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Combiner function for hybrid key encapsulation mechanisms (Hybrid KEMs)

draft-ounsworth-cfrg-kem-combiners-00

Document | Type | Active Internet-Draft (individual) | |
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Author | Mike Ounsworth | ||

Last updated | 2022-11-25 | ||

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draft-ounsworth-cfrg-kem-combiners-00

CFRG M. Ounsworth Internet-Draft Entrust Intended status: Informational 25 November 2022 Expires: 29 May 2023 Combiner function for hybrid key encapsulation mechanisms (Hybrid KEMs) draft-ounsworth-cfrg-kem-combiners-00 Abstract 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. 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/draft-ounsworth-cfrg-kem-combiners/. Discussion of this document takes place on the Limited Additional Mechanisms for PKIX and SMIME (lamps) Working Group mailing list (mailto:spasm@ietf.org), which is archived at https://mailarchive.ietf.org/arch/browse/spasm/. Subscribe at https://www.ietf.org/mailman/listinfo/spasm/. Source for this draft and an issue tracker can be found at https://github.com/EntrustCorporation/draft-ounsworth-cfrg-kem- combiners. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Ounsworth Expires 29 May 2023 [Page 1] Internet-Draft KEM Combiner November 2022 Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on 29 May 2023. Copyright Notice Copyright (c) 2022 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/ license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Table of Contents 1. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Key Encapsulation Mechanisms . . . . . . . . . . . . . . 3 2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1. KEM/PSK hybrids . . . . . . . . . . . . . . . . . . . . . 4 2.2. PQ/Traditional hybrid KEMs . . . . . . . . . . . . . . . 4 2.3. KEM-based AKE . . . . . . . . . . . . . . . . . . . . . . 4 3. KEM Combiner . . . . . . . . . . . . . . . . . . . . . . . . 4 4. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 6 5. Security Considerations . . . . . . . . . . . . . . . . . . . 6 6. References . . . . . . . . . . . . . . . . . . . . . . . . . 6 6.1. Normative References . . . . . . . . . . . . . . . . . . 6 6.2. Informative References . . . . . . . . . . . . . . . . . 6 Appendix A. Security Analysis . . . . . . . . . . . . . . . . . 8 A.1. Dual PRF . . . . . . . . . . . . . . . . . . . . . . . . 8 A.2. KEM primitives . . . . . . . . . . . . . . . . . . . . . 9 A.3. RSA-KEM . . . . . . . . . . . . . . . . . . . . . . . . . 9 A.4. Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . 11 A.5. Edwards Curve Diffie-Hellman (X25519 / X448) . . . . . . 12 A.6. CRYSTALS-Kyber . . . . . . . . . . . . . . . . . . . . . 13 Appendix B. Intellectual Property Considerations . . . . . . . . 13 Appendix C. Contributors and Acknowledgements . . . . . . . . . 13 C.1. Making contributions . . . . . . . . . . . . . . . . . . 14 Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 14 Ounsworth Expires 29 May 2023 [Page 2] Internet-Draft KEM Combiner November 2022 1. 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 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here. This document is consistent with all terminology defined in [I-D.driscoll-pqt-hybrid-terminology]. 1.1. 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 [PQCAPI]. def kemKeyGen() -> (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. 2. Introduction The need for a KEM combiner function arises in three different contexts within IETF security protocols: 1. KEM / PSK hybrids where the output of a KEM is combined with a static pre-shared key. 2. Post-quantum / tradtional hybrid KEMs. 3. KEM-based authenticated key exchanges (AKEs). This document normalizes a mechanisms for combining the output of two KEMs. Ounsworth Expires 29 May 2023 [Page 3] Internet-Draft KEM Combiner November 2022 2.1. 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 [RFC8696] and IKEv2 [RFC8784]. 2.2. PQ/Traditional hybrid KEMs A post-quantum / traditional hybrid key encapsulation mechanism (hybrid KEM) as defined in [I-D.driscoll-pqt-hybrid-terminology] 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 [I-D.ietf-tls-hybrid-design], IKEv2 [I-D.ietf-ipsecme-ikev2-multiple-ke], X.509; PKIX; CMS [I-D.ounsworth-pq-composite-kem], OpenPGP (CITE once Aron's draft is up), JOSE / COSE (CITE once Orie's drafts are up). 2.3. 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 [I-D.ietf-lamps-cmp-updates], .. TODO: cite others. 3. 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. ss = kemCombiner(ss1, ss2) Ounsworth Expires 29 May 2023 [Page 4] Internet-Draft KEM Combiner November 2022 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 [RFC8784]. 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 Appendix A.1 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: ss = kemCombiner(ss1, ss2) = KDF( H(ss1) || H(ss2) ) Figure 1: general KEM combiner construction 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 Appendix A.2 for security analysis on the safety of using this combiner with RSA-KEM [RFC5990], elliptic curve Diffie-Hellman [SEC1], Edwards curve Diffie-Hellman [RFC7748], and CRYSTALS-Kyber [I-D.cfrg-schwabe-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. ss = kemCombiner(ss1, ss2) = KDF( ss1 || ss2 ) Figure 2: simplified KEM combiner construction when both KEMs are known to provide strong output This simplified combiner proposed as a KEM combiner, for example in [I-D.ietf-tls-hybrid-design]. In the case that more than two shared secrets need to be combined, the above construction can be extended in the obvious way: ss = kemCombiner(ss1, ss2, ss3, ... ) = KDF( H(ss1) || H(ss2) || H(ss3) ... ) Figure 3: KEM combiner construction for combining more than two shared secrets Ounsworth Expires 29 May 2023 [Page 5] Internet-Draft KEM Combiner November 2022 4. IANA Considerations None. 5. Security Considerations This work assumes that the KEM combiner defined in Section 3 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. 6. References 6.1. Normative References [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.rfc-editor.org/info/rfc2119>. [RFC5990] Randall, J., Kaliski, B., Brainard, J., and S. Turner, "Use of the RSA-KEM Key Transport Algorithm in the Cryptographic Message Syntax (CMS)", RFC 5990, DOI 10.17487/RFC5990, September 2010, <https://www.rfc-editor.org/info/rfc5990>. [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, <https://www.rfc-editor.org/info/rfc7748>. [SEC1] "Standards for Efficient Cryptography Group, SEC1: Elliptic Curve Cryptography", May 2009, <<https://www.secg.org/sec1-v2.pdf>>. 6.2. Informative References Ounsworth Expires 29 May 2023 [Page 6] Internet-Draft KEM Combiner November 2022 [Aviram2022] Aviram, N., Dowling, B., Komargodski, I., Paterson, K. G., Ronen, E., and E. Yogev, "Practical (Post-Quantum) Key Combiners from One-Wayness and Applications to TLS.", 2022, <https://eprint.iacr.org/2022/065>. [Bellare2015] Bellare, M. and A. Lysyanskaya, "Symmetric and Dual PRFs from Standard Assumptions: A Generic Validation of an HMAC Assumption.", 2015, <https://eprint.iacr.org/2015/1198>. [I-D.cfrg-schwabe-kyber] Schwabe, P. and B. Westerbaan, "Kyber Post-Quantum KEM", Work in Progress, Internet-Draft, draft-cfrg-schwabe- kyber-01, 23 September 2022, <https://www.ietf.org/archive/id/draft-cfrg-schwabe-kyber- 01.txt>. [I-D.driscoll-pqt-hybrid-terminology] D, F., "Terminology for Post-Quantum Traditional Hybrid Schemes", Work in Progress, Internet-Draft, draft- driscoll-pqt-hybrid-terminology-01, 20 October 2022, <https://www.ietf.org/archive/id/draft-driscoll-pqt- hybrid-terminology-01.txt>. [I-D.ietf-ipsecme-ikev2-multiple-ke] Tjhai, C., Tomlinson, M., Bartlett, G., Fluhrer, S., Van Geest, D., Garcia-Morchon, O., and V. Smyslov, "Multiple Key Exchanges in IKEv2", Work in Progress, Internet-Draft, draft-ietf-ipsecme-ikev2-multiple-ke-10, 9 November 2022, <https://www.ietf.org/archive/id/draft-ietf-ipsecme-ikev2- multiple-ke-10.txt>. [I-D.ietf-lamps-cmp-updates] Brockhaus, H., von Oheimb, D., and J. Gray, "Certificate Management Protocol (CMP) Updates", Work in Progress, Internet-Draft, draft-ietf-lamps-cmp-updates-23, 29 June 2022, <https://www.ietf.org/archive/id/draft-ietf-lamps- cmp-updates-23.txt>. [I-D.ietf-tls-hybrid-design] Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key exchange in TLS 1.3", Work in Progress, Internet-Draft, draft-ietf-tls-hybrid-design-05, 28 August 2022, <https://www.ietf.org/archive/id/draft-ietf-tls-hybrid- design-05.txt>. Ounsworth Expires 29 May 2023 [Page 7] Internet-Draft KEM Combiner November 2022 [I-D.ounsworth-pq-composite-kem] Ounsworth, M. and J. Gray, "Composite KEM For Use In Internet PKI", Work in Progress, Internet-Draft, draft- ounsworth-pq-composite-kem-00, 11 July 2022, <https://www.ietf.org/archive/id/draft-ounsworth-pq- composite-kem-00.txt>. [PQCAPI] Project, N. P.-Q. C., "PQC - API notes", November 2022, <https://csrc.nist.gov/CSRC/media/Projects/Post-Quantum- Cryptography/documents/example-files/api-notes.pdf>. [RFC3447] Jonsson, J. and B. Kaliski, "Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1", RFC 3447, DOI 10.17487/RFC3447, February 2003, <https://www.rfc-editor.org/info/rfc3447>. [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, <https://www.rfc-editor.org/info/rfc8174>. [RFC8411] Schaad, J. and R. Andrews, "IANA Registration for the Cryptographic Algorithm Object Identifier Range", RFC 8411, DOI 10.17487/RFC8411, August 2018, <https://www.rfc-editor.org/info/rfc8411>. [RFC8696] Housley, R., "Using Pre-Shared Key (PSK) in the Cryptographic Message Syntax (CMS)", RFC 8696, DOI 10.17487/RFC8696, December 2019, <https://www.rfc-editor.org/info/rfc8696>. [RFC8784] Fluhrer, S., Kampanakis, P., McGrew, D., and V. Smyslov, "Mixing Preshared Keys in the Internet Key Exchange Protocol Version 2 (IKEv2) for Post-quantum Security", RFC 8784, DOI 10.17487/RFC8784, June 2020, <https://www.rfc-editor.org/info/rfc8784>. Appendix A. Security Analysis A.1. 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 [Aviram2022]. 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 [Bellare2015], [Aviram2022]. Ounsworth Expires 29 May 2023 [Page 8] Internet-Draft KEM Combiner November 2022 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 [I-D.ietf-tls-hybrid-design]. As the academic literature evolves, it may become appropriate to obsolete this document with a KEM combiner based on a provably secure dual PRF. A.2. 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 Appendix A.1 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 Section 3, Figure 2. 1. A malicious encapsulater cannot control the length of the KEM output (shared secret) that will be derived by the decapsulater. 2. 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(). A.3. RSA-KEM RSA encryption [RFC3447] can be promoted into a KEM as per [RFC5990] which defines a key transport based on RSA-KEM. Ounsworth Expires 29 May 2023 [Page 9] Internet-Draft KEM Combiner November 2022 1. Generate a random integer z between 0 and n-1 (see note), and convert z to a byte string Z of length nLen, most significant byte first: z = RandomInteger (0, n-1) Z = IntegerToString (z, nLen) 2. Encrypt the random integer z using the recipient's public key (n,e), and convert the resulting integer c to a ciphertext C, a byte string of length nLen: c = z^e mod n C = IntegerToString (c, nLen) 3. Derive a key-encrypting key KEK of length kekLen bytes from the byte string Z using the underlying key derivation function: KEK = KDF (Z, kekLen) 4. Wrap the keying data K with the key-encrypting key KEK using the underlying key-wrapping scheme to obtain wrapped keying data WK: WK = Wrap (KEK, K) 5. Concatenate the ciphertext C and the wrapped keying data WK to obtain the encrypted keying data EK: EK = C || WK 6. Output the encrypted keying data EK. 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. Ounsworth Expires 29 May 2023 [Page 10] Internet-Draft KEM Combiner November 2022 Here the transported symmetric key, KEK, is the KEM output (shared secret) ss as defined in Section 1.1. 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 Section 3, Figure 2. 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. A.4. Elliptic Curve Diffie-Hellman (ECDH) The elliptic curve Diffie-Hellman key exchange [SEC1] 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 [SEC1] section 6.1.3: 1. Use one of the Diffie-Hellman primitives specified in Section 3.3 to derive a shared secret field element z ∈ Fq from U's secret key d_U established during the key deployment procedure and V's public key Q_V obtained during the key deployment procedure. If the Diffie- Hellman primitive outputs “invalid”, output “invalid” and stop. Decide whether to use the “standard” elliptic curve Diffie-Hellman primitive or the elliptic curve cofactor Diffie-Hellman primitive according to the convention established during the setup procedure. 2. Convert z ∈ Fq to an octet string Z using the conversion routine specified in Section 2.3.5. 3. Use the key derivation function KDF established during the setup procedure to generate keying data K of length keydatalen octets from Z and [SharedInfo]. If the key derivation function outputs “invalid”, output “invalid” and stop. 4. Output K. Ounsworth Expires 29 May 2023 [Page 11] Internet-Draft KEM Combiner November 2022 Other key exchange methods defined in [SEC1] 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 Appendix A.2. Therefore elliptic curve Diffie-Hellman is considered to be suitable for use with the simplified KEM combiner defined in Section 3, Figure 2. A.5. Edwards Curve Diffie-Hellman (X25519 / X448) The elliptic curve Diffie-Hellman key exchange [SEC1] 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 [RFC7748] section 6.1, the X25519 key exchange is defined as: Alice generates 32 random bytes in a[0] to a[31] and transmits K_A = X25519(a, 9) to Bob, where 9 is the u-coordinate of the base point and is encoded as a byte with value 9, followed by 31 zero bytes. Bob similarly generates 32 random bytes in b[0] to b[31], computes K_B = X25519(b, 9), and transmits it to Alice. Using their generated values and the received input, Alice computes X25519(a, K_B) and Bob computes X25519(b, K_A). Both now share K = X25519(a, X25519(b, 9)) = X25519(b, X25519(a, 9)) as a shared secret. Both MAY check, without leaking extra information about the value of K, whether K is the all-zero value and abort if so (see below). Alice and Bob can then use a key-derivation function that includes K, K_A, and K_B to derive a symmetric key. 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 Ounsworth Expires 29 May 2023 [Page 12] Internet-Draft KEM Combiner November 2022 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 Appendix A.2. Therefore Edwards curve Diffie-Hellman, X25519 and X448, are considered to be suitable for use with the simplified KEM combiner defined in Section 3, Figure 2. A.6. CRYSTALS-Kyber The CRYSTALS-Kyber kemEncaps() is defined as follows [I-D.cfrg-schwabe-kyber]: 1.Compute 1. m = H(seed) 2. (Kbar, cpaSeed) = G(m || H(pk)) 3. cpaCipherText = InnerEnc(m, publicKey, cpaSeed) 2.Return 1. cipherText = cpaCipherText 2. sharedSecret = KDF(KBar || H(cpaCipherText)) with definitions as per [I-D.cfrg-schwabe-kyber]. 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 Section 3, Figure 2. Appendix B. Intellectual Property Considerations None. Appendix C. 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: Ounsworth Expires 29 May 2023 [Page 13] Internet-Draft KEM Combiner November 2022 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" - [RFC8411]. C.1. 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 Author's Address Mike Ounsworth Entrust Limited 2500 Solandt Road – Suite 100 Ottawa, Ontario K2K 3G5 Canada Email: mike.ounsworth@entrust.com Ounsworth Expires 29 May 2023 [Page 14]