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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 a comprehensible and easy to implement Keccak-based KEM combiner to join an arbitrary number of key shares, that is compatible with NIST SP 800-56Cr2 when viewed as a key derivation function. The combiners defined here are practical multi-PRFs and are CCA-secure as long as at least one of the ingredient KEMs is. 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 the ciphertext representing an encapsulated key, 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 / traditional hybrid KEMs where output of a post-quantum KEM is combined with the output of a classical key transport or key exchange algorithm. KEM-based authenticated key exchanges (AKEs) where the output of two or more KEMs performed in different directions are combined. This document normalizes a mechanism for combining the output of two or more 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 , JOSE / COSE (CITE once Orie's drafts are up). The traditional algorithm may in fact be a key transport or key agreement scheme, but since simple transformations exist to turn both of those schemes into KEMs, this document assumes that all cryptograhpic algorithms satisfy the KEM interface described in .

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 .

KEM Combiner construction A KEM combiner is a function that takes in two or more shared secrets ss_i and returns a combined shared secret ss.

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 multi-PRF as a KEM combiner, meaning that the combiner has the properties of a PRF when keyed by any of its single inputs. The following simple yet generic construction can be used in all IETF protocols that need to combine the output of two or more KEMs:

where: KDF represents a suitable choice of a cryptographic key derivation function, k_i represent the constant-length input keys and is discussed in , fixedInfo is some protocol-specific KDF binding, counter parameter is instantiation-specific and is discussed in . outputBits determines the length of the output keying material, || represents concatenation. In several possible practical instantiations are listed that are in compliance with NIST SP-800 56Cr2 . The shared secret ss MAY be used directly as a symmetric key, for example as a MAC key or as a Key Encryption Key (KEK).

k_i construction Following the guidance of Giacon et al. , we wish for a KEM combiner that is CCA-secure as long as at least one of the ingredient KEMs is. In order to protect against chosen ciphertext attacks, it is necessary to include both the shared secret ss_i and its corresponding ciphertext ct_i. In addition, each k_i MUST be constant in length to prevent abuse of the underlying hash function, therefore the secret shares ss_i || ct_i MUST be hashed prior to inclusion in the overall construction described in :

In the case of a PSK there is no associated ciphertext present. ~~~ BEGIN EDNOTE ~~~ Each k_i MUST be constant in length, therefore it MAY be the concatenation of the secret share ss_i and ciphertext ct_i only if they are guaranteed to be constant length:

For all other cases, it is REQUIRED to concatenate and hash them first:

Including the ciphertext guarantees that the combined kem is IND-CCA secure as long as one of the ingredient KEMs is, as stated by . Any protocols making use of this construction MUST either hash all inputs ss_i || ct_i, or justify that any un-hashed inputs will always be fixed length. ~~~ END EDNOTE ~~~

Protocol binding The fixedInfo string is a fixed-length string containing some context-specific information. The intention is to prevent cross-context attacks by making this key derivation unique to its protocol context. The fixedInfo string MUST have a definite structure depending on the protocol where all parts are fixed length. This prevents a variable length structure from creating collisions between two different instances. In cases some variable length input is necessary, such as the representation of a user ID, a public key, or an OID, then hashing or padding MAY be used. fixedInfo MUST NOT include the shared secrets and ciphertexts, as they are already represented in the KDF input. The parameter fixedInfo MAY contain any of the following information: Public information about the communication parties, such as their identifiers. The public keys or certificates contributed by each party to the key-agreement transaction. Other public information shared between communication parties before or during the transaction, such as nonces. An indication of the protocol or application employing the key-derivation method. Protocol-related information, such as a label or session identifier. An indication of the key-agreement scheme and/or key-derivation method used. An indication of the domain parameters associated with the asymmetric key pairs employed for key establishment. An indication of other parameter or primitive choices. An indication of how the derived keying material should be parsed, including an indication of which algorithm(s) will use the (parsed) keying material. This is a non-comprehensive list, further information can be found in paragraph 5.8.2 of NIST SP800-56Ar3 .

Practical instantiations The KDF must be instantiated with cryptographically-secure choices for KDF and H. The following are RECOMMENDED Keccak-based instatiations, but other choices MAY be made for example to allow for future cryptographic agility. A protocol using a different instantiation MUST justify that it will behave as a multi-PRF given fixed length inputs. Each instance defines a function to be used as KDF, a hash H function to derive the k_i, and optionally a counter. KDF = SHA3-256 and H = SHA3-256, with hashSize = 256 bit. KDF = SHA3-512 and H = SHA3-512, with hashSize = 512 bit. KDF = KMAC128 and H = SHA3-256, with hashSize = 128 bit. KDF = KMAC256 and H = SHA3-512, with hashSize = 256 bit.

Hash-and-counter based construction Options 1 and 2 instantiate the KDF using SHA3, specified in NIST FIPS 202 . To generate an outputBits long secret share ss: the counter MUST be initialized with the value 0x00000001. The hash is performed over the string defined in , and repeated ceil(outputBits/hashSize) times. For each iteration the counter MUST be increased by 0x01. The strings are concatenated ordered by counter. The leftmost outputBits are returned as ss. An implementation MUST NOT overflow and reuse the counter and an error MUST be returned when producing more than 2^32 consecutive hashes.

KMAC based construction Options 3 and 4 are KMAC-based, as specified in NIST SP 800-185 . To instantiate the function: The context S MUST be the utf-8 string "KDF". The key K MUST be a context-specific string of at least hashSize bits, and it MAY be used as an additional option to perform context separation, in scenarios where fixedInfo is not sufficient. The parameter counter MUST be the fixed value 0x00000001. To derive a shared secret ss of desired length, KMAC is called a single time with the input string X defined in and length L being outputBits. This is compatible with the one-step KDF definition given in NIST SP800-56Cr2 , Section 4.

IANA Considerations None.

Security Considerations The proposed instantiations in are practical multi-PRFs since this specification limits to the use of Keccak-based constructions. The sponge construction was proven to be indifferentiable from a random oracle . 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 Keccak 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 k_i from a random bit string, an adversary has to correctly guess all key shares k_i entirely. The proposed construction in with the instantiations in preserves IND-CCA2 of any of its ingredient KEMs, i.e. the newly formed combined KEM is IND-CCA2 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 split-key pseudorandom function as defined in . That is, Keccak behaves like a random function if at least one input shared secret ss_i is picked uniformly at random. Our construction can thus be seen as an instantiation of the IND-CCA2 preserving Example 3 in Figure 1 of , up to some reordering of input shared secrets ss_i and ciphertexts ct_i and their potential compression H(ss_i || ct_i) by a cryptographic hash function.

&RFC2119; &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.wussler-openpgp-pqc;

Acknowledgements This document incorporates contributions and comments from a large group of experts. The authors 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, and Andreas Huelsing. 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" - .