybrid KEM using the KitchenSink combiner based on ML-KEM-768 and X25519.
  3. QSF-SHA3-256-ML-KEM-1024-P-384 {{qsf-p384}}: A hybrid KEM using the QSF combiner based on ML-KEM-1024 and P-384.

Each instance specifies the PQ and traditional KEMs being combined, the combiner construction from {{constructions}}, the label to use for domain separation in the combiner function, as well as the XOF and KDF functions to use throughout.

QSF-SHA3-256-ML-KEM-768-P-256 {#qsf-p256}

This hybrid KEM is heavily based on {{XWING}}. In particular, it has the same exact design but uses P-256 instead of X25519 as the the traditional component of the algorithm. It has the following parameters.

  • label: QSF-SHA3-256-ML-KEM-768-P-256
  • XOF: SHAKE-256 {{FIPS202}}
  • KDF: SHA3-256 {{FIPS202}}
  • Combiner: QSF-KEM.SharedSecret
  • Nseed: 65
  • Npk: 1217
  • Nsk: 32
  • Nct: 1121

QSF-SHA3-256-ML-KEM-768-P-256 depends on P-256 as a nominal prime-order group {{FIPS186}} (secp256r1) {{ANSIX9.62}}, where Ne = 33 and Ns = 32, with the following functions:

  • Order(): Return 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551.
  • Identity(): As defined in {{ANSIX9.62}}.
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to {{random-scalar}} for implementation guidance.
  • SerializeElement(A): Implemented using the compressed Elliptic-Curve-Point-to-Octet-String method according to {{SEC1}}, yielding a 33-byte output. Additionally, this function validates that the input element is not the group identity element.
  • DeserializeElement(buf): Implemented by attempting to deserialize a 33-byte input string to a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to {{SEC1}}, and then performs public-key validation as defined in section 3.2.2.1 of {{SEC1}}. This includes checking that the coordinates of the resulting point are in the correct range, that the point is on the curve, and that the point is not the point at infinity. (As noted in the specification, validation of the point order is not required since the cofactor is 1.) If any of these checks fail, deserialization returns an error.
  • SerializeScalar(s): Implemented using the Field-Element-to-Octet-String conversion according to {{SEC1}}.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 32-byte string using Octet-String-to-Field-Element from {{SEC1}}. This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1].

The rest of this section specifies the key generation, encapsulation, and decapsulation procedures for this hybrid KEM.

Key generation

QSF-SHA3-256-ML-KEM-768-P-256 KeyGen works as follows.

def expandDecapsulationKey(sk):
  expanded = SHAKE256(sk, 96)
  (pq_PK, pq_SK) = ML-KEM-768.KeyGen_internal(expanded[0:32], expanded[32:64])
  trad_SK = P-256.Scalar(expanded[64:96])
  trad_PK = P-256.ScalarMultBase(trad_SK)
  return (pq_SK, trad_SK, pq_PK, trad_PK)

def KeyGen():
  sk = random(32)
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(sk)
  return sk, concat(pq_PK, trad_PK)

Similarly, QSF-SHA3-256-ML-KEM-768-P-256 DeriveKey works as follows:

def DeriveKey(seed):
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(seed)
  return sk, concat(pq_PK, trad_PK)

Encapsulation

Given an encapsulation key pk, QSF-SHA3-256-ML-KEM-768-P-256 Encaps proceeds as follows.

def Encaps(pk):
  pq_PK = pk[0:1184]
  trad_PK = pk[1184:1217]
  (pq_SS, pq_CT) = ML-KEM-768.Encaps(pq_PK)
  ek = P-256.RandomScalar()
  trad_CT = P-256.ScalarBaseMult(ek)
  trad_SS = P-256.ScalarMult(trad_PK, ek)
  ss = SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)
  ct = concat(pq_CT, trad_CT)
  return (ss, ct)

pk is a 1217-byte encapsulation key resulting from KeyGen().

Encaps() returns the 32-byte shared secret ss and the 1121-byte ciphertext ct.

Note that Encaps() may raise an error if ML-KEM-768.Encaps fails, e.g., if it does not pass the check of {{FIPS203}} §7.2.

Derandomized

For testing, it is convenient to have a deterministic version of encapsulation. In such cases, an implementation can provide the following derandomized function.

def EncapsDerand(pk, randomness):
  pq_PK = pk[0:1184]
  trad_PK = pk[1184:1217]
  (pq_SS, pq_CT) = ML-KEM-768.EncapsDerand(pq_PK, randomness[0:32])
  ek = randomness[32:65]
  trad_CT = P-256.ScalarMultBase(ek)
  trad_SS = P-256.ScalarMult(ek, trad_PK)
  ss = SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)
  ct = concat(pq_CT, trad_CT)
  return (ss, ct)

Note that randomness MUST be 65 bytes.

Decapsulation

Given a decapsulation key sk and ciphertext ct, QSF-SHA3-256-ML-KEM-768-P-256 Decaps proceeds as follows.

def Decaps(sk, ct):
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(sk)
  pq_CT = ct[0:1088]
  trad_CT = ct[1088:1121]
  pq_SS = ML-KEM-768.Decapsulate(pq_SK, pq_CT)
  trad_SS = P-256.ScalarMult(trad_SK, trad_CT)
  return SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)

ct is the 1121-byte ciphertext resulting from Encaps() and sk is a 32-byte decapsulation key resulting from KeyGen().

Decaps() returns the 32 byte shared secret.

Security properties

The inlined DH-KEM is instantiated over the elliptic curve group P-256: as shown in {{CDM23}}, this gives the traditional KEM maximum binding properties (MAL-BIND-K-CT, MAL-BIND-K-PK).

ML-KEM-768 as standardized in {{FIPS203}}, when using the 64-byte seed key format as is here, provides MAL-BIND-K-CT security and LEAK-BIND-K-PK security, as demonstrated in {{SCHMIEG2024}}.

Therefore this concrete instance provides MAL-BIND-K-PK and MAL-BIND-K-CT security.

This implies via {{KSMW2024}} that this instance also satisfies

  • MAL-BIND-K,CT-PK
  • MAL-BIND-K,PK-CT
  • LEAK-BIND-K-PK
  • LEAK-BIND-K-CT
  • LEAK-BIND-K,CT-PK
  • LEAK-BIND-K,PK-CT
  • HON-BIND-K-PK
  • HON-BIND-K-CT
  • HON-BIND-K,CT-PK
  • HON-BIND-K,PK-CT

KitchenSink-HKDF-SHA-256-ML-KEM-768-X25519 {#ks-x25519}

KitchenSink-HKDF-SHA-256-ML-KEM-768-X25519 has the following parameters.

  • label: KitchenSink-HKDF-SHA-256-ML-KEM-768-X25519
  • XOF: SHAKE-256 {{FIPS202}}
  • KDF: HKDF-SHA-256 {{HKDF}}
  • Combiner: KitchenSink-KEM.SharedSecret
  • Nseed: 96
  • Npk: 1216
  • Nsk: 32
  • Nct: 1120

KitchenSink-HKDF-SHA-256-ML-KEM-768-X25519 depends on a prime-order group implemented using Curve25519 and X25519 {{!RFC7748}}. Additionally, it uses a modified version of HKDF in the combiner, denoted LabeledHKDF, defined below.

QSF-SHA3-256-ML-KEM-1024-P-384 has the following parameters.

  • label: QSF-SHA3-256-ML-KEM-768-P-256
  • XOF: SHAKE-256 {{FIPS202}}
  • KDF: SHA3-256 {{FIPS202}}
  • Combiner: QSF-KEM.SharedSecret
  • Nseed: 112
  • Npk: 1629
  • Nsk: 32
  • Nct: 1629

QSF-SHA3-256-ML-KEM-1024-P-384 depends on P-384 as a nominal prime-order group {{FIPS186}} (secp256r1) {{ANSIX9.62}}, where Ne = 61 and Ns = 48, with the following functions:

  • Order(): Return 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf 581a0db248b0a77aecec196accc52973
  • Identity(): As defined in {{ANSIX9.62}}.
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to {{random-scalar}} for implementation guidance.
  • SerializeElement(A): Implemented using the compressed Elliptic-Curve-Point-to-Octet-String method according to {{SEC1}}, yielding a 61-byte output. Additionally, this function validates that the input element is not the group identity element.
  • DeserializeElement(buf): Implemented by attempting to deserialize a 61-byte input string to a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to {{SEC1}}, and then performs public-key validation as defined in section 3.2.2.1 of {{SEC1}}. This includes checking that the coordinates of the resulting point are in the correct range, that the point is on the curve, and that the point is not the point at infinity. (As noted in the specification, validation of the point order is not required since the cofactor is 1.) If any of these checks fail, deserialization returns an error.
  • SerializeScalar(s): Implemented using the Field-Element-to-Octet-String conversion according to {{SEC1}}.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 48-byte string using Octet-String-to-Field-Element from {{SEC1}}. This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1].

The rest of this section specifies the key generation, encapsulation, and decapsulation procedures for this hybrid KEM.

Key generation

QSF-SHA3-256-ML-KEM-1024-P-384 KeyGen works as follows.

def expandDecapsulationKey(sk):
  expanded = SHAKE256(sk, 112)
  (pq_PK, pq_SK) = ML-KEM-1024.KeyGen_internal(expanded[0:32], expanded[32:64])
  trad_SK = P-384.Scalar(expanded[64:112])
  trad_PK = P-384.ScalarMultBase(trad_SK)
  return (pq_SK, trad_SK, pq_PK, trad_PK)

def KeyGen():
  sk = random(32)
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(sk)
  return sk, concat(pq_PK, trad_PK)

Similarly, QSF-SHA3-256-ML-KEM-1024-P-384 DeriveKey works as follows:

def DeriveKey(seed):
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(seed)
  return sk, concat(pq_PK, trad_PK)

Encapsulation

Given an encapsulation key pk, QSF-SHA3-256-ML-KEM-1024-P-384 Encaps proceeds as follows.

def Encaps(pk):
  pq_PK = pk[0:1568]
  trad_PK = pk[1568:1629]
  (pq_SS, pq_CT) = ML-KEM-1024.Encaps(pq_PK)
  ek = P-384.RandomScalar()
  trad_CT = P-384.ScalarBaseMult(ek)
  trad_SS = P-384.ScalarMult(trad_PK, ek)
  ss = SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)
  ct = concat(pq_CT, trad_CT)
  return (ss, ct)

pk is a 1629-byte encapsulation key resulting from KeyGen().

Encaps() returns the 32-byte shared secret ss and the 1629-byte ciphertext ct.

Note that Encaps() may raise an error if ML-KEM-1024.Encaps fails, e.g., if it does not pass the check of {{FIPS203}} §7.2.

Derandomized

For testing, it is convenient to have a deterministic version of encapsulation. In such cases, an implementation can provide the following derandomized function.

def EncapsDerand(pk, randomness):
  pq_PK = pk[0:1568]
  trad_PK = pk[1568:1629]
  (pq_SS, pq_CT) = ML-KEM-1024.EncapsDerand(pq_PK, randomness[0:32])
  ek = randomness[32:80]
  trad_CT = P-384.ScalarMultBase(ek)
  trad_SS = P-384.ScalarMult(ek, trad_PK)
  ss = SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)
  ct = concat(pq_CT, trad_CT)
  return (ss, ct)

Note that randomness MUST be 80 bytes.

Decapsulation

Given a decapsulation key sk and ciphertext ct, QSF-SHA3-256-ML-KEM-1024-P-384 Decaps proceeds as follows.

def Decaps(sk, ct):
  (pq_SK, trad_SK, pq_PK, trad_PK) = expandDecapsulationKey(sk)
  pq_CT = ct[0:1568]
  trad_CT = ct[1568:1629]
  pq_SS = ML-KEM-1024.Decapsulate(pq_SK, pq_CT)
  trad_SS = P-384.ScalarMult(trad_SK, trad_CT)
  return SHA3-256(pq_SS, trad_SS, trad_CT, trad_PK, label)

ct is the 1629-byte ciphertext resulting from Encaps() and sk is a 32-byte decapsulation key resulting from KeyGen().

Decaps() returns the 32-byte shared secret.

Security properties

The inlined DH-KEM is instantiated over the elliptic curve group P-384: as shown in {{CDM23}}, this gives the traditional KEM maximum binding properties (MAL-BIND-K-CT, MAL-BIND-K-PK).

ML-KEM-1024 as standardized in {{FIPS203}}, when using the 64-byte seed key format as is here, provides MAL-BIND-K-CT security and LEAK-BIND-K-PK security, as demonstrated in {{SCHMIEG2024}}.

Therefore this concrete instance provides MAL-BIND-K-PK and MAL-BIND-K-CT security.

This implies via {{KSMW2024}} that this instance also satisfies

  • MAL-BIND-K,CT-PK
  • MAL-BIND-K,PK-CT
  • LEAK-BIND-K-PK
  • LEAK-BIND-K-CT
  • LEAK-BIND-K,CT-PK
  • LEAK-BIND-K,PK-CT
  • HON-BIND-K-PK
  • HON-BIND-K-CT
  • HON-BIND-K,CT-PK
  • HON-BIND-K,PK-CT

Random Scalar Generation {#random-scalar}

Two popular algorithms for generating a random integer uniformly distributed in the range [0, G.Order() -1] are as follows:

Rejection Sampling

Generate a random byte array with Ns bytes, and attempt to map to a Scalar by calling DeserializeScalar in constant time. If it succeeds, return the result. If it fails, try again with another random byte array, until the procedure succeeds. Failure to implement DeserializeScalar in constant time can leak information about the underlying corresponding Scalar.

As an optimization, if the group order is very close to a power of 2, it is acceptable to omit the rejection test completely. In particular, if the group order is p, and there is an integer b such that |p - 2b| is less than 2(b/2), then RandomScalar can simply return a uniformly random integer of at most b bits.

Wide Reduction

Generate a random byte array with l = ceil(((3 * ceil(log2(G.Order()))) / 2) / 8) bytes, and interpret it as an integer; reduce the integer modulo G.Order() and return the result. See {{Section 5 of !HASH-TO-CURVE=RFC9380}} for the underlying derivation of l.

Security Considerations

Hybrid KEM constructions aim to provide security by combining two or more schemes so that security is preserved if all but one schemes are replaced by an arbitrarily bad scheme. Informally, these hybrid KEMs are secure if the KDF is secure, and either the elliptic curve is secure, or the post-quantum KEM is secure: this is the 'hybrid' property.

More precisely for the concrete instantiations in this document, if SHA3-256, SHA3-512, and SHAKE-256 may be modelled as a random oracle, then the IND-CCA security of QSF constructions is bounded by the IND-CCA security of ML-KEM, and the gap-CDH security of secp256n1, see {{XWING}}.

IND-CCA security

Also known as IND-CCA2 security for general public key encryption, for KEMs that encapsulate a new random 'message' each time.

The notion of INDistinguishability against Chosen-Ciphertext Attacks (IND-CCA) [RS92] is now widely accepted as the standard security notion for asymmetric encryption schemes. IND-CCA security requires that no efficient adversary can recognize which of two messages is encrypted in a given ciphertext, even if the two candidate messages are chosen by the adversary himself.

Ciphertext second preimage resistant (C2PRI) security / ciphertext collision resistance (CCR)

The notion where, even if a KEM has broken IND-CCA security (either due to construction, implementation, or other), its internal structure, based on the Fujisaki-Okamoto transform, guarantees that it is impossible to find a second ciphertext that decapsulates to the same shared secret K: this notion is known as ciphertext second preimage resistance (C2SPI) for KEMs {{XWING}}. The same notion has also been described as chosen ciphertext resistance elsewhere {{CDM23}}.

Binding properties

TODO

X-BIND-K-PK security

TODO

X-BIND-K-CT security

Ciphertext second preimage resistance for KEMs ([C2PRI]{{XWING}}). Related to the ciphertext collision-freeness of the underlying PKE scheme of a FO-transform KEM. Also called ciphertext collision resistance.

Domain Separation {#domain-separation}

ASCII-encoded bytes provide oracle cloning {{BDG2020}} in the security game via domain separation. The IND-CCA security of hybrid KEMs often relies on the KDF function KDF to behave as an independent random oracle, which the inclusion of the label achieves via domain separation {{GHP2018}}.

By design, the calls to KDF in these constructions and usage anywhere else in higher level protoocl use separate input domains unless intentionally duplicating the 'label' per concrete instance with fixed paramters. This justifies modeling them as independent functions even if instantiated by the same KDF. This domain separation is achieved by using prefix-free sets of label values. Recall that a set is prefix-free if no element is a prefix of another within the set.

Length diffentiation is sometimes used to achieve domain separation but as a technique it is [brittle and prone to misuse]{{BDG2020}} in practice so we favor the use of an explicit post-fix label.

Fixed-length

Variable-length secrets are generally dangerous. In particular, using key material of variable length and processing it using hash functions may result in a timing side channel. In broad terms, when the secret is longer, the hash function may need to process more blocks internally. In some unfortunate circumstances, this has led to timing attacks, e.g. the Lucky Thirteen [LUCKY13] and Raccoon [RACCOON] attacks.

Furthermore, [AVIRAM] identified a risk of using variable-length secrets when the hash function used in the key derivation function is no longer collision-resistant.

If concatenation were to be used with values that are not fixed-length, a length prefix or other unambiguous encoding would need to be used to ensure that the composition of the two values is injective and requires a mechanism different from that specified in this document.

Therefore, this specification MUST only be used with algorithms which have fixed-length shared secrets (after the variant has been fixed by the algorithm identifier in the NamedGroup negotiation in Section 3.1).

Out of Scope

Considerations that were considered and not included in these designs:

More than two component KEMs

Design team decided to restrict the space to only two components, a traditional and a post-quantum KEM.

Parameterized output length

Not analyzed as part of any security proofs in the literature, and a complicatation deemed unnecessary.

Protocol-specific labels / info

The concrete instantiations have specific labels, protocol-specific information is out of scope.

Other Component Primitives

There is demand for other hybrid variants that either use different primitives (RSA, NTRU, Classic McEliece, FrodoKEM), parameters, or that use a combiner optimized for a specific use case. Other use cases could be covered in subsequent documents and not included here.

IANA Considerations

TODO

HPKE

TODO

--- back

Acknowledgments

{:numbered="false"}

TODO acknowledge.