Internet DRAFT - draft-cfrg-schwabe-kyber
draft-cfrg-schwabe-kyber
None P. Schwabe
Internet-Draft MPI-SPI & Radboud University
Intended status: Informational B. Westerbaan
Expires: 2 October 2023 Cloudflare
31 March 2023
Kyber Post-Quantum KEM
draft-cfrg-schwabe-kyber-02
Abstract
This memo specifies a preliminary version ("draft00", "v3.02") of
Kyber, an IND-CCA2 secure Key Encapsulation Method.
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://bwesterb.github.io/draft-schwabe-cfrg-kyber/draft-cfrg-
schwabe-kyber.html. Status information for this document may be
found at https://datatracker.ietf.org/doc/draft-cfrg-schwabe-kyber/.
Source for this draft and an issue tracker can be found at
https://github.com/bwesterb/draft-schwabe-cfrg-kyber.
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document authors. All rights reserved.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Warning on stability . . . . . . . . . . . . . . . . . . 3
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 4
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4. The field GF(q) . . . . . . . . . . . . . . . . . . . . . . . 5
4.1. Size . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2. Compression . . . . . . . . . . . . . . . . . . . . . . . 6
5. The ring Rq . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.1. Operations . . . . . . . . . . . . . . . . . . . . . . . 7
5.1.1. Size of polynomials . . . . . . . . . . . . . . . . . 7
5.1.2. Addition and subtraction . . . . . . . . . . . . . . 7
5.1.3. Multiplication . . . . . . . . . . . . . . . . . . . 8
6. Vector and matrices . . . . . . . . . . . . . . . . . . . . . 12
6.1. Operations on vectors . . . . . . . . . . . . . . . . . . 12
6.2. Dot product and matrix multiplication . . . . . . . . . . 12
6.3. Transpose . . . . . . . . . . . . . . . . . . . . . . . . 12
7. Symmetric cryptographic primitives . . . . . . . . . . . . . 13
8. Serialization . . . . . . . . . . . . . . . . . . . . . . . . 13
8.1. OctetsToBits . . . . . . . . . . . . . . . . . . . . . . 13
8.2. Encode and Decode . . . . . . . . . . . . . . . . . . . . 14
8.2.1. Polynomials . . . . . . . . . . . . . . . . . . . . . 14
8.2.2. Vectors . . . . . . . . . . . . . . . . . . . . . . . 14
9. Sampling of polynomials . . . . . . . . . . . . . . . . . . . 14
9.1. Uniformly . . . . . . . . . . . . . . . . . . . . . . . . 14
9.1.1. sampleMatrix . . . . . . . . . . . . . . . . . . . . 15
9.2. From a binomial distribution . . . . . . . . . . . . . . 15
9.2.1. sampleNoise . . . . . . . . . . . . . . . . . . . . . 16
10. Inner malleable public-key encryption scheme . . . . . . . . 16
10.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 16
10.2. Key generation . . . . . . . . . . . . . . . . . . . . . 16
10.3. Encryption . . . . . . . . . . . . . . . . . . . . . . . 17
10.4. Decryption . . . . . . . . . . . . . . . . . . . . . . . 18
11. Kyber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
11.1. Key generation . . . . . . . . . . . . . . . . . . . . . 19
11.2. Encapsulation . . . . . . . . . . . . . . . . . . . . . 19
11.3. Decapsulation . . . . . . . . . . . . . . . . . . . . . 19
12. Parameter sets . . . . . . . . . . . . . . . . . . . . . . . 20
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13. Machine-readable specification . . . . . . . . . . . . . . . 22
14. Security Considerations . . . . . . . . . . . . . . . . . . . 28
15. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 28
16. References . . . . . . . . . . . . . . . . . . . . . . . . . 28
16.1. Normative References . . . . . . . . . . . . . . . . . . 28
16.2. Informative References . . . . . . . . . . . . . . . . . 29
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 30
Appendix B. Change Log . . . . . . . . . . . . . . . . . . . . . 30
B.1. Since draft-schwabe-cfrg-kyber-01 . . . . . . . . . . . . 30
B.2. Since draft-schwabe-cfrg-kyber-00 . . . . . . . . . . . . 30
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 30
1. Introduction
Kyber is NIST's pick for a post-quantum key agreement [nistr3].
Kyber is not a Diffie-Hellman (DH) style non-interactive key
agreement, but instead, Kyber is a Key Encapsulation Method (KEM). A
KEM is a three-tuple of algorithms (_KeyGen_, _Encapsulate_,
_Decapsulate_):
* _KeyGen_ takes no inputs and generates a private key and a public
key;
* _Encapsulate_ takes as input a public key and produces as ouput a
ciphertext and a shared secret;
* _Decapsulate_ takes as input a ciphertext and a private key and
produces a shared secret.
Like DH, a KEM can be used as an unauthenticated key-agreement
protocol, for example in TLS [hybrid]. However, unlike DH, a KEM-
based key agreement is _interactive_, because the party executing
Encapsulate can compute its protocol message (the ciphertext) only
after having received the input (public key) from the party running
_KeyGen_ and _Decapsulate_.
A KEM can be transformed into a PKE scheme using HPKE [RFC9180].
1.1. Warning on stability
*NOTE* This draft is not stable and does not (yet) match the final
NIST standard expected in 2024. Currently it matches Kyber as
submitted to round 3 of the NIST PQC process [KyberV302].
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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. Overview
Kyber is an IND-CCA2 secure KEM. It is constructed by applying a
Fujisaki-Okamato style transformation on InnerPKE, which is the
underlying IND-CPA secure Public Key Encryption scheme. We cannot
use InnerPKE directly, as its ciphertexts are malleable.
F.O. transform
InnerPKE ----------------------> Kyber
IND-CPA IND-CCA2
Kyber is a lattice-based scheme. More precisely, its security is
based on the learning-with-errors-and-rounding problem in module
lattices (MLWER). The underlying polynomial ring R (defined in
Section 5) is chosen such that multiplication is very fast using the
number theoretic transform (NTT, see Section 5.1.3.1).
An InnerPKE private key is a vector _s_ over R of length k which is
_small_ in a particular way. Here k is a security parameter akin to
the size of a prime modulus. For Kyber512, which targets AES-128's
security level, the value of k is 2, for Kyber768 (AES-192 security
level) k is 3, and for Kyber1024 (AES-256 security level) k is 4.
The public key consists of two values:
* _A_ a k-by-k matrix over R sampled uniformly at random _and_
* _t = A s + e_, where e is a suitably small masking vector.
Distinguishing between such A s + e and a uniformly sampled t is the
decision module learning-with-errors (MLWE) problem. If that is
hard, then it is also hard to recover the private key from the public
key as that would allow you to distinguish between those two.
To save space in the public key, A is recomputed deterministically
from a 256bit seed _rho_. Strictly speaking, A is not uniformly
random anymore, but it's computationally indistuinguishable from it.
A ciphertext for a message m under this public key is a pair (c_1,
c_2) computed roughly as follows:
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c_1 = Compress(A^T r + e_1, d_u)
c_2 = Compress(t^T r + e_2 + Decompress(m, 1), d_v)
where
* e_1, e_2 and r are small blinds;
* Compress(-, d) removes some information, leaving d bits per
coefficient and Decompress is an "approximate inverse" of
Compress;
* d_u, d_v are scheme parameters; and
* superscript T denotes transposition, so _A^T_ is the transpose of
A, see Section 6.3 and _t^T r_ is the dot product of t and r, see
Section 6.2.
Distinguishing such a ciphertext and uniformly sampled (c_1, c_2) is
an example of the full MLWER problem, see Section 4.4 of [KyberV302].
To decrypt the ciphertext, one computes
m = Compress(Decompress(c_2, d_v) - s^T Decompress(c_1, d_u), 1).
It it not straight-forward to see that this formula is correct. In
fact, there is negligable but non-zero probability that a ciphertext
does not decrypt correctly given by the DFP column in Table 4. This
failure probability can be computed by a careful automated analysis
of the probabilities involved, see kyber_failure.py of [SecEst].
To define all these operations precisely, we first define the field
of coefficients for our polynomial ring; what it means to be small;
and how to compress. Then we define the polynomial ring R; its
operations and in particular the NTT. We continue with the different
methods of sampling and (de)serialization. Then, we first define
InnerPKE and finally Kyber proper.
4. The field GF(q)
Kyber is defined over GF(q) = Z/qZ, the integers modulo q = 13*2^8+1
= 3329.
4.1. Size
To define the size of a field element, we need a signed modulo. For
any odd m, we write
a smod m
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for the unique integer b with -(m-1)/2 < b <= (m-1)/2 and b = a
modulo m.
To avoid confusion, for the more familiar modulo we write umod; that
is
a umod m
is the unique integer b with 0 <= b < m and b = a modulo m.
Now we can define the norm of a field element:
|| a || = abs(a smod q)
Examples:
3325 smod q = -4 || 3325 || = 4
-3320 smod q = 9 || -3320 || = 9
// TODO (#23) Should we define smod and umod at all, since we don't
// use it.
//
// -- Bas
4.2. Compression
In several parts of the algorithm, we will need a method to compress
fied elements down into d bits. To do this, we use the following
method.
For any positive integer d, integer x and integer 0 <= y < 2^d, we
define
Compress(x, d) = Round( (2^d / q) x ) umod 2^d
Decompress(y, d) = Round( (q / 2^d) y )
where Round(x) rounds any fraction to the nearest integer going up
with ties.
Note that in Section 6.1 we extend Compress and Decompress to
polynomials and vectors of polynomials.
These two operations have the following properties:
* 0 <= Compress(x, d) < 2^d
* 0 <= Decompress(y, d) < q
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* Compress(Decompress(y, d), d) = y
* If Decompress(Compress(x, d), d) = x', then || x' - x || <=
Round(q/2^(d+1))
* If x = x' modulo q, then Compress(x, d) = Compress(x', d)
For implementation efficiency, these can be computed as follows.
Compress(x, d) = Div( (x << d) + q/2), q ) & ((1 << d) - 1)
Decompress(y, d) = (q*y + (1 << (d-1))) >> d
where Div(x, a) = Floor(x / a).
// TODO Do we want to include the proof that this is correct? Do we
// need to define >> and <<?
//
// -- Bas
5. The ring Rq
Kyber is defined over a polynomial ring Rq = GF(q)[x]/(x^n+1) where
n=256 (and q=3329). Elements of Rq are tuples of 256 integers modulo
q. We will call them polynomials or elements interchangeably.
A tuple a = (a_0, ..., a_255) represents the polynomial
a_0 + a_1 x + a_2 x^2 + ... + a_255 x^255.
With polynomial coefficients, vector and matrix indices, we will
start counting at zero.
5.1. Operations
5.1.1. Size of polynomials
For a polynomial a = (a_0, ..., a_255) in R, we write:
|| a || = max_i || a_i ||
Thus a polynomial is considered large if one of its components is
large.
5.1.2. Addition and subtraction
Addition and subtraction of elements is componentwise. Thus
(a_0, ..., a_255) + (b_0, ..., b_255) = (a_0 + b_0, ..., a_255 + b_255),
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and
(a_0, ..., a_255) - (b_0, ..., b_255) = (a_0 - b_0, ..., a_255 - b_255),
where addition/subtractoin in each component is computed modulo q.
5.1.3. Multiplication
Multiplication is that of polynomials (convolution) with the
additional rule that x^256=-1. To wit
(a_0, ..., a_255) \* (b_0, ..., b_255)
= (a_0 * b_0 - a_255 * b_1 - ... - a_1 * b_255,
a_0 * b_1 + a_1 * b_0 - a_255 * b_2 - ... - a_2 * b_255,
...
a_0 * b_255 + ... + a_255 * b_0)
We will not use this schoolbook multiplication to compute the
product. Instead we will use the more efficient, number theoretic
transform (NTT), see Section 5.1.3.1.
5.1.3.1. Background on the Number Theoretic Transform (NTT)
The modulus q was chosen such that 256 divides into q-1. This means
that there are zeta with
zeta^128 = -1 modulo q
With such a zeta, we can almost completely split the polynomial
x^256+1 used to define R over GF(q):
x^256 + 1 = x^256 - zeta^128
= (x^128 - zeta^64)(x^128 + zeta^64)
= (x^128 - zeta^64)(x^128 - zeta^192)
= (x^64 - zeta^32)(x^64 + zeta^32)
(x^64 - zeta^96)(x^64 + zeta^96)
...
= (x^2 - zeta)(x^2 + zeta)(x^2 - zeta^65)(x^2 + zeta^65)
... (x^2 - zeta^127)(x^2 + zeta^127)
Note that the powers of zeta that appear in the second, fourth, ...,
and final lines are in binary:
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0100000 1100000
0010000 1010000 0110000 1110000
0001000 1001000 0101000 1101000 0011000 1011000 0111000 1111000
...
0000001 1000001 0100001 1100001 0010001 1010001 0110001 ... 1111111
That is: brv(2), brv(3), brv(4), ..., where brv(x) denotes the 7-bit
bitreversal of x. The final line is brv(64), brv(65), ..., brv(127).
These polynomials x^2 +- zeta^i are irreducible and coprime, hence by
the Chinese Remainder Theorem for commutative rings, we know
R = GF(q)[x]/(x^256+1) -> GF(q)[x]/(x^2-zeta) x ... x GF(q)[x]/(x^2+zeta^127)
given by a |-> ( a mod x^2 - zeta, ..., a mod x^2 + zeta^127 ) is an
isomorphism. This is the Number Theoretic Transform (NTT).
Multiplication on the right is much easier: it's almost
componentwise, see Section 5.1.3.3.
A propos, the the constant factors that appear in the moduli in order
can be computed efficiently as follows (all modulo q):
-zeta = -zeta^brv(64) = -zeta^{1 + 2 brv(0)}
zeta = zeta^brv(64) = -zeta^{1 + 2 brv(1)}
-zeta^65 = -zeta^brv(65) = -zeta^{1 + 2 brv(2)}
zeta^65 = zeta^brv(65) = -zeta^{1 + 2 brv(3)}
-zeta^33 = -zeta^brv(66) = -zeta^{1 + 2 brv(4)}
zeta^33 = zeta^brv(66) = -zeta^{1 + 2 brv(5)}
...
-zeta^127 = -zeta^brv(127) = -zeta^{1 + 2 brv(126)}
zeta^127 = zeta^brv(127) = -zeta^{1 + 2 brv(127)}
To compute a multiplication in R efficiently, one can first use the
NTT, to go to the right "into the NTT domain"; compute the
multiplication there and move back with the inverse NTT.
The NTT can be computed efficiently by performing each binary split
of the polynomial separately as follows:
a |-> ( a mod x^128 - zeta^64, a mod x^128 + zeta^64 ),
|-> ( a mod x^64 - zeta^32, a mod x^64 + zeta^32,
a mod x^64 - zeta^96, a mod x^64 + zeta^96 ),
et cetera
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If we concatenate the resulting coefficients, expanding the
definitions, for the first step we get:
a |-> ( a_0 + zeta^64 a_128, a_1 + zeta^64 a_129,
...
a_126 + zeta^64 a_254, a_127 + zeta^64 a_255,
a_0 - zeta^64 a_128, a_1 - zeta^64 a_129,
...
a_126 - zeta^64 a_254, a_127 - zeta^64 a_255)
We can see this as 128 applications of the linear map CT_64, where
CT_i: (a, b) |-> (a + zeta^i b, a - zeta^i b) modulo q
for the appropriate i in the following order, pictured in the case of
n=16:
-x----------------x--------x---
-|-x--------------|-x------|-x-
-|-|-x------------|-|-x----x-|-
-|-|-|-x----------|-|-|-x----x-
-|-|-|-|-x--------x-|-|-|--x---
-|-|-|-|-|-x--------x-|-|--|-x-
-|-|-|-|-|-|-x--------x-|--x-|-
-|-|-|-|-|-|-|-x--------x----x-
-x-|-|-|-|-|-|-|--x--------x---
---x-|-|-|-|-|-|--|-x------|-x-
-----x-|-|-|-|-|--|-|-x----x-|-
-------x-|-|-|-|--|-|-|-x----x-
---------x-|-|-|--x-|-|-|--x---
-----------x-|-|----x-|-|--|-x-
-------------x-|------x-|--x-|-
---------------x--------x----x-
For n=16 there are 3 levels with 1, 2 and 4 row groups respectively.
For the full n=256, there are 7 levels with 1, 2, 4, 8, 16, 32 and 64
row groups respectively. The appropriate power of zeta in the first
level is brv(1)=64. The second level has brv(2) and brv(3) as powers
of zeta for the top and bottom row group respectively, and so on.
The CT_i is known as a Cooley-Tukey butterfly. Its inverse is given
by the Gentleman-Sande butterfly:
GS_i: (a, b) |-> ( (a+b)/2, zeta^-i (a-b)/2 ) modulo q
The inverse NTT can be computed by replacing CS_i by GS_i and
flipping the diagram horizontally.
// TODO (#8) This section gives background not necessary for the
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// implementation. Should we keep it?
//
// -- Bas
5.1.3.1.1. Optimization notes
The modular divisions by two in the InvNTT can be collected into a
single modular division by 128.
zeta^-i can be computed as -zeta^(128-i), which allows one to use the
same precomputed table of powers of zeta for both the NTT and InvNTT.
// TODO Add hints on lazy Montgomery reduction? Including
// https://eprint.iacr.org/2020/1377.pdf
//
// -- Bas
5.1.3.2. NTT and InvNTT
As primitive 256th root of unity we use zeta=17.
As before, brv(i) denotes the 7-bit bitreversal of i, so brv(1)=64
and brv(91)=109.
The NTT is a linear bijection R -> R given by the matrix:
[ zeta^{ (2 brv(i>>1) + 1) (j>>1) } if i=j mod 2
(NTT)_{ij} = [
[ 0 otherwise
Recall that we start counting rows and columns at zero. The NTT can
be computed more efficiently as described in section Section 5.1.3.1.
The inverse of the NTT is called InvNTT. It is given by the matrix:
[ zeta^{ 256 - (2 brv(j>>1) + 1) (i>>1) } if i=j mod 2
128 (InvNTT)_{ij} = [
[ 0 otherwise
Examples:
NTT(1, 1, 0, ..., 0) = (1, 1, ..., 1, 1)
NTT(0, 1, 2, ..., 255) = (2429, 2845, 425, 1865, ..., 2502, 2134, 2717, 2303)
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5.1.3.3. Multiplication in NTT domain
For elements a, b in R, we write a o b for multiplication in the NTT
domain. That is: a * b = InvNTT(NTT(a) o NTT(b)). Concretely:
[ a_i b_i + zeta^{2 brv(i >> 1) + 1} a_{i+1} b_{i+1} if i even
(a o b)_i = [
[ a_{i-1} b_i + a_i b_{i-1} otherwise
6. Vector and matrices
6.1. Operations on vectors
Recall that Compress(x, d) maps a field element x into {0, ...,
2^d-1}. In Kyber d is at most 11 and so we can interpret Compress(x,
d) as a field element again.
In this way, we can extend Compress(-, d) to polynomials by applying
to each coefficient separately and in turn to vectors by applying to
each polynomial. That is, for a vector v and polynomial p:
Compress(p, d)_i = Compress(p_i, d)
Compress(v, d)_i = Compress(v_i, d)
6.2. Dot product and matrix multiplication
We will also use "o", from section Section 5.1.3.3, to denote the dot
product and matrix multiplication in the NTT domain. Concretely:
1. For two length k vector v and w, we write
v o w = v_0 o w_0 + ... + v_{k-1} o w_{k-1}
2. For a k by k matrix A and a length k vector v, we have
(A o v)_i = A_i o v,
where A_i denotes the (i+1)th row of the matrix A as we start
counting at zero.
6.3. Transpose
For a matrix A, we denote by A^T the tranposed matrix. To wit:
A^T_ij = A_ji.
We define Decompress(-, d) for vectors and polynomials in the same
way.
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7. Symmetric cryptographic primitives
Kyber makes use of various symmertic primitives PRF, XOF, KDF, H and
G, where
XOF(seed) = SHAKE-128(seed)
PRF(seed, counter) = SHAKE-256(seed || counter)
KDF(prekey) = SHAKE-256(msg)[:32]
H(msg) = SHA3-256(msg)
G(msg) = (SHA3-512(msg)[:32], SHA3-512(msg)[32:])
Here counter is an octet; seed is 32 octets; prekey is 64 octets; and
the length of msg varies.
On the surface, they look different, but they are all based on the
same flexible Keccak XOF that uses the f1600 permutation, see
[fips202]:
XOF(seed) = Keccak[256](seed || 1111, .)
PRF(seed, ctr) = Keccak[512](seed || ctr || 1111, .)
KDF(prekey) = Keccak[512](prekey || 1111, 256)
H(msg) = Keccak[512](msg || 01, 256)
G(msg) = (Keccak[1024](msg || 01, 512)[:32],
Keccak[1024](msg || 01, 512)[32:])
Keccak[c] = Sponge[Keccak-f[1600], pad10*1, 1600-c]
The reason five different primitives are used is to ensure domain
separation, which is crucial for security, cf. [hashToCurve] ยง2.2.5.
Additionally, a smaller sponge capacity is used for performance where
permissable by the security requirements.
8. Serialization
8.1. OctetsToBits
For any list of octets a_0, ..., a_{s-1}, we define OctetsToBits(a),
which is a list of bits of length 8s, defined by
OctetsToBits(a)_i = ((a_(i>>3)) >> (i umod 8)) umod 2.
Example:
OctetsToBits(12,45) = (0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0)
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8.2. Encode and Decode
For an integer 0 < w <= 12, we define Decode(a, w), which converts
any list a of w*l/8 octets into a list of length l with values in {0,
..., 2^w-1} as follows.
Decode(a, w)_i = b_{wi} + b_{wi+1} 2 + b_{wi+2} 2^2 + ... + b_{wi+w-1} 2^{w-1},
where b = OctetsToBits(a).
Encode(-, w) is the unique inverse of Decode(-, w)
8.2.1. Polynomials
A polynomial p is encoded by passing its coefficients to Encode:
EncodePoly(p, w) = Encode(p_0, p_1, ..., p_{n-1}, w)
DecodePoly(-, w) is the unique inverse of EncodePoly(-, w).
8.2.2. Vectors
A vector v of length k over R is encoded by concatenating the
coefficients in the obvious way:
EncodeVec(v, w) = Encode((v_0)_0, ..., (v_0)_{n-1},
(v_1)_{0}, ..., (v_1)_{n-1},
..., (v_{k-1})_{n-1}, w)
DecodeVec(-, w) is the unique inverse of EncodeVec(-, w).
9. Sampling of polynomials
9.1. Uniformly
The polynomials in the matrix A are sampled uniformly and
deterministically from an octet stream (XOF) using rejection sampling
as follows.
Three octets b_0, b_1, b_2 are read from the stream at a time. These
are interpreted as two 12-bit unsigned integers d_1, d_2 via
d_1 + d_2 2^12 = b_0 + b_1 2^8 + b_2 2^16
This creates a stream of 12-bit ds. Of these, the elements >= q are
ignored. From the resultant stream, the coefficients of the
polynomial are taken in order. In Python:
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def sampleUniform(stream):
cs = []
while True:
b = stream.read(3)
d1 = b[0] + 256*(b[1] % 16)
d2 = (b[1] >> 4) + 16*b[2]
for d in [d1, d2]:
if d >= q: continue
cs.append(d)
if len(cs) == n: return Poly(cs)
Example:
sampleUniform(SHAKE-128('')) = (3199, 697, 2212, 2302, ..., 255, 846, 1)
9.1.1. sampleMatrix
Now, the _k_ by _k_ matrix _A_ over _R_ is derived deterministically
from a 32-octet seed _rho_ using sampleUniform as follows.
sampleMatrix(rho)_{ij} = sampleUniform(XOF(rho || octet(j) || octet(i))
That is, to derive the polynomial at the _i_th row and _j_th column,
sampleUniform is called with the 34-octet seed created by first
appending the octet _j_ and then the octet _i_ to _rho_. Recall that
we start counting rows and columns from zero.
As the NTT is a bijection, it does not matter whether we interpret
the polynomials of the sampled matrix in the NTT domain or not. For
efficiency, we do interpret the sampled matrix in the NTT domain.
9.2. From a binomial distribution
Noise is sampled from a centered binomial distribution Binomial(2eta,
1/2) - eta deterministically as follows.
An octet array a of length 64*eta is converted to a polynomial CBD(a,
eta)
CBD(a, eta)_i = b_{2i eta} + b_{2i eta + 1} + ... + b_{2i eta + eta-1}
- b_{2i eta + eta} - ... - b_{2i eta + 2eta - 1},
where b = OctetsToBits(a).
Examples:
CBD((0, 1, 2, ..., 127), 2) = (0, 0, 1, 0, 1, 0, ..., 3328, 1, 0, 1)
CBD((0, 1, 2, ..., 191), 3) = (0, 1, 3328, 0, 2, ..., 3328, 3327, 3328, 1)
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9.2.1. sampleNoise
A _k_ component small vector _v_ is derived from a seed 32-octet seed
_sigma_, an offset _offset_ and size _eta_ as follows:
sampleNoise(sigma, eta, offset)_i = CBD(PRF(sigma, octet(i+offset)), eta)
Recall that we start counting vector indices at zero.
10. Inner malleable public-key encryption scheme
We are ready to define the IND-CPA secure Public-Key Encryption
scheme that underlies Kyber. It is unsafe to use this underlying
scheme directly as its ciphertexts are malleable. Instead, a Public-
Key Encryption scheme can be constructed on top of Kyber by using
HPKE [RFC9180].
10.1. Parameters
We have already been introduced to the following parameters:
_q_ Order of field underlying _R_.
_n_ Length of polynomials in _R_.
_zeta_ Primitive root of unity in GF(q), used for NTT in R.
_XOF_, _H_, _G_, _PRF_, _KDF_ Various symmetric primitives.
_k_ Main security parameter: the number of rows and columns in the
matrix _A_.
Additionally, Kyber takes the following parameters
_eta1_, _eta2_ Size of small coefficients used in the private key
and noise vectors.
_d_u_, _d_v_ How many bits to retain per coefficient of the _u_ and
_v_ components of the ciphertext.
The values of these parameters are given in Section 12.
10.2. Key generation
InnerKeyGen(seed) takes a 32 octet *seed* and deterministically
produces a keypair as follows.
1. Set (rho, sigma) = G(seed).
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2. Derive
1. AHat = sampleMatrix(rho).
2. s = sampleNoise(sigma, eta1, 0)
3. e = sampleNoise(sigma, eta1, k)
3. Compute
1. sHat = NTT(s)
2. tHat = AHat o sHat + NTT(e)
4. Return
1. publicKey = EncodeVec(tHat, 12) || rho
2. privateKey = EncodeVec(sHat, 12)
Note that in essence we're simply computing t = A s + e.
10.3. Encryption
InnerEnc(msg, publicKey, seed) takes a 32-octet seed, and
deterministically encrypts the 32-octet msg for the InnerPKE public
key publicKey as follows.
1. Split publicKey into
1. k*(n/8)*12-octet tHatPacked
2. 32-octet rho
2. Parse tHat = DecodeVec(tHatPacked, 12)
3. Derive
1. AHat = sampleMatrix(rho)
2. r = sampleNoise(seed, eta1, 0)
3. e_1 = sampleNoise(seed, eta2, k)
4. e_2 = sampleNoise(seed, eta2, 2k)_0
4. Compute
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1. rHat = NTT(r)
2. u = InvNTT(AHat^T o rHat) + e_1
3. v = InvNTT(tHat o rHat) + e_2 + Decompress(DecodePoly(msg,
1), 1)
4. c_1 = EncodeVec(Compress(u, d_u), d_u)
5. c_2 = EncodePoly(Compress(v, d_v), d_v)
5. Return
1. cipherText = c_1 || c_2
10.4. Decryption
InnerDec(cipherText, privateKey) takes an InnerPKE private key
privateKey and decrypts a cipher text cipherText as follows.
1. Split cipherText into
1. d_u*k*n/8-octet c_1
2. d_v*n/8-octet c_2
2. Parse
1. u = Decompress(DecodeVec(c_1, d_u), d_u)
2. v = Decompress(DecodePoly(c_2, d_v), d_v)
3. sHat = DecodeVec(privateKey, 12)
3. Compute
1. m = v - InvNTT(sHat o NTT(u))
4. Return
1. plainText = EncodePoly(Compress(m, 1), 1)
11. Kyber
Now we are ready to define Kyber itself.
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11.1. Key generation
A Kyber keypair is derived deterministically from a 64-octet seed as
follows.
1. Split seed into
1. A 32-octet cpaSeed
2. A 32-octet z
2. Compute
1. (cpaPublicKey, cpaPrivateKey) = InnerKeyGen(cpaSeed)
2. h = H(cpaPublicKey)
3. Return
1. publicKey = cpaPublicKey
2. privateKey = cpaPrivateKey || cpaPublicKey || h || z
11.2. Encapsulation
Kyber encapsulation takes a public key and a 32-octet seed and
deterministically generates a shared secret and ciphertext for the
public key as follows.
1. Compute
1. m = H(seed)
2. (Kbar, cpaSeed) = G(m || H(publicKey))
3. cpaCipherText = InnerEnc(m, publicKey, cpaSeed)
2. Return
1. cipherText = cpaCipherText
2. sharedSecret = KDF(KBar || H(cpaCipherText))
11.3. Decapsulation
Kyber decapsulation takes a private key and a cipher text and returns
a shared secret as follows.
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1. Split privateKey into
1. A 12*k*n/8-octet cpaPrivateKey
2. A 12*k*n/8+32-octet cpaPublicKey
3. A 32-octet h
4. A 32-octet z
2. Compute
1. m2 = InnerDec(cipherText, cpaPrivateKey)
2. (KBar2, cpaSeed2) = G(m2 || h)
3. cipherText2 = InnerEnc(m2, cpaPublicKey, cpaSeed2)
4. K1 = KDF(KBar2 || H(cipherText))
5. K2 = KDF(z || H(cipherText))
3. In constant-time, set K = K1 if cipherText == cipherText2 else
set K = K2.
4. Return
1. sharedSecret = K
12. Parameter sets
+======+=======+=================================+
| Name | Value | Description |
+======+=======+=================================+
| q | 3329 | Order of base field |
+------+-------+---------------------------------+
| n | 256 | Degree of polynomials |
+------+-------+---------------------------------+
| zeta | 17 | nth root of unity in base field |
+------+-------+---------------------------------+
Table 1: Common parameters to all versions of
Kyber
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+===========+===================+
| Primitive | Instantiation |
+===========+===================+
| XOF | SHAKE-128 |
+-----------+-------------------+
| H | SHA3-256 |
+-----------+-------------------+
| G | SHA3-512 |
+-----------+-------------------+
| PRF(s,b) | SHAKE-256(s || b) |
+-----------+-------------------+
| KDF | SHAKE-256 |
+-----------+-------------------+
Table 2: Instantiation of
symmetric primitives in Kyber
+============+===================================================+
| Name | Description |
+============+===================================================+
| k | Dimension of module |
+------------+---------------------------------------------------+
| eta1, eta2 | Size of "small" coefficients used in the private |
| | key and noise vectors. |
+------------+---------------------------------------------------+
| d_u | How many bits to retain per coefficient of u, the |
| | private-key independent part of the ciphertext |
+------------+---------------------------------------------------+
| d_v | How many bits to retain per coefficient of v, the |
| | private-key dependent part of the ciphertext. |
+------------+---------------------------------------------------+
Table 3: Description of kyber parameters
+===============+===+======+======+=====+=====+=====+========+
| Parameter set | k | eta1 | eta2 | d_u | d_v | sec | DFP |
+===============+===+======+======+=====+=====+=====+========+
| Kyber512 | 2 | 3 | 2 | 10 | 4 | I | 2^-139 |
+---------------+---+------+------+-----+-----+-----+--------+
| Kyber768 | 3 | 2 | 2 | 10 | 4 | III | 2^-164 |
+---------------+---+------+------+-----+-----+-----+--------+
| Kyber1024 | 4 | 2 | 2 | 11 | 5 | V | 2^-174 |
+---------------+---+------+------+-----+-----+-----+--------+
Table 4: Kyber parameter sets with NIST security level
(sec) and decryption failure probability (DFP)
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13. Machine-readable specification
# WARNING This is a specification of Kyber; not a production ready
# implementation. It is slow and does not run in constant time.
import io
import hashlib
import functools
import collections
from math import floor
q = 3329
nBits = 8
zeta = 17
eta2 = 2
n = 2**nBits
inv2 = (q+1)//2 # inverse of 2
params = collections.namedtuple('params', ('k', 'du', 'dv', 'eta1'))
params512 = params(k = 2, du = 10, dv = 4, eta1 = 3)
params768 = params(k = 3, du = 10, dv = 4, eta1 = 2)
params1024 = params(k = 4, du = 11, dv = 5, eta1 = 2)
def smod(x):
r = x % q
if r > (q-1)//2:
r -= q
return r
# Rounds to nearest integer with ties going up
def Round(x):
return int(floor(x + 0.5))
def Compress(x, d):
return Round((2**d / q) * x) % (2**d)
def Decompress(y, d):
assert 0 <= y and y <= 2**d
return Round((q / 2**d) * y)
def BitsToWords(bs, w):
assert len(bs) % w == 0
return [sum(bs[i+j] * 2**j for j in range(w))
for i in range(0, len(bs), w)]
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def WordsToBits(bs, w):
return sum([[(b >> i) % 2 for i in range(w)] for b in bs], [])
def Encode(a, w):
return bytes(BitsToWords(WordsToBits(a, w), 8))
def Decode(a, w):
return BitsToWords(WordsToBits(a, 8), w)
def brv(x):
""" Reverses a 7-bit number """
return int(''.join(reversed(bin(x)[2:].zfill(nBits-1))), 2)
class Poly:
def __init__(self, cs=None):
self.cs = (0,)*n if cs is None else tuple(cs)
assert len(self.cs) == n
def __add__(self, other):
return Poly((a+b) % q for a,b in zip(self.cs, other.cs))
def __neg__(self):
return Poly(q-a for a in self.cs)
def __sub__(self, other):
return self + -other
def __str__(self):
return f"Poly({self.cs}"
def __eq__(self, other):
return self.cs == other.cs
def NTT(self):
cs = list(self.cs)
layer = n // 2
zi = 0
while layer >= 2:
for offset in range(0, n-layer, 2*layer):
zi += 1
z = pow(zeta, brv(zi), q)
for j in range(offset, offset+layer):
t = (z * cs[j + layer]) % q
cs[j + layer] = (cs[j] - t) % q
cs[j] = (cs[j] + t) % q
layer //= 2
return Poly(cs)
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def RefNTT(self):
# Slower, but simpler, version of the NTT.
cs = [0]*n
for i in range(0, n, 2):
for j in range(n // 2):
z = pow(zeta, (2*brv(i//2)+1)*j, q)
cs[i] = (cs[i] + self.cs[2*j] * z) % q
cs[i+1] = (cs[i+1] + self.cs[2*j+1] * z) % q
return Poly(cs)
def InvNTT(self):
cs = list(self.cs)
layer = 2
zi = n//2
while layer < n:
for offset in range(0, n-layer, 2*layer):
zi -= 1
z = pow(zeta, brv(zi), q)
for j in range(offset, offset+layer):
t = (cs[j+layer] - cs[j]) % q
cs[j] = (inv2*(cs[j] + cs[j+layer])) % q
cs[j+layer] = (inv2 * z * t) % q
layer *= 2
return Poly(cs)
def MulNTT(self, other):
""" Computes self o other, the multiplication of self and other
in the NTT domain. """
cs = [None]*n
for i in range(0, n, 2):
a1 = self.cs[i]
a2 = self.cs[i+1]
b1 = other.cs[i]
b2 = other.cs[i+1]
z = pow(zeta, 2*brv(i//2)+1, q)
cs[i] = (a1 * b1 + z * a2 * b2) % q
cs[i+1] = (a2 * b1 + a1 * b2) % q
return Poly(cs)
def Compress(self, d):
return Poly(Compress(c, d) for c in self.cs)
def Decompress(self, d):
return Poly(Decompress(c, d) for c in self.cs)
def Encode(self, d):
return Encode(self.cs, d)
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def sampleUniform(stream):
cs = []
while True:
b = stream.read(3)
d1 = b[0] + 256*(b[1] % 16)
d2 = (b[1] >> 4) + 16*b[2]
assert d1 + 2**12 * d2 == b[0] + 2**8 * b[1] + 2**16*b[2]
for d in [d1, d2]:
if d >= q:
continue
cs.append(d)
if len(cs) == n:
return Poly(cs)
def CBD(a, eta):
assert len(a) == 64*eta
b = WordsToBits(a, 8)
cs = []
for i in range(n):
cs.append((sum(b[:eta]) - sum(b[eta:2*eta])) % q)
b = b[2*eta:]
return Poly(cs)
def XOF(seed, j, i):
# TODO #5 proper streaming SHAKE128
return io.BytesIO(hashlib.shake_128(seed + bytes([j, i])).digest(
length=1344))
def PRF(seed, nonce):
# TODO #5 proper streaming SHAKE256
assert len(seed) == 32
return io.BytesIO(hashlib.shake_256(seed + bytes([nonce])
).digest(length=2000))
def G(seed):
h = hashlib.sha3_512(seed).digest()
return h[:32], h[32:]
def H(msg): return hashlib.sha3_256(msg).digest()
def KDF(msg): return hashlib.shake_256(msg).digest(length=32)
class Vec:
def __init__(self, ps):
self.ps = tuple(ps)
def NTT(self):
return Vec(p.NTT() for p in self.ps)
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def InvNTT(self):
return Vec(p.InvNTT() for p in self.ps)
def DotNTT(self, other):
""" Computes the dot product <self, other> in NTT domain. """
return sum((a.MulNTT(b) for a, b in zip(self.ps, other.ps)),
Poly())
def __add__(self, other):
return Vec(a+b for a,b in zip(self.ps, other.ps))
def Compress(self, d):
return Vec(p.Compress(d) for p in self.ps)
def Decompress(self, d):
return Vec(p.Decompress(d) for p in self.ps)
def Encode(self, d):
return Encode(sum((p.cs for p in self.ps), ()), d)
def __eq__(self, other):
return self.ps == other.ps
def EncodeVec(vec, w):
return Encode(sum([p.cs for p in vec.ps], ()), w)
def DecodeVec(bs, k, w):
cs = Decode(bs, w)
return Vec(Poly(cs[n*i:n*(i+1)]) for i in range(k))
def DecodePoly(bs, w):
return Poly(Decode(bs, w))
class Matrix:
def __init__(self, cs):
""" Samples the matrix uniformly from seed rho """
self.cs = tuple(tuple(row) for row in cs)
def MulNTT(self, vec):
""" Computes matrix multiplication A*vec in the NTT domain. """
return Vec(Vec(row).DotNTT(vec) for row in self.cs)
def T(self):
""" Returns transpose of matrix """
k = len(self.cs)
return Matrix((self.cs[j][i] for j in range(k))
for i in range(k))
def sampleMatrix(rho, k):
return Matrix([[sampleUniform(XOF(rho, j, i))
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for j in range(k)] for i in range(k)])
def sampleNoise(sigma, eta, offset, k):
return Vec(CBD(PRF(sigma, i+offset).read(64*eta), eta)
for i in range(k))
def constantTimeSelectOnEquality(a, b, ifEq, ifNeq):
# WARNING! In production code this must be done in a
# data-independent constant-time manner, which this implementation
# is not. In fact, many more lines of code in this
# file are not constant-time.
return ifEq if a == b else ifNew
def InnerKeyGen(seed, params):
assert len(seed) == 32
rho, sigma = G(seed)
A = sampleMatrix(rho, params.k)
s = sampleNoise(sigma, params.eta1, 0, params.k)
e = sampleNoise(sigma, params.eta1, params.k, params.k)
sHat = s.NTT()
eHat = e.NTT()
tHat = A.MulNTT(sHat) + eHat
pk = EncodeVec(tHat, 12) + rho
sk = EncodeVec(sHat, 12)
return (pk, sk)
def InnerEnc(pk, msg, seed, params):
assert len(msg) == 32
tHat = DecodeVec(pk[:-32], params.k, 12)
rho = pk[-32:]
A = sampleMatrix(rho, params.k)
r = sampleNoise(seed, params.eta1, 0, params.k)
e1 = sampleNoise(seed, eta2, params.k, params.k)
e2 = sampleNoise(seed, eta2, 2*params.k, 1).ps[0]
rHat = r.NTT()
u = A.T().MulNTT(rHat).InvNTT() + e1
m = Poly(Decode(msg, 1)).Decompress(1)
v = tHat.DotNTT(rHat).InvNTT() + e2 + m
c1 = u.Compress(params.du).Encode(params.du)
c2 = v.Compress(params.dv).Encode(params.dv)
return c1 + c2
def InnerDec(sk, ct, params):
split = params.du * params.k * n // 8
c1, c2 = ct[:split], ct[split:]
u = DecodeVec(c1, params.k, params.du).Decompress(params.du)
v = DecodePoly(c2, params.dv).Decompress(params.dv)
sHat = DecodeVec(sk, params.k, 12)
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return (v - sHat.DotNTT(u.NTT()).InvNTT()).Compress(1).Encode(1)
def KeyGen(seed, params):
assert len(seed) == 64
z = seed[32:]
pk, sk2 = InnerKeyGen(seed[:32], params)
h = H(pk)
return (pk, sk2 + pk + h + z)
def Enc(pk, seed, params):
assert len(seed) == 32
m = H(seed)
Kbar, r = G(m + H(pk))
ct = InnerEnc(pk, m, r, params)
K = KDF(Kbar + H(ct))
return (ct, K)
def Dec(sk, ct, params):
sk2 = sk[:12 * params.k * n//8]
pk = sk[12 * params.k * n//8 : 24 * params.k * n//8 + 32]
h = sk[24 * params.k * n//8 + 32 : 24 * params.k * n//8 + 64]
z = sk[24 * params.k * n//8 + 64 : 24 * params.k * n//8 + 96]
m2 = InnerDec(sk, ct, params)
Kbar2, r2 = G(m2 + h)
ct2 = InnerEnc(pk, m2, r2, params)
return constantTimeSelectOnEquality(
ct2, ct,
KDF(Kbar2 + H(ct)), # if ct == ct2
KDF(z + H(ct)), # if ct != ct2
)
14. Security Considerations
TODO Security (#18)
15. IANA Considerations
TODO (#17)
16. References
16.1. Normative References
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[fips202] National Institute of Standards and Technology, "FIPS PUB
202: SHA-3 Standard: Permutation-Based Hash and
Extendable-Output Functions", n.d.,
<https://nvlpubs.nist.gov/nistpubs/fips/
nist.fips.202.pdf>.
[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>.
16.2. Informative References
[hashToCurve]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
16, 15 June 2022, <https://datatracker.ietf.org/doc/html/
draft-irtf-cfrg-hash-to-curve-16>.
[hybrid] Stebila, D., Fluhrer, S., and S. Gueron, "Hybrid key
exchange in TLS 1.3", Work in Progress, Internet-Draft,
draft-stebila-tls-hybrid-design-03, 12 February 2020,
<https://datatracker.ietf.org/doc/html/draft-stebila-tls-
hybrid-design-03>.
[KyberV302]
Avanzi, R., Bos, J., Ducas, L., Kiltz, E., Lepoint, T.,
Lyubashevsky, V., Schanck, J., Schwabe, P., Seiler, G.,
and D. Stehle, "CRYSTALS-Kyber, Algorithm Specification
And Supporting Documentation (version 3.02)", 2021,
<https://pq-crystals.org/kyber/data/kyber-specification-
round3-20210804.pdf>.
[nistr3] The NIST PQC Team, "PQC Standardization Process:
Announcing Four Candidates to be Standardized, Plus Fourth
Round Candidates", n.d., <https://csrc.nist.gov/News/2022/
pqc-candidates-to-be-standardized-and-round-4>.
[RFC9180] Barnes, R., Bhargavan, K., Lipp, B., and C. Wood, "Hybrid
Public Key Encryption", RFC 9180, DOI 10.17487/RFC9180,
February 2022, <https://www.rfc-editor.org/rfc/rfc9180>.
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[SecEst] Ducas, L. and J. Schanck, "CRYSTALS security estimate
scripts", n.d.,
<https://github.com/pq-crystals/security-estimates>.
Appendix A. Acknowledgments
The authors would like to thank C. Wood, Florence D., I. Liusvaara,
J. Crawford, J. Schanck, M. Thomson, and N. Sullivan for their
input and assistance.
Appendix B. Change Log
*RFC Editor's Note:* Please remove this section prior to
publication of a final version of this document.
B.1. Since draft-schwabe-cfrg-kyber-01
* Fix various typos.
* Move sections around.
* Elaborate domain separation and encoding of nonces in symmetric
primitives.
* Add explicit formula for InvNTT.
* Add acknowledgements.
B.2. Since draft-schwabe-cfrg-kyber-00
* Test specification against NIST test vectors.
* Fix two unintentional mismatches between this document and the
reference implementation:
1. KDF uses SHAKE-256 instead of SHAKE-128.
2. Reverse order of seed. (z comes at the end.)
* Elaborate text in particular introduction, and symmetric key
section.
Authors' Addresses
Peter Schwabe
MPI-SPI & Radboud University
Email: peter@cryptojedi.org
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Bas Westerbaan
Cloudflare
Email: bas@cloudflare.com
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