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title: "Kemeleon Encodings" abbrev: "Kemeleon" category: info

docname: draft-veitch-kemeleon-latest submissiontype: IETF # also: "independent", "editorial", "IAB", or "IRTF" number: date: consensus: true v: 3

area: AREA

workgroup: WG Working Group

keyword: Internet-Draft venue:

group: WG

type: Working Group

mail: [email protected]

arch: https://example.com/WG

github: "ssveitch/draft-kemeleon" latest: "https://ssveitch.github.io/draft-kemeleon/draft-kemeleon.html"

author:

fullname: Felix Günther
organization: IBM Research Europe - Zurich
email: [email protected]

normative:

FIPS203: DOI.10.6028/NIST.FIPS.203 RFC9380:

ELL2: title: "Elligator Squared: Uniform Points on Elliptic Curves of Prime Order as Uniform Random Strings" target: https://eprint.iacr.org/2014/043 date: 2014 author: - ins: M. Tibouchi name: Mehdi Tibouchi

GSV24: title: Obfuscated Key Exchange target: https://eprint.iacr.org/2024/1086 date: 2024 author: - ins: F. Günther name: Felix Günther - ins: D. Stebila name: Douglas Stebila - ins: S. Veitch name: Shannon Veitch

informative:

OBFS4: title: obfs4 (The obfourscator) target: https://gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/lyrebird/-/blob/HEAD/doc/obfs4-spec.txt

--- abstract

This document specifies Kemeleon encoding algorithms for encoding ML-KEM public keys and ciphertexts as random bytestrings. Kemeleon encodings provide obfuscation of public 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.

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Introduction {#intro}

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 include protocols used for censorship circumvention in Tor {{OBFS4}}, password-authenticated key exchange (PAKE) protocols {{?CPACE=I-D.irtf-cfrg-cpace}} {{?OPAQUE=I-D.irtf-cfrg-opaque}}, and private set intersection (PSI) {{?ECDH-PSI=I-D.ecdh-psi}}.

For the post-quantum transition, an analogous encoding for (ML-)KEM public keys and ciphertexts to random bytestrings is required. This document specifies such an encoding, Kemeleon, for ML-KEM public keys and ciphertexts. Kemeleon was introduced in {{GSV24}} for building an (post-quantum) "obfuscated" KEM whose public 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 public keys results in smaller representations than in the original ML-KEM specification.

Conventions and Definitions {#conventions}

{::boilerplate bcp14-tagged}

Notation / ML-KEM Background {#notation}

A KEM consists of three algorithms:

  • KeyGen() -> (pk, sk): A probabilistic key generation algorithm that, with no input, generates a public key pk and a secret key sk.
  • Encaps(pk) -> (c, K): A probabilistic encapsulation algorithm that takes as input a public key pk, and outputs a ciphertext ct and shared secret key K.
  • Decaps(sk, c) -> K: A decapsulation algorithm that takes as input a secret key sk and ciphertext c, and outputs a shared secret key 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)
  • 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 a public key, pk, (termed an encapsulation key in {{FIPS203}}) and a private key, sk, (decapsulation key). Public 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(pk) (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.

Kemeleon Encoding {#kemeleon}

At a high level, the constructions in this document instantiate the following functions:

  • EncodePk(pk) -> epk is the (possibly randomized) encoding algorithm that on input a public key, outputs an obfuscated public key or an error.
  • DecodePk(epk) -> pk is the deterministic decoding algorithm that on input an obfuscated public key, outputs a public 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.

Common Functions {#common-func}

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

Encoding Public Keys {#pk-encoding}

The following algorithms encode ML-KEM public 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(pk = (t, rho)):
   r = VectorEncode(t)
   if r == err:
      return err
   else:
      return concat(r,rho)

Kemeleon.DecodePk(epk):
   r,rho = epk # rho is fixed length
   t = VectorDecode(r)
   return (t, rho)

Encoding Ciphertexts {#ctxt-encoding}

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 public 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)

Non-Rejection Sampling Variant {#no-rejection}

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 public 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(ec):
   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)

Faster Arithmetic Variant {#faster}

[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, {{fast-succ-prob}} gives success probabilities for public key and ciphertext encodings:

Parameter Pk 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
{: #fast-succ-prob title="Success probabilities for faster Kemeleon encoding"}

Deterministic Encoding {#deterministic}

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., not using the original key K to derive randomness and for other purposes).

The randomness used to encode a public key MAY be stored alongside the corresponding secret key, if it is subsequently needed. See {{randomness-security}} for relevant discussion on keeping this randomness secret.

Summary of Encodings {#comparison}

Algorithm / Parameter Output size (bytes) Success probability Additional considerations
Kemeleon - ML-KEM512 pk: 781, ctxt: 877 pk: 0.56, ctxt: 0.51 Large int (750B) arithmetic
Kemeleon - ML-KEM768 pk: 1156, ctxt: 1252 pk: 0.83, ctxt: 0.77 Large int (1150B) arithmetic
Kemeleon - ML-KEM1024 pk: 1530, ctxt: 1658 pk: 0.62, ctxt: 0.57 Large int (1500B) arithmetic
:----------------------- -------------------: -------------------: ------------------------:
KemeleonNR - ML-KEM512 pk: 797, ctxt: 1140 pk: 1.00, ctxt: 1.00 Large int (1123B) arithmetic
KemeleonNR - ML-KEM768 pk: 1172, ctxt: 1514 pk: 1.00, ctxt: 1.00 Large int (1498B) arithmetic
KemeleonNR - ML-KEM1024 pk: 1546, ctxt: 1889 pk: 1.00, ctxt: 1.00 Large int (1872B) arithmetic
:----------------------- -------------------: -------------------: ------------------------:
KemeleonFT - ML-KEM512 pk: 781, ctxt: 877 pk: 0.49, ctxt: 0.45 Smaller int (235B) arithmetic
KemeleonFT - ML-KEM768 pk: 1156, ctxt: 1252 pk: 0.29, ctxt: 0.25 Smaller int (355B) arithmetic
KemeleonFT - ML-KEM1024 pk: 1530, ctxt: 1658 pk: 0.53, ctxt: 0.47 Smaller int (475B) arithmetic
{: #summary-encoding title="Summary of Kemeleon Variants, NR = No Reject, FT = Faster"}

Security Considerations {#security}

This section contains additional security considerations about the Kemeleon encodings described in this document.

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.

Randomness Sampling {#randomness-security}

Both public 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.

Timing Side-Channels {#timing-security}

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 public keys or ciphertexts may leak information about the use of Kemeleon encodings, as might the overhead of the encoding itself.

IANA Considerations

This document has no IANA actions.

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