Network Working Group F. Günther Internet-Draft IBM Research Europe - Zurich Intended status: Informational D. Stebila Expires: 13 October 2025 University of Waterloo S. Veitch ETH Zurich 11 April 2025 Kemeleon Encodings draft-irtf-cfrg-kemeleon-00 Abstract This document specifies Kemeleon encoding algorithms for encoding ML- KEM encapsulation keys and ciphertexts as random bytestrings. Kemeleon encodings provide obfuscation of encapsulation keys and ciphertexts, relying on module LWE assumptions. This document specifies a number of variants of these encodings, with differing failure rates, output sizes, and performance profiles. About This Document This note is to be removed before publishing as an RFC. The latest revision of this draft can be found at https://ssveitch.github.io/draft-kemeleon/draft-irtf-cfrg- kemeleon.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draft-irtf-cfrg-kemeleon/. Source for this draft and an issue tracker can be found at https://github.com/ssveitch/draft-kemeleon. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. 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." Günther, et al. Expires 13 October 2025 [Page 1] Internet-Draft Kemeleon April 2025 This Internet-Draft will expire on 13 October 2025. Copyright Notice Copyright (c) 2025 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. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Conventions and Definitions . . . . . . . . . . . . . . . . . 3 3. Notation / ML-KEM Background . . . . . . . . . . . . . . . . 3 4. Kemeleon Encoding . . . . . . . . . . . . . . . . . . . . . . 4 4.1. Common Functions . . . . . . . . . . . . . . . . . . . . 4 4.2. Encoding Encapsulation Keys . . . . . . . . . . . . . . . 6 4.3. Encoding Ciphertexts . . . . . . . . . . . . . . . . . . 7 4.4. Non-Rejection Sampling Variant . . . . . . . . . . . . . 8 4.5. Faster Arithmetic Variant . . . . . . . . . . . . . . . . 9 4.6. Summary of Encodings . . . . . . . . . . . . . . . . . . 10 5. Additional Considerations for Applications . . . . . . . . . 10 5.1. Deterministic Encoding . . . . . . . . . . . . . . . . . 10 5.2. Relation to Hash-to-Curve . . . . . . . . . . . . . . . . 11 5.3. Modifying ML-KEM Algorithms . . . . . . . . . . . . . . . 11 6. Security Considerations . . . . . . . . . . . . . . . . . . . 12 6.1. Computational Assumptions . . . . . . . . . . . . . . . . 12 6.2. Randomness Sampling . . . . . . . . . . . . . . . . . . . 12 6.3. Timing Side-Channels . . . . . . . . . . . . . . . . . . 13 7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 13 8. References . . . . . . . . . . . . . . . . . . . . . . . . . 13 8.1. Normative References . . . . . . . . . . . . . . . . . . 13 8.2. Informative References . . . . . . . . . . . . . . . . . 13 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 14 1. Introduction ML-KEM [FIPS203] is a post-quantum key-encapsulation mechanism (KEM) recently standardized by NIST, Many applications are transitioning from classical Diffie-Hellman (DH) based solutions to constructions based on ML-KEM. The use of Elligator and related Hash-to-Curve [RFC9380] algorithms are ubiquitous in DH-based protocols where DH shares are required to be encoded as, and look indistinguishable from, random bytestrings. For example, applications using Elligator Günther, et al. Expires 13 October 2025 [Page 2] Internet-Draft Kemeleon April 2025 include protocols used for censorship circumvention in Tor [OBFS4], password-authenticated key exchange (PAKE) protocols [CPACE] [OPAQUE], and private set intersection (PSI) [ECDH-PSI]. For the post-quantum transition, an analogous encoding for (ML-)KEM encapsulation keys and ciphertexts to random bytestrings is required. This document specifies such an encoding, Kemeleon, for ML-KEM encapsulation keys and ciphertexts. Kemeleon was introduced in [GSV24] for building an (post-quantum) "obfuscated" KEM whose encapsulation keys and ciphertexts are indistinguishable from random. Beyond the original construction, this document additionally specifies variants that avoid the encoding failing or the use of large integer computations, or allow for a deterministic encoding. Aside from these variants, it is notable that the Kemeleon encodings of encapsulation keys results in smaller representations than in the original ML-KEM specification. 2. Conventions and Definitions 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. 3. Notation / ML-KEM Background A KEM consists of three algorithms: * KeyGen() -> (ek, dk): A probabilistic key generation algorithm that, with no input, generates an encapsulation key ek and a decapsulation key dk. * Encaps(ek) -> (c, K): A probabilistic encapsulation algorithm that takes as input an encapsulation key ek, and outputs a ciphertext ct and shared secret K. * Decaps(dk, c) -> K: A decapsulation algorithm that takes as input a decapsulation key dk and ciphertext c, and outputs a shared secret K. The following variables and functions are adopted from [FIPS203]: * q = 3329, n = 256 * Compress_d : x -> round((2d/q)*x) mod 2d (Equation 4.7) * Decompress_d : y -> round((q/2d)*y) (Equation 4.8) Günther, et al. Expires 13 October 2025 [Page 3] Internet-Draft Kemeleon April 2025 * remaining parameters k, d_u, d_v, etc. are defined by the respective ML-KEM parameter set -- this document writes du and dv in place of d_u, d_v in pseudocode ML-KEM.KeyGen() (Section 7.1 [FIPS203]) produces an encapsulation key, ek and a decapsulation key, dk. Encapsulation keys consist of byte-encoded vectors of coefficients in Z_q, where each coefficient is encoded in 12 bits, together with a 32-byte seed for generating the matrix A. ML-KEM.Encaps(ek) (Section 7.2 [FIPS203]) produces ciphertexts consisting of byte-encoded compressed vectors of cofficients, where each coefficient in Z_q is compressed by a certain number of bits (depending on the ML-KEM parameter set). The following terms and notation are used throughout this document: * msb(x) refers to the most significant bit of the value x * a[i] denotes the ith position of a vector a of coefficients * concat(x0, ..., xN): returns the concatenation of bytestrings. 4. Kemeleon Encoding At a high level, the constructions in this document instantiate the following functions: * EncodePk(ek) -> eek is the (possibly randomized) encoding algorithm that on input an encapsulation key, outputs an obfuscated encapsulation key or an error. * DecodePk(eek) -> ek is the deterministic decoding algorithm that on input an obfuscated encapsulation key, outputs an encapsulation key. * EncodeCtxt(c) -> ec is the (possibly randomized) encoding algorithm that on input a ciphertext, outputs an obfuscated ciphertext or an error. * DecodeCtxt(ec) -> c is the deterministic decoding algorithm that on input an obfuscated ciphertext, outputs a ciphertext. 4.1. Common Functions The following function maps a vector of k*n coefficients modulo q to a large integer, rejecting if the most significant bit of the integer is 1. Günther, et al. Expires 13 October 2025 [Page 4] Internet-Draft Kemeleon April 2025 VectorEncode(a): r = 0 for i from 1 to k*n: r += q^(i-1)*a[i] if msb(r) == 1: return err else: return r VectorDecode(r): for i from 1 to k*n: a[i] = r % q r = r // q return a The following algorithm samples an uncompressed pre-image of a coefficient c at random, where u is the decompressed value of c. The mapping is based on the Compress_d, Decompress_d algorithms from (Section 4.2.1 [FIPS203]). Günther, et al. Expires 13 October 2025 [Page 5] Internet-Draft Kemeleon April 2025 SamplePreimage(d,u,c): if d == 10: if Compress_d(u + 2) == c: rand <--$ [-1,0,1,2] else: rand <--$ [-1,0,1] return u + rand if d == 11: if Compress_d(u + 1) == c: rand <--$ [0,1] else if Compress_d(u - 1) == c: rand <--$ [-1,0] else: rand = 0 return u + rand if d == 5: if c == 0: rand <--$ [-52,...,52] else: rand <--$ [-51,...,52] return u + rand if d == 4: if c == 0: rand <--$ [-104,...,104] else: rand <--$ [-104,...,103] return u + rand else: return err 4.2. Encoding Encapsulation Keys The following algorithms encode ML-KEM encapsulation keys as random bytestrings. rho is the public seed used to generate the public matrix A [FIPS203]. This is already a random 32-byte string, so it is returned alongside the encoded value of t. t is a vector of k polynomials with n coefficients, but in the following pseudocode t is treated as a vector of k*n coefficients. Kemeleon.EncodePk(ek = (t, rho)): r = VectorEncode(t) if r == err: return err else: return concat(r,rho) Günther, et al. Expires 13 October 2025 [Page 6] Internet-Draft Kemeleon April 2025 Kemeleon.DecodePk(eek): r,rho = eek # rho is fixed length t = VectorDecode(r) return (t, rho) While the encapsulation key encoding algorithm has some failure probability, the decoding algorithm can never fail. Otherwise, a failure in decoding would provide a distinguishing factor between Kemeleon-encoded values and random bitstrings. 4.3. Encoding Ciphertexts ML-KEM ciphertexts consist of two components: c_1, a vector of k polynomials with n coefficients mod 2^du, and c_2, a polynomial with n coefficients mod 2^dv. The coefficients of these polynomials are not uniformly distributed, as a result of the compression step in encapsulation. The following encoding function decompresses and recovers a random preimage of this compression step in order to recover the uniform distribution of coefficients. Then, the same vector encoding step used for encapsulation keys is applied. For the second ciphertext component, rejection sampling is performed to retain uniformity, rather than decompressing. Kemeleon.EncodeCtxt(c = (c_1,c_2)): u = Decompress_du(c_1) for i from 1 to k*n: u[i] = SamplePreimage(du,u[i],c_1[i]) r = VectorEncode(u) if r == err: return err for i from 1 to n: if c_2[1] == 0: return err with prob. 1/ceil(q/(2^dv)) return concat(r,c_2) Kemeleon.DecodeCtxt(ec): r,c_2 = ec # c_2 is fixed length u = VectorDecode(r) c_1 = Compress_du(u) return (c_1,c_2) Günther, et al. Expires 13 October 2025 [Page 7] Internet-Draft Kemeleon April 2025 4.4. Non-Rejection Sampling Variant Applying a technique from [ELL2] (Section 3.4), the original Kemeleon construction can be adapted to avoid rejection sampling. This results in larger output sizes, but the encoding algorithm never fails. Applying the technique from [ELL2], where r is the encoded vector before rejection occurs in VectorEncode, we then choose m at random from [0,floor((2^(b+t)-r)/(q^(k*n)))], where b = log_2(q^(k*n)) and t is a security parameter, and return r + m*q^(k*n). This variant results in encoded values whose statistical distance from uniform is at most 2^-t. This results in an increased output size of t bits, where t is the security parameter. For example, with t=128, this increases the output size by 16 bytes. For encapsulation key encodings, one can immediately replace VectorEncode and VectorDecode calls with calls to the following algorithms. VectorEncodeNR(a): r = 0 t = sec_param # e.g. t = 128, 256, ... b = log_2(q^(k*n)) for i from 1 to k*n: r += q^(i-1)*a[i] m <--$ [0,...,floor((2^(b+t)-r)/(q^(k*n)))] return r + m*q^(k*n) VectorDecodeNR(a): a = a % q^(k*n) for i from 1 to k*n: a[i] = r % q r = r // q return a Notably, the random value m need not be transmitted alongside the encoded values. For ciphertext encodings, one must also avoid rejection sampling based on coefficients of the second component of the ciphertext. Therefore, the new ciphertext encoding must decompress and VectorEncodeNR the second component of the ciphertext. This more significantly increases the size of the encoded ciphertext. Günther, et al. Expires 13 October 2025 [Page 8] Internet-Draft Kemeleon April 2025 Kemeleon.EncodeCtxtNR(c = (c_1,c_2)): u = Decompress_du(c_1) for i from 1 to k*n: u[i] = SamplePreimage(du,u[i],c_1[i]) v = Decompress_dv(c_2) for i from 1 to n: v[i] = SamplePreimage(dv,v[i],c_2[i]) w = [u,v] # treat u,v as a singular vector of (k+1)*n coefficients r = VectorEncodeNR(w) # this call should use k+1 rather than k when accumulating to a large integer return r Kemeleon.DecodeCtxtNR(r): w = VectorDecodeNR(r) u,v = w # u, v are fixed length c_1 = Compress_du(u) c_2 = Compress_dv(v) return (c_1,c_2) 4.5. Faster Arithmetic Variant OPEN ISSUE: Is the faster variant of interest? If so, the following can be extended with a complete description. Observing that q = 3329 = 13*2^8+1, a variant of Kemeleon with faster integer arithmetic can be specified. First, the encoding rejects any polynomial with a coefficient equal to q-1 = 3328. This ensures that all arithmetic can be computed with values modulo q-1 = 13*2^8. Then, note that rather than accumulating values to a large integer mod q^(k*n), it is only required to accumulate values to an integer mod 13^(k*n), while keeping track of the 8 lower order bits of each coefficient. The output size of the encoding does not change, but this results in an increased rejection rate. In particular, Table 1 gives success probabilities for encapsulation key and ciphertext encodings: +=============+========================+==========================+ | Parameter | ek success probability | ctxt success probability | +=============+========================+==========================+ | ML-KEM-512 | 0.49 | 0.45 | +-------------+------------------------+--------------------------+ | ML-KEM-768 | 0.29 | 0.25 | +-------------+------------------------+--------------------------+ | ML-KEM-1024 | 0.53 | 0.47 | +-------------+------------------------+--------------------------+ Table 1: Success probabilities for faster Kemeleon encoding Günther, et al. Expires 13 October 2025 [Page 9] Internet-Draft Kemeleon April 2025 4.6. Summary of Encodings +==============+=============+=============+===================+ | Algorithm / | Output size | Success | Additional | | Parameter | (bytes) | probability | considerations | +==============+=============+=============+===================+ | Kemeleon - | ek: 781, | ek: 0.56, | Large int (750B) | | ML-KEM512 | ctxt: 877 | ctxt: 0.51 | arithmetic | +--------------+-------------+-------------+-------------------+ | Kemeleon - | ek: 1156, | ek: 0.83, | Large int (1150B) | | ML-KEM768 | ctxt: 1252 | ctxt: 0.77 | arithmetic | +--------------+-------------+-------------+-------------------+ | Kemeleon - | ek: 1530, | ek: 0.62, | Large int (1500B) | | ML-KEM1024 | ctxt: 1658 | ctxt: 0.57 | arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonNR - | ek: 797, | ek: 1.00, | Large int (1123B) | | ML-KEM512 | ctxt: 1140 | ctxt: 1.00 | arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonNR - | ek: 1172, | ek: 1.00, | Large int (1498B) | | ML-KEM768 | ctxt: 1514 | ctxt: 1.00 | arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonNR - | ek: 1546, | ek: 1.00, | Large int (1872B) | | ML-KEM1024 | ctxt: 1889 | ctxt: 1.00 | arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonFT - | ek: 781, | ek: 0.49, | Smaller int | | ML-KEM512 | ctxt: 877 | ctxt: 0.45 | (235B) arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonFT - | ek: 1156, | ek: 0.29, | Smaller int | | ML-KEM768 | ctxt: 1252 | ctxt: 0.25 | (355B) arithmetic | +--------------+-------------+-------------+-------------------+ | KemeleonFT - | ek: 1530, | ek: 0.53, | Smaller int | | ML-KEM1024 | ctxt: 1658 | ctxt: 0.47 | (475B) arithmetic | +--------------+-------------+-------------+-------------------+ Table 2: Summary of Kemeleon Variants, NR = No Reject, FT = Faster 5. Additional Considerations for Applications This section contains additional considerations and comments related to using Kemeleon encodings in different applications. 5.1. Deterministic Encoding The randomness used in Kemeleon ciphertext encodings MAY be derived in a deterministic manner. To do so, following a call to Encap which returns a KEM key K and a ciphertext c, the following steps can be taken: Günther, et al. Expires 13 October 2025 [Page 10] Internet-Draft Kemeleon April 2025 * Using a key derivation function (KDF), derive from the key K a new key K' and a seed for randomness rnd. * The seed rnd can be used to generate the randomness required when encoding the ciphertext c. * Use K' in place of K wherever applicable in the remainder of the protocol/system. * Upon any call to Decap, apply the same KDF to derive the new key K', as required. Deriving a new KEM key for use in the remainder of a system is crucial in order to ensure key separation (i.e., the implementation MUST NOT use the original key K to derive randomness and for other purposes). The randomness used to encode an encapsulation key MAY be stored alongside the corresponding decapsulation key, if it is subsequently needed. See Section 6.2 for relevant discussion on keeping this randomness secret. 5.2. Relation to Hash-to-Curve While the functionality of Kemeleon is similar to hash-to-curve [RFC9380] (mapping arbitrary byte strings to public keys/ ciphertexts), the applications where hash-to-curve is used do not immediately follow in the KEM-based setting because having such an encapsulation key (without dk) or ciphertext (without dk or ek) does not appear to provide the same functionality, since it is not clear how to continue working with the element in the same way that can be done with an elliptic curve point. 5.3. Modifying ML-KEM Algorithms In applications that _only_ require Kemeleon-encoded values _and_ where the underlying ML-KEM implementation can be modified, the ciphertext encoding algorithm (and ML-KEM encapsulation/decapsulation algorithms) MAY be adapted as follows for improved efficiency. In particular, the compression step in the ML-KEM encapsulation algorithm can be omitted, and therefore, the decompression step in the Kemeleon algorithm can be omitted. In the implementation of ML- KEM, the compression step (lines 22-23 of Algorithm 14 [FIPS203]) and corresponding decompression step (lines 3-4 of Algorithm 15 [FIPS203]) can be omitted from the encapsulation/decapsulation algorithms in ML-KEM. In this case, the Kemeleon encoding algorithm for ciphertexts would omit the Decompress and SamplePreimage steps and immediately apply VectorEncode: Günther, et al. Expires 13 October 2025 [Page 11] Internet-Draft Kemeleon April 2025 Kemeleon.EncodeCtxt(c = (c_1,c_2)): w = [c_1,c_2] # treat c_1,c_2 as a singular vector of (k+1)*n coefficients r = VectorEncode(w) # this call should use k+1 rather than k when accumulating to a large integer return r Decoding is adapted analogously. Kemeleon.DecodeCtxt(ec): w = VectorDecode(r) c_1,c_2 = w # c_1, c_2 are fixed length return (c_1,c_2) Naturally, these algorithms can use VectorEncodeNR, VectorDecodeNR, if the non-rejecting variant is desirable. 6. Security Considerations This section contains additional security considerations about the Kemeleon encodings described in this document. 6.1. Computational Assumptions In general, the obfuscation properties of the Kemeleon encodings depend on module LWE assumptions similar to those underlying the IND- CCA security of ML-KEM; see [GSV24] for the detailed security analysis of the original Kemeleon encoding. In particular, the notions of public key and ciphertext uniformity capture the indistinguishability of Kemeleon-encoded encapsulation keys and ciphertexts from random bitstrings, respectively. Both require the module LWE assumption to hold in order for Kemeleon to maintain its uniformity properties. Furthermore, distinguishing a pair of a Kemeleon-encoded encapsulation key and a Kemeleon-encoded ciphertext from uniformly random bitstrings also reduces to a module LWE assumption. 6.2. Randomness Sampling Both encapsulation key and ciphertext encodings in the original Kemeleon encoding are randomized. The randomness (or seed used to generate randomness) used in Kemeleon encodings MUST be kept secret. In particular, public randomness enables distinguishing a Kemeleon- encoded value from a random bytestring: Decoding the value in question and re-encoding it with the public randomness will yield the original value if it was Kemeleon-encoded. Günther, et al. Expires 13 October 2025 [Page 12] Internet-Draft Kemeleon April 2025 6.3. Timing Side-Channels Beyond timing side-channel considerations for ML-KEM itself, care should be taken when using Kemeleon encodings, in particular those with a non-zero failure probability. Rejecting and re-generating encapsulation keys or ciphertexts may leak information about the use of Kemeleon encodings, as might the overhead of the encoding itself. Additionally, the algorithms required to perform big integer arithmetic may leak information via timing. 7. IANA Considerations This document has no IANA actions. 8. References 8.1. Normative References [ELL2] Tibouchi, M., "Elligator Squared: Uniform Points on Elliptic Curves of Prime Order as Uniform Random Strings", 2014, . [FIPS203] "Module-lattice-based key-encapsulation mechanism standard", National Institute of Standards and Technology (U.S.), DOI 10.6028/nist.fips.203, August 2024, . [GSV24] Günther, F., Stebila, D., and S. Veitch, "Obfuscated Key Exchange", 2024, . [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, . [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, . [RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, August 2023, . 8.2. Informative References Günther, et al. Expires 13 October 2025 [Page 13] Internet-Draft Kemeleon April 2025 [CPACE] Abdalla, M., Haase, B., and J. Hesse, "CPace, a balanced composable PAKE", Work in Progress, Internet-Draft, draft- irtf-cfrg-cpace-13, 14 October 2024, . [ECDH-PSI] wangyuchen, Chang, Lu, Y., Hong, C., and J. Peng, "PSI based on ECDH", Work in Progress, Internet-Draft, draft- ecdh-psi-00, 21 October 2024, . [OBFS4] "obfs4 (The obfourscator)", n.d., . [OPAQUE] Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The OPAQUE Augmented PAKE Protocol", Work in Progress, Internet-Draft, draft-irtf-cfrg-opaque-18, 21 November 2024, . Authors' Addresses Felix Günther IBM Research Europe - Zurich Email: mail@felixguenther.info Douglas Stebila University of Waterloo Email: dstebila@uwaterloo.ca Shannon Veitch ETH Zurich Email: shannon.veitch@inf.ethz.ch Günther, et al. Expires 13 October 2025 [Page 14]