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Security CFRG Internet-Draft The migration to post-quantum cryptography often calls for performing multiple key encapsulations in parallel and then combining their outputs to derive a single shared secret. This document defines the KEM combiner KDF( H(ss1) || H(ss2) ) which is considered to be a dual PRF in practice, even though not provably secure. This mechanism simplifies to KDF( ss1 || ss2 ) when used with a KEM which internally uses a KDF to produce its shared secret. RSA-KEM, ECDH, Edwards curve DH, and CRYSTALS-Kyber are shown to meet this criteria and therefore be safe to use with the simplified KEM combiner. Status information for this document may be found at . Discussion of this document takes place on the Limited Additional Mechanisms for PKIX and SMIME (lamps) Working Group mailing list (), which is archived at . Subscribe at . Source for this draft and an issue tracker can be found at .

Terminology The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 when, and only when, they appear in all capitals, as shown here. This document is consistent with all terminology defined in .

Key Encapsulation Mechanisms For the purposes of this document, we consider a Key Encapsulation Mechanism (KEM) to be any asymmetric cryptographic scheme comprised of algorithms satisfying the following interfaces .

(pk, sk) def kemEncaps(pk) -> (ct, ss) def kemDecaps(ct, sk) -> ss ]]>

where pk is public key, sk is secret key, ct is ciphertext, and ss is shared secret. KEMs are typically used in cases where two parties, hereby refereed to as the "encapsulater" and the "decapsulater", wish to establish a shared secret via public key cryptography, where the decapsulater has an asymmetric key pair and has previously shared the public key with the encapsulater.

Introduction The need for a KEM combiner function arises in three different contexts within IETF security protocols: KEM / PSK hybrids where the output of a KEM is combined with a static pre-shared key. Post-quantum / tradtional hybrid KEMs. KEM-based authenticated key exchanges (AKEs). This document normalizes a mechanisms for combining the output of two KEMs.

KEM/PSK hybrids As a post-quantum stop-gap, several IETF protocols have added extensions to allow for mixing a pre-shared key (PSK) into an (EC)DH based key exchange. Examples include CMS and IKEv2 .

PQ/Traditional hybrid KEMs A post-quantum / traditional hybrid key encapsulation mechanism (hybrid KEM) as defined in as

PQ/T Hybrid Key Encapsulation Mechanism:

A Key Encapsulation Mechanism (KEM) made up of two or more component KEM algorithms where at least one is a post-quantum algorithm and at least one is a traditional algorithm.

Building a PQ/T hybrid KEM requires a secure function which combines the output of both component KEMs to form a single output. Several IETF protocols are adding PQ/T hybrid KEM mechanisms as part of their overall post-quantum migration strategies, examples include TLS 1.3 , IKEv2 , X.509; PKIX; CMS , OpenPGP (CITE once Aron's draft is up), JOSE / COSE (CITE once Orie's drafts are up).

KEM-based AKE The need for a KEM-based authenticated key establishment arises, for example, when two communicating parties each have long-term KEM keys (for example in X.509 certificates), and wish to involve both KEM keys in deriving a mutually-authenticated shared secret. In particular this will arise for any protocol that needs to provide post-quantum replacements for static-static (elliptic curve) Diffie-Hellman mechanisms. Examples include a KEM replacement for CMP's DHBasedMac , .. TODO: cite others.

KEM Combiner A KEM combiner is a function that takes in two shared secrets and returns a combined shared secret, where all values are byte arrays.

This document assumes that shared secrets are the output of a KEM, but without loss of generality they may also be any other source of cryptographic key material, such as pre-shared keys (PSKs), with PQ/PSK being a quantum-safe migration strategy being made available by some protocols, see for example IKEv2 in . In general it is desirable to use a dual PRF, a dual-input PRF which is keyed off either input, as a KEM combiner (see for a discussion of dual PRFs). We take the following construction as a dual PRF in practice, and therefore suitable for use in all IETF protocols that need to combine the output of two KEMs:

where KDF represents a suitable choice of cryptographic key derivation function, H represents a cryptographic hash function, ss1 and ss2 represent the outputs of the first and second KEMs, and || represents concatenation. KDF and H are assumed to behave as random oracles. See for security analysis on the safety of using this combiner with RSA-KEM , elliptic curve Diffie-Hellman , Edwards curve Diffie-Hellman , and CRYSTALS-Kyber . All of these cryptographic algorithms are found to have a KDF or cryptographic hash as the last step before output of the shared secret, and therefore the KEM combiner construction may be simplified to the following when used with combinations of the analyzed cryptographic algorithms.

This simplified combiner proposed as a KEM combiner, for example in . In the case that more than two shared secrets need to be combined, the above construction can be extended in the obvious way:

IANA Considerations None.

Security Considerations This work assumes that the KEM combiner defined in is a dual PRF in practice, despite the lack of security proof. As the academic literature progresses towards provably secure dual PRFs, this document may need to be obsoleted in favour of a more robust primitive. It is important to note that the analysis herein pertains to the specific cryptographic algorithms analyzed and do not necessarily generalize to other constructions, even if they are based on the same underlying primitive. Specifically it depends on whether the shared secret produced by the KEM is the output of a fixed and secure key derivation function which makes the length and value of the shared secret beyond the direct control of the initiator of the KEM.

&RFC2119; &RFC5990; &RFC7748; &RFC3447; &RFC8174; &RFC8411; &RFC8784; &RFC8696; &I-D.ietf-lamps-cmp-updates; &I-D.driscoll-pqt-hybrid-terminology; &I-D.ietf-tls-hybrid-design; &I-D.ietf-ipsecme-ikev2-multiple-ke; &I-D.ounsworth-pq-composite-kem; &I-D.cfrg-schwabe-kyber;

Security Analysis

Dual PRF Dual PRFs are a active area of research. A dual PRF is a function which is a PRF when keyed by either of its two inputs – guaranteeing pseudo-randomness if one of the keys is compromised or even maliciously chosen by an adversary . As of publication of this document, no dual PRFs have been standardized for use. In practice we often use HMACs or HKDFs to serve the role of a dual PRF even though they have never been proved to be dual PRFs , . In essence, this document assumes that KDF( H(input1) || H(input2)) is a dual PRF in practice, for suitable choices of key derivation function KDF and hash function H, despite not having formal security proofs. It has been proposed as a KEM combiner, for example in . As the academic literature evolves, it may become appropriate to obsolete this document with a KEM combiner based on a provably secure dual PRF.

KEM primitives In modern cryptographic design, KEM algorithms seek to have indistinguishability under adaptive chosen ciphertext attack (IND-CCA2). FFor hybrid KEMs we desire the additional property that even if one input is controlled by an attacker, then combiner leaks no information about the other input. There are two ways to achieve such a hybrid KEM combiner; either by designing a combiner that is robust to one of the inputs being maliciously-chosen, called a dual PRF. See for a discussion about the current state of dual PRF research. Or alternatively by only allowing the hybridization of KEMs where a malicious kemEncaps() algorithm cannot control the shared secret derived by the victim's kemDecaps() algorithm and then combining the shared secrets in a trivial way. The following sections analyze commonly-used KEM algorithms to show that they have the following two properties, and are therefore suitable for use with the simplified KEM combiner presented in , . A malicious encapsulater cannot control the length of the KEM output (shared secret) that will be derived by the decapsulater. A malicious encapsulater cannot control the value of the KEM output (shared secret) that will be derived by the decapsulater. We define a KEM output to be "controlled by an attacker" if a maliciously-written kemEncaps() function can cause the victim's kemDecaps() algorithm to produce a shared secret either of a length chosen by the attacker, or to take on a given value with higher probability than can be obtained via rejection sampling on the shared secret output of kemEncaps().

RSA-KEM RSA encryption can be promoted into a KEM as per which defines a key transport based on RSA-KEM.

where Steps 1 - 3 define "RSA-KEM", which is considered here. Steps 4 - 6 define "Key Transport based on RSA-KEM" and is out of scope for this analysis as we assume that any RSA-KEM construction intended for use in a hybrid KEM would use the KEK output from Step 3 as the final shared secret. Here the transported symmetric key, KEK, is the KEM output (shared secret) ss as defined in . The encapsulater must choose a key derivation function KDF and declare it in the RSA-KEM parameters. The decapsulater may refuse to perform the the decapsulation if it does not like the encapsulater's choice of KDF, therefore it can be modeled as a random oracle producing an output of fixed length. The attacker is free to choose z, but assuming a strong choice of KDF, they cannot control either the length or the value KEK beyond what can be obtained by rejection sampling, thus satisfying properties 1 and 2 and defined in {.sec-kemprimitives}}. Therefore RSA-KEM is considered to be suitable for use with the simplified KEM combiner defined in , . Security note: This analysis applies to the specific RSA-KEM construction defined above. This result is not intended to generalize to all RSA-based key transport mechanisms, as they may not have the same cryptographic properties.

Elliptic Curve Diffie-Hellman (ECDH) The elliptic curve Diffie-Hellman key exchange can be promoted into a KEM in a straightforward way by assuming an ephemeral-static (ES) mode where def kemEncaps(pk) -> (ct, ss) includes generation of an ephemeral key pair, the public key being included as part of the ciphertext ct and the private key being discarded upon completion of the encapsulation. According to section 6.1.3:

Other key exchange methods defined in follow a similar construction. The attacker is free to choose a private key d_U which yields a shared secret Z, but cannot force Z to take on a chosen value without solving the elliptic curve discrete logarithm problem or performing rejection sampling. Assuming a strong choice of KDF, the attacker cannot control either the length or the value of KEK beyond what can be obtained by rejection sampling, thus satisfying properties 1 and 2 and defined in . Therefore elliptic curve Diffie-Hellman is considered to be suitable for use with the simplified KEM combiner defined in , .

Edwards Curve Diffie-Hellman (X25519 / X448) The elliptic curve Diffie-Hellman key exchange can be promoted into a KEM in a straightforward way by assuming an ephemeral-static (ES) mode where def kemEncaps(pk) -> (ct, ss) includes generation of an ephemeral key pair, the public key being included as part of the ciphertext ct and the private key being discarded upon completion of the encapsulation. According to section 6.1, the X25519 key exchange is defined as:

The X448 key exchange follows a similar construction. The attacker is free to choose a private key b which yields a shared secret K, but cannot force K to take on a chosen value without solving the elliptic curve discrete logarithm problem or performing rejection sampling. Assuming a strong choice of KDF, the attacker cannot control either the length or the value of the derived symmetric key beyond what can be obtained by rejection sampling, thus satisfying properties 1 and 2 and defined in . Therefore Edwards curve Diffie-Hellman, X25519 and X448, are considered to be suitable for use with the simplified KEM combiner defined in , .

CRYSTALS-Kyber The CRYSTALS-Kyber kemEncaps() is defined as follows :

with definitions as per . Here the hash functions G, H and the key derivation function KDF are in theory chosen by the encapsulater, but in practice are fixed by the Kyber specification to be H: SHA3-256, G: SHA3-512, and KDF: SHAKE-256, which can be modeled as random oracles. The attacker is free to choose m, but not the decapsulater's public key pk, nor do they have control of cpaCipherText so long as InnerEnc remains IND-CPA secure. Therefore the attacker cannot control the value or length of K beyond what can be obtained by rejection sampling. Therefore CRYSTALS-Kyber is considered to be suitable for use with the simplified KEM combiner defined in , .

Intellectual Property Considerations None.

Contributors and Acknowledgements This document incorporates contributions and comments from a large group of experts. The Editors would especially like to acknowledge the expertise and tireless dedication of the following people, who attended many long meetings and generated millions of bytes of electronic mail and VOIP traffic over the past years in pursuit of this document: Douglas Stebila, Nimrod Aviram. We are grateful to all, including any contributors who may have been inadvertently omitted from this list. This document borrows text from similar documents, including those referenced below. Thanks go to the authors of those documents. "Copying always makes things easier and less error prone" - .

Making contributions To be removed before publication. Additional contributions to this draft are welcome. Please see the working copy of this draft at, as well as open issues at: https://github.com/EntrustCorporation/draft-ounsworth-cfrg-kem-combiners