Kemeleon Encodings

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.¶

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]).¶

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)

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)

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.¶

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:¶

4.6. Summary of Encodings

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:¶

• 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:¶

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.¶

6.1. 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.¶

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.¶

8.1. Normative References

[ELL2]

Tibouchi, M., "Elligator Squared: Uniform Points on Elliptic Curves of Prime Order as Uniform Random Strings", 2014, <https://eprint.iacr.org/2014/043>.

[FIPS203]

"Module-lattice-based key-encapsulation mechanism standard", National Institute of Standards and Technology (U.S.), DOI 10.6028/nist.fips.203, August 2024, <https://doi.org/10.6028/nist.fips.203>.

[GSV24]

Günther, F., Stebila, D., and S. Veitch, "Obfuscated Key Exchange", 2024, <https://eprint.iacr.org/2024/1086>.

[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/rfc/rfc2119>.

[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/rfc/rfc8174>.

[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, <https://www.rfc-editor.org/rfc/rfc9380>.

8.2. Informative References

[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, <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-cpace-13>.

[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, <https://datatracker.ietf.org/doc/html/draft-ecdh-psi-00>.

[OBFS4]

"obfs4 (The obfourscator)", n.d., <https://gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/lyrebird/-/blob/HEAD/doc/obfs4-spec.txt>.

[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, <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-opaque-18>.