CRYSTALS-Kyber
Algorithm Specifications And Supporting Documentation(version 3.02)
Roberto Avanzi, Joppe Bos, Léo Ducas, Eike Kiltz, Tancrède Lepoint,
Vadim Lyubashevsky, John M. Schanck, Peter Schwabe, Gregor Seiler, Damien Stehlé
August 4, 2021

1

Changelog
In the following we list the changes from Kyber as submitted to round-2 of the the NIST PQC projectin April 2019 and Kyber with round-3 tweaks together with brief explanations of the motivation for thechanges.

Changes to the core Kyber design• Increase noise parameter for Kyber512
In the round-2 submission of Kyber, the decryption error for Kyber512 was rather conservative, whileat the same time, there were requests to increase the Core-SVP hardness of this parameter set. It
therefore made sense to increase the binomial error distribution for Kyber512. Instead of increasing itfrom 2 to 3 across the board, we chose to use the fact that the rounding of the ciphertext coefficients to
10-bit integers also effectively adds some noise. We therefore increased the binomial noise from 2 to 3of the parameters in the key generation procedure and of the parameter r in the encryption procedure,
but not of errors e1 and e2 .
Relying on the rounding noise to add error is akin to the LWR assumption, but our reliance on it isquite small. First, it only adds 6 bits of Core-SVP hardness, and second, we are adding noise androunding, which presumably has less algebraic structure than just rounding. In short, without the
LWR assumption, our new parameter set for Kyber512 still has 112 bits of core-SVP hardness as
before, while with a weak version of the LWR assumption, it has 118 bits.
• Reduce ciphertext compression of Kyber512
By making the noise larger, the probability of decryption failures also increases. To compensate, wedrop one less bit in the “second” ciphertext element, which increases our ciphertext size from 736 bytes
to 768 bytes, for a 2−139 decryption error.
• More efficient uniform sampling of the public matrix A
Instead of sampling uniformly-random integers modulo 3329 by using rejection sampling on a 2-byte
integer, we now use rejection sampling on a 12-bit integer. While the new rejection rate per coefficientis higher (i.e. ≈ 20%), the total number of required bits and the running time of key generation are
noticeably smaller.

Changes to the specification and supporting documentation• Updated specification to match the round-3 parameters.
• Updated performance numbers.
• ARM Cortex-M4 numbers are now included in the performance-analysis section. The corresponding
software (to be dropped into pqm4 [59]) is included in the submission package.
• A major update of the security analysis, in particular a much more detailed analysis of the stateof-the art understanding of attacks beyond “core-SVP” hardness and a updated discussion of attacks
exploiting decryption failures.

2Contents
1 Written specification
1.1 Preliminaries and notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Specification of Kyber.CPAPKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Specification of Kyber.CCAKEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Kyber parameter sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Design rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4
4
7
10
10
12

2 Performance analysis
15
2.1 Implementation considerations and tradeoffs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Performance of Kyber on Intel Haswell CPUs . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Performance of Kyber on ARM Cortex-M4 CPUs . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Known Answer Test values

18

4 Expected security strength19
4.1 Security definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Rationale of our security estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3 Security Assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.3.1 Tight reduction from MLWE in the ROM . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.2 Non-tight reduction from MLWE in the QROM . . . . . . . . . . . . . . . . . . . . . . 20
4.4 Estimated security strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 Additional security properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5.1 Forward secrecy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5.2 Side-channel attacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.5.3 Multi-target attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5.4 Misuse resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Analysis with respect to known attacks
25
5.1 Attacks against the underlying MLWE problem . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1.1 Attacks against LWE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1.2 Primal attack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.1.3 Dual attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.1.4 Core-SVP hardness of Kyber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2 Beyond core-SVP hardness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.1 A tentative gate-count estimate accounting for recent progress . . . . . . . . . . . . . 27
5.3 Approximations, overheads, and foreseeable improvements . . . . . . . . . . . . . . . . . . . . 29
5.3.1 Algebraic attacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4 Attacks against symmetric primitives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.5 Attacks exploiting decryption failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6 Advantages and limitations
33
6.1 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 Comparison to SIDH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.3 Comparison to code-based KEMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4 Comparison to other lattice-based schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.4.1 Schemes that build a KEM directly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.4.2 LWE based schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.4.3 Ring-LWE based schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.4.4 NTRU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.4.5 Different Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.4.6 Deterministic Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7 Brief discussion of relevant results since Nov. 2017

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1

Written specification

Kyber is an IND-CCA2-secure key-encapsulation mechanism (KEM), which has first been described in [24].
The security of Kyber is based on the hardness of solving the learning-with-errors problem in module lattices(MLWE problem [67]). The construction of Kyber follows a two-stage approach: we first introduce an INDCPA-secure public-key encryption scheme encrypting messages of a fixed length of 32 bytes, which we call
Kyber.CPAPKE. We then use a slightly tweaked Fujisaki–Okamoto (FO) transform [47] to construct the
IND-CCA2-secure KEM. Whenever we want to emphasize that we are speaking about the IND-CCA2-secure
KEM, we will refer to it as Kyber.CCAKEM.
In Subsection 1.1 we give preliminaries and fix notation. In Subsection 1.2 we give a full specification of Kyber.CPAPKE. Subsection 1.3 gives details of the transform that we use in Kyber to obtain
Kyber.CCAKEM from Kyber.CPAPKE. Subsection 1.4 lists the parameters that we propose for different
security levels. Finally, Subsection 1.5 explains the design rationale behind Kyber.

1.1

Preliminaries and notation.

Bytes and byte arrays. Inputs and outputs to all API functions of Kyber are byte arrays. To simplify
notation, we denote by by B the set {0, . . . , 255}, i.e., the set of 8-bit unsigned integers (bytes). Consequently
we denote by B k the set of byte arrays of length k and by B ∗ the set of byte arrays of arbitrary length (orbyte streams). For two byte arrays a and b we denote by (akb) the concatenation of a and b. For a byte arraya we denote by a + k the byte array starting at byte k of a (with indexing starting at zero). For example,
let a by a byte array of length `, let b be another byte array and let c = (akb) be the concatenation of a andb; then b = a + `. When it is more convenient to work with an array of bits than an array of bytes we makethis conversion explicit via the BytesToBits function that takes as input an array of ` bytes and produces as
output an array of 8` bits. Bit βi at position i of the output bit array is obtained from byte bi/8 at position
i/8 of the input array by computing βi = (bi/8 /2(i mod 8) ) mod 2 .
Polynomial rings and vectors. We denote by R the ring Z[X]/(X n +1) and by Rq the ring Zq [X]/(X n +1),
00where n = 2n −1 such that X n + 1 is the 2n -th cyclotomic polynomial. Throughout this document, thevalues of n, n0 and q are fixed to n = 256, n0 = 9, and q = 3329. Regular font letters denote elements in R or
Rq (which includes elements in Z and Zq ) and bold lower-case letters represent vectors with coefficients in
R or Rq . By default, all vectors will be column vectors. Bold upper-case letters are matrices. For a vectorv (or matrix A), we denote by vT (or AT ) its transpose. For a vector v we write v[i] to denote it’s i-th
entry (with indexing starting at zero); for a matrix A we write A[i][j] to denote the entry in row i, columnj (again, with indexing starting at zero).
Modular reductions. For an even (resp. odd) positive integer α, we define r0 = r mod± α to be the0≤ r0 ≤ α−1unique element r0 in the range − α2 < r0 ≤ α2 (resp. − α−122 ) such that r = r mod α. For any+
00
positive integer α, we define r = r mod α to be the unique element r in the range 0 ≤ r0 < α such thatr0 = r mod α. When the exact representation is not important, we simply write r mod α.
Rounding. For an element x ∈ Q we denote by dxc rounding of x to the closest integer with ties being
rounded up.
Sizes of elements. For an element w ∈ Zq , we write kwk∞ to mean |w mod± q|. We now define the `∞and `2 norms for w = w0 + w1 X + . . . + wn−1 X n−1 ∈ R:p
kwk∞ = max kwi k∞ , kwk = kw0 k2∞ + . . . + kwn−1 k2∞ .
iSimilarly, for w = (w1 , . . . , wk ) ∈ Rk , we define
kwk∞ = max kwi k∞ , kwk =ip

kw1 k2 + . . . + kwk k2 .

Sets and Distributions. For a set S, we write s ← S to denote that s is chosen uniformly at random from
S. If S is a probability distribution, then this denotes that s is chosen according to the distribution S.
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Compression and Decompression. We now define a function Compressq (x, d) that takes an elementx ∈ Zq and outputs an integer in {0, . . . , 2d − 1}, where d < dlog2 (q)e. We furthermore define a function
Decompressq , such thatx0 = Decompressq Compressq (x, d), d(1)
is an element close to x – more specifically|x0 − x mod± q| ≤ Bq :=

l q k
.
2d+1

(2)

The functions satisfying these requirements are defined as:
Compressq (x, d) = d(2d /q) · xc mod+ 2d ,Decompressq (x, d) = d(q/2d ) · xc .
When Compressq or Decompressq is used with x ∈ Rq or x ∈ Rqk , the procedure is applied to eachcoefficient individually.
The main reason for defining the Compressq and Decompressq functions is to be able to discard some
low-order bits in the ciphertext which do not have much effect on the correctness probability of decryption– thus reducing the size of ciphertexts.
The Compressq and Decompressq are also used for a purpose other than compression – namely to performthe usual LWE error correction during encryption and decryption. More precisely, in line 20 of the encryption
procedure (Algorithm 5) the Decompressq function is used to create error tolerance gaps by sending themessage bit 0 to 0 and 1 to dq/2c. Later, on line 4 of the decryption procedure (Algorithm 6), the Compressq
function is used to decrypt to a 1 if v − sT u is closer to dq/2c than to 0, and decrypt to a 0 otherwise.
Symmetric primitives. The design of Kyber makes use of a pseudorandom function PRF : B 32 × B → B ∗and of an extendable output function XOF : B ∗ × B × B → B ∗ . Kyber also makes use of two hash functions
H : B ∗ → B 32 and G : B ∗ → B 32 × B 32 and of a key-derivation function KDF : B ∗ → B ∗ .
NTTs, multiplication, and bitreversed order. A very efficient way to perform multiplications in Rq is
via the so-called number-theoretic transform (NTT).
For our prime q = 3329 with q − 1 = 28 · 13, the base field Zq contains primitive 256-th roots of unity but
not primitive 512-th roots. Therefore, the defining polynomial X 256 + 1 of R factors into 128 polynomialsof degree 2 modulo q and the NTT of a polynomial f ∈ Rq is a vector of 128 polynomials of degree one.
Simple in-place implementations of the NTT without reordering outputs these polynomials in bit-reversedorder and we define the NTT in this way. Concretely, let ζ = 17 be the first primitive 256-th root of unity
modulo q, and {ζ, ζ 3 , ζ 5 , . . . , ζ 255 } the set of all the 256-th roots of unity. The polynomial X 256 + 1 cantherefore be written as
127
127
Y
Y
2
2i+1
256
X
+1=
(X − ζ
)=(X 2 − ζ 2br7 (i)+1 ),
i=0

i=0

where br7 (i) for i = 0, . . . , 127 is the bit reversal of the unsigned 7-bit integer i. This latter ordering of thefactors is useful for compatibility with the idiosyncrasies of AVX instructions. Then the NTT of f ∈ Rq isgiven by(f mod X 2 − ζ 2br7 (0)+1 , . . . , f mod X 2 − ζ 2br7 (127)+1 ).
(3)This vector of linear polynomials is then serialized to a vector in Z256
in the canonical way. Moreover, inqorder to not introduce additional data types and facilitate in-place implementations of the NTT we define
NTT : Rq → Rq to be the bijection that maps f ∈ Rq to the polynomial with the aforementioned coefficient
vector. Hence,
NTT(f ) = fˆ = fˆ0 + fˆ1 X + · · · + fˆ255 X 255

5

with
fˆ2i =

127X

f2j ζ (2br7 (i)+1)j ,(4)

f2j+1 ζ (2br7 (i)+1)j .

(5)

j=0

fˆ2i+1 =

127
X
j=0

We would like to stress that even though we write fˆ as a polynomial in Rq , it has no algebraic meaning
as such. The natural algebraic representation of NTT(f ) = fˆ is as 128 polynomials of degree 1 as in (3)using the definitions for fˆi from (4) and (5). That is,NTT(f ) = fˆ = (fˆ0 + fˆ1 X, fˆ2 + fˆ3 X, . . . , fˆ254 + fˆ255 X).
Using NTT and its inverse NTT−1 we can compute the product f · g of two elements f, g ∈ Rq very efficiently as NTT−1 (NTT(f ) ◦ NTT(g)) where NTT(f ) ◦ NTT(g) = fˆ◦ ĝ = ĥ denotes the basecase multiplicationconsisting of the 128 productsĥ2i + ĥ2i+1 X = (fˆ2i + fˆ2i+1 X)(ĝ2i + ĝ2i+1 X) mod X 2 − ζ 2br7 (i)+1of linear polynomials.
When we apply NTT or NTT−1 to a vector or matrix of elements of Rq , then this means that therespective operation is applied to each entry individually. When we apply ◦ to matrices or vectors it meansthat we perform a usual matrix multiplication, but that the individual products of entries are the above
basecase multiplications.
Throughout the document we will write NTT and NTT−1 whenever we refer to the concrete functions as
defined above and use normal-font NTT whenever we refer to the general technique.
Uniform sampling in Rq . Kyber uses a deterministic approach to sample elements in Rq that are
statistically close to a uniformly random distribution. For this sampling we use a function Parse : B ∗ → Rq ,which receives as input a byte stream B = b0 , b1 , b2 , . . . and computes the NTT-representation â = â0 +â1 X + · · · + ân−1 X n−1 ∈ Rq of a ∈ Rq . Parse is described in Algorithm 1 (note that this description assumes
that q = 3329).
Algorithm 1 Parse : B ∗ → Rqn
Input: Byte stream B = b0 , b1 , b2 · · · ∈ B ∗Output: NTT-representation â ∈ Rq of a ∈ Rq
i := 0j := 0while j < n do
d1 := bi + 256 · (bi+1 mod+ 16)
d2 := bbi+1 /16c + 16 · bi+2
if d1 < q thenâj := d1
j := j + 1
end ifif d2 < q and j < n thenâj := d2
j := j + 1
end ifi := i + 3
end while
return â0 + â1 X + · · · + ân−1 X n−1

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The intuition behind the function Parse is that if the input byte array is statistically close to a uniformly
random byte array, then the output polynomial is statistically close to a uniformly random element of Rq . It
represents a uniformly random polynomial in Rq because NTT is bijective and thus maps polynomials withuniformly random coefficients to polynomials with again uniformly random coefficients.
Sampling from a binomial distribution. Noise in Kyber is sampled from a centered binomial distribution Bη for η = 2 or η = 3. We define Bη as follows:
Sample (a1 , . . . , aη , b1 , . . . , bη ) ← {0, 1}2ηand output

ηX(ai − bi ).

i=1

When we write that a polynomial f ∈ Rq or a vector of such polynomials is sampled from Bη , we mean thateach coefficient is sampled from Bη .
For the specification of Kyber we need to define how a polynomial f ∈ Rq is sampled according to Bη
deterministically from 64η bytes of output of a pseudorandom function (we fix n = 256 in this description).
This is done by the function CBD (for “centered binomial distribution”) defined as described in Algorithm 2.
Algorithm 2 CBDη : B 64η → Rq
Input: Byte array B = (b0 , b1 , . . . , b64η−1 ) ∈ B 64η
Output: Polynomial f ∈ Rq
(β0 , . . . , β512η−1 ) := BytesToBits(B)for i from0 to 255 do
Pη−1
a := j=0 β2iη+jPη−1b := j=0 β2iη+η+j
fi := a − b
end forreturn f0 + f1 X + f2 X 2 + · · · + f255 X 255
Encoding and decoding. There are two data types that Kyber needs to serialize to byte arrays: byte
arrays and (vectors of) polynomials. Byte arrays are trivially serialized via the identity, so we need to define
how we serialize and deserialize polynomials. In Algorithm 3 we give a pseudocode description of the function
Decode` , which deserializes an array of 32` bytes into a polynomial f = f0 + f1 X + · · · + f255 X 255 (we again
fix n = 256 in this description) with each coefficient fi in {0, . . . , 2` − 1}. We define the function Encode` as
the inverse of Decode` . Whenever we apply Encode` to a vector of polynomials we encode each polynomial
individually and concatenate the output byte arrays.
Algorithm 3 Decode` : B 32` → Rq
Input: Byte array B ∈ B 32`
Output: Polynomial f ∈ Rq
(β0 , . . . , β256`−1 ) := BytesToBits(B)for i fromto 255 do
P0`−1
fi := j=0 βi`+j 2j
end forreturn f0 + f1 X + f2 X 2 + · · · + f255 X 255

1.2

Specification of Kyber.CPAPKE

Kyber.CPAPKE is similar to the LPR encryption scheme that was introduced (for Ring-LWE) by Lyubashevsky, Peikert, and Regev in the presentation of [72] at Eurocrypt 2010 [73]; the description is also in thefull version of the paper [74, Sec. 1.1]. The roots of this scheme go back to the first LWE-based encryption
7

scheme presented by Regev in [94, 95], with the main difference being that the underlying ring is not Zq andboth the secret and the error vectors have small coefficients. The idea of using a polynomial ring (instead ofZq ) goes back to the NTRU cryptosystem presented by Hoffstein, Pipher, and Silverman in [54], while thesymmetry between the secret and the error was already employed in very similar cryptographic schemes in[8, 71] with the security justification from [12].
The main difference from the LPR encryption scheme is to use Module-LWE instead of Ring-LWE. Also,
we adopt the approach taken by Alkım, Ducas, Pöppelmann and Schwabe in [10] for the generation of the
public matrix A. Furthermore, we shorten ciphertexts by rounding off the low bits as in learning-withrounding-based schemes [15, Eq. 2.1], which is a common technique for reducing ciphertext size also in
LWE-based schemes (c.f. [84, 90]).
Parameters. Kyber.CPAPKE is parameterized by integers n, k, q, η1 , η2 , du , and dv . As stated before,
throughout this document n is always 256 and q is always 3329.
Using the notation of Subsection 1.1 we give the definition of key generation, encryption, and decryptionof the Kyber.CPAPKE public-key encryption scheme in Algorithms 4, 5, and 6. A more high-level view ofthese algorithms is given in the comments.
Algorithm 4 Kyber.CPAPKE.KeyGen(): key generation
Output: Secret key sk ∈ B 12·k·n/8
Output: Public key pk ∈ B 12·k·n/8+32
1: d ← B 32
2: (ρ, σ) := G(d)3: N := 0
4: for i from 0 to k − 1 do
5:for j from 0 to k − 1 do
6:Â[i][j] := Parse(XOF(ρ, j, i))7:
end for
8: end for
9: for i from 0 to k − 1 do
10:s[i] := CBDη1 (PRF(σ, N ))11:
N := N + 1
12: end for
13: for i from 0 to k − 1 do
14:e[i] := CBDη1 (PRF(σ, N ))15:
N := N + 1
16: end for
17: ŝ := NTT(s)18: ê := NTT(e)19: t̂ := Â ◦ ŝ + ê+
20: pk := (Encode12 (t̂ mod q)kρ)+
21: sk := Encode12 (ŝ mod q)
22: return (pk , sk ). Generate matrix Â ∈ Rqk×k in NTT domain. Sample s ∈ Rqk from Bη1. Sample e ∈ Rqk from Bη1. pk := As + e. sk := s8

Algorithm 5 Kyber.CPAPKE.Enc(pk , m, r): encryption
Input: Public key pk ∈ B 12·k·n/8+32
Input: Message m ∈ B 32
Input: Random coins r ∈ B 32
Output: Ciphertext c ∈ B du ·k·n/8+dv ·n/8
1: N := 0
2: t̂ := Decode12 (pk )3: ρ := pk + 12 · k · n/8
4: for i from 0 to k − 1 do
5:for j from 0 to k − 1 do
6:ÂT [i][j] := Parse(XOF(ρ, i, j))7:
end for
8: end for
9: for i from 0 to k − 1 do
10:r[i] := CBDη1 (PRF(r, N ))11:
N := N + 1
12: end for
13: for i from 0 to k − 1 do
14:
e1 [i] := CBDη2 (PRF(r, N ))15:
N := N + 1
16: end for
17: e2 := CBDη2 (PRF(r, N ))18: r̂ := NTT(r)−1
19: u := NTT (ÂT ◦ r̂) + e1
−1 T
20: v := NTT (t̂ ◦ r̂) + e2 + Decompressq (Decode1 (m), 1)
21: c1 := Encodedu (Compressq (u, du ))22: c2 := Encodedv (Compressq (v, dv ))23: return c = (c1 kc2 ). Generate matrix Â ∈ Rqk×k in NTT domain. Sample r ∈ Rqk from Bη1. Sample e1 ∈ Rqk from Bη2. Sample e2 ∈ Rq from Bη2. u := AT r + e1
. v := t r + e2 + Decompressq (m, 1)T. c := (Compressq (u, du ), Compressq (v, dv ))Algorithm 6 Kyber.CPAPKE.Dec(sk , c): decryption
Input: Secret key sk ∈ B 12·k·n/8
Input: Ciphertext c ∈ B du ·k·n/8+dv ·n/8
Output: Message m ∈ B 32
1: u := Decompressq (Decodedu (c), du )2: v := Decompressq (Decodedv (c + du · k · n/8), dv )3: ŝ := Decode12 (sk )−1
4: m := Encode1 (Compressq (v − NTT (ŝT ◦ NTT(u)), 1))
5: return m

9. m := Compressq (v − sT u, 1))1.3

Specification of Kyber.CCAKEM

We construct the Kyber.CCAKEM IND-CCA2-secure KEM from the IND-CPA-secure public-key encryption
scheme described in the previous subsection via a slightly tweaked Fujisaki–Okamoto transform [47]. In
Algorithms 7, 8, and 9 we define key generation, encapsulation, and decapsulation of Kyber.CCAKEM.
Algorithm 7 Kyber.CCAKEM.KeyGen()
Output: Public key pk ∈ B 12·k·n/8+32
Output: Secret key sk ∈ B 24·k·n/8+96
1: z ← B 32
0
2: (pk , sk ) := Kyber.CPAPKE.KeyGen()0
3: sk := (sk kpk kH(pk )kz)
4: return (pk , sk )Algorithm 8 Kyber.CCAKEM.Enc(pk )Input: Public key pk ∈ B 12·k·n/8+32
Output: Ciphertext c ∈ B du ·k·n/8+dv ·n/8
Output: Shared key K ∈ B ∗1: m ← B 32
2: m ← H(m)3: (K̄, r) := G(mkH(pk ))4: c := Kyber.CPAPKE.Enc(pk , m, r)
5: K := KDF(K̄kH(c))
6: return (c, K). Do not send output of system RNG

Algorithm 9 Kyber.CCAKEM.Dec(c, sk )Input: Ciphertext c ∈ B du ·k·n/8+dv ·n/8
Input: Secret key sk ∈ B 24·k·n/8+96
Output: Shared key K ∈ B ∗1: pk := sk + 12 · k · n/8
2: h := sk + 24 · k · n/8 + 32 ∈ B 32
3: z := sk + 24 · k · n/8 + 64
4: m0 := Kyber.CPAPKE.Dec(sk , c)
5: (K̄ 0 , r 0 ) := G(m0 kh)
6: c0 := Kyber.CPAPKE.Enc(pk , m0 , r 0 )7: if c = c0 then8:
return K := KDF(K̄ 0 kH(c))
9: else10:
return K := KDF(zkH(c))
11: end if
12: return K

1.4

Kyber parameter sets

We define three parameter sets for Kyber, which we call Kyber512, Kyber768, and Kyber1024. The
parameters are listed in Table 1. Note that the table also lists the derived parameter δ, which is theprobability that decapsulation of a valid Kyber.CCAKEM ciphertext fails. The parameters were obtained
via the following approach:10

Table 1: Parameter sets for Kybernkq

256 2 3329
256 3 3329
256 4 3329

Kyber512
Kyber768
Kyber1024

η1 η2 (du , dv )

δ

3
2
2

−139

2
2
2

(10, 4)(10, 4)(11, 5)

2
2−164
2−174

• n is set to 256 because the goal is to encapsulate keys with 256 bits of entropy (i.e., use a plaintext
size of 256 bits in Kyber.CPAPKE.Enc). Smaller values of n would require to encode multiple key bits
into one polynomial coefficient, which requires lower noise levels and therefore lowers security. Larger
values of n would reduce the capability to easily scale security via parameter k.
• We choose q as a small prime satisfying n | (q − 1); this is required to enable the fast NTT-based
multiplication. There are two smaller primes for which this property holds, namely 257 and 769.
However, for those primes we would not be able to achieve negligible failure probability required for
CCA security, so we chose the next largest, i.e., q = 3329.
• k is selected to fix the lattice dimension as a multiple of n; changing k is the main mechanism in Kyber
to scale security (and as a consequence, efficiency) to different levels.
• The remaining parameters η1 , η2 , du and dv were chosen to balance between security (see Section 4),
ciphertext size, and failure probability. Note that all three parameter sets achieve a failure probabilityof < 2−128 with some margin. We discuss this in more detail in Subsections 1.5 and 5.5.
• The parameter η1 defines the noise of s and e in Algorithm 4 and of r in Algorithm 5. The parameterη2 defines the noise of e1 and e2 in Algorithm 5. We discuss the reason for setting η1 > η2 in the
Kyber512 parameter set in Section 1.5.
The failure probability δ is computed with the help of the Kyber.py Python script which is available
online at https://github.com/pq-crystals/security-estimates. For the theoretical background of that
script see [24, Theorem 1].
Instantiating PRF, XOF, H, G, and KDF. What is still missing to complete the specification of Kyberis the instantiation of the symmetric primitives. We instantiate all of those primitives with functions fromthe FIPS-202 standard [81] as follows:• We instantiate XOF with SHAKE-128;• we instantiate H with SHA3-256;• we instantiate G with SHA3-512;• we instantiate PRF(s, b) with SHAKE-256(s||b); and• we instantiate KDF with SHAKE-256.
“90s” variant of Kyber. In the 90s variant of Kyber• we instantiate XOF(ρ, i, j) with AES-256 in CTR mode, where ρ is used as the key and ikj is zeropadded to a 12-byte nonce. The counter of CTR mode is initialized to zero.
• we instantiate H with SHA-256;• we instantiate G with SHA-512;• we instantiate PRF(s, b) with AES-256 in CTR mode, where s is used as the key and b is zero-padded
to a 12-byte nonce. The counter of CTR mode is initialized to zero.
• we instantiate KDF with SHA-256.
11

1.5

Design rationale

The design of Kyber is based on the module version [67] of the Ring-LWE LPR encryption scheme [72]with bit-dropping [84, 90]. It is also enhanced by many of the improvements of preceding implementations oflattice-based encryption schemes such as NewHope [10]. In NewHope (and all other Ring-LWE schemes),
operations were of the form As+e where all the variables were polynomials in some ring. The main differencein Kyber is that A is now a matrix (with a small dimension like 3) over a constant-size polynomial ringand s, e are vectors over the same ring. We refer to this as a scheme over “module lattices.”
The use of Module-LWE. Previous proposals of LWE-based cryptosystems either used the very structured
Ring-LWE problem (as, for example, NewHope [10]) or standard LWE (as, for example, Frodo [23]). The
main advantage of structured LWE variants based on polynomial rings is efficiency in terms of both speed
and key and ciphertext sizes. The disadvantages are concerns that the additional structure might enable
more efficient attacks and that tradeoffs between efficiency and security can be scaled only rather coarsely.
The advantages of standard LWE is the lack of structure and easy scalability, but those come at the cost ofsignificantly decreasing efficiency. Module-LWE offers a trade-off between these two extremes. In the specificcase of the Module-LWE parameters used in Kyber, we obtain somewhat reduced structure compared to
Ring-LWE, much better scalability, and—when encrypting messages of a fixed size of 256 bits—performancevery similar to Ring-LWE-based schemes.
Active security. In [25], Bos, Costello, Naehrig, and Stebila used a passively secure KEM to migrate TLS totransitional post-quantum security (i.e., post-quantum confidentiality, but only pre-quantum authentication).
Subsequent work, like NewHope [10] or Frodo [23] followed up and proposed more efficient and moreconservative instantiations of the underlying passively secure KEM. One advantage of passively secure KEMs
is that they can accept a higher failure probability (which allows to either increase security by increasing
noise or decreasing public-key and ciphertext size). The other advantage is that they do not require a
CCA transform, and therefore come with faster decapsulation. Despite these advantages, Kyber is defined
as an IND-CCA2 secure KEM only. For many applications like public-key encryption (via a KEM-DEM
construction) or in authenticated key exchange active security is mandatory. However, also in use cases (likekey exchange in TLS) that do not strictly speaking require active security, using an actively secure KEM
has advantages. Most notably, it allows (intentional or accidental) caching of ephemeral keys. Furthermore,
the CCA transform of Kyber protects against certain bugs in implementations. Specifically, passively
secure schemes will not notice if the communication partner uses “wrong” noise, for example, all-zero noise.
Such a bug in the encapsulation of Kyber will immediately be caught by the re-encryption step during
decapsulation. As a conclusion, we believe that the overhead of providing CCA security is not large enoughto justify saving it and making the scheme less robust.
The role of the NTT. Multiplication in Rq based on the number-theoretic transform (NTT) has multiple
advantages: it is extremely fast, does not require additional memory (like, for example, Karatsuba or Toom
multiplication) and can be done in very little code space. Consequently, it has become common practice tochoose parameters of lattice-based crypto to support this very fast multiplication algorithm. Some schemes
go further and make the NTT part of the definition of the scheme. A prominent example is again NewHope,
which samples the public value a in NTT domain and also sends public keys and ciphertexts in NTT domain
to save 2 NTTs. NewHope was not the first scheme to do this; for earlier examples see [70, 90, 96].
In Kyber we also decided to make the NTT part of the definition of the scheme, but only in the samplingof A and the public key, not for the format of the ciphertext. A consequence of this decision is that the
NTT appears in the specification of Kyber.CPAPKE. Note that multiplications by A have to use the NTT,
simply because Â is sampled in NTT domain1 . Similarly, multiplications by the public-key t̂ have to use
the NTT, because the public key is transmitted in NTT domain. As a consequence, implementations will
also want to use the NTT for all other multiplications, so we make those invocations of NTT and NTT−1
also explicit in Alg. 4, Alg. 5, and Alg. 6. Note that also the secret key sk is stored in NTT domain.
We could have chosen to not make the NTT part of the definition of Kyber, which would have increased
simplicity of the description. The cost for this increased simplicity would have been k 2 additional NTT
operations in both key generation and encapsulation, which would result in a noticeable slowdown. We
could also have chosen to not encode the public key in NTT domain; however this would also have required
1 An alternative would be to apply NTT−1 to Â but that would counteract the whole point of sampling A in NTT domain.

12

additional NTTs. Finally, we could have chosen to also send the ciphertext in NTT domain; however, thiswould be incompatible with ciphertext compression via the Compressq function.
Against all authority. For the generation of the public uniformly random matrix A, we decided to adopt
the “against-all-authority” approach of NewHope. This means that the matrix is not a system parameterbut instead generated freshly as part of every public key. There are two advantages to this approach: First,
this avoids discussions about how exactly a uniformly random system parameter was generated. Second, it
protects against the all-for-the-price-of-one attack scenario of an attacker using a serious amount of computation to find a short basis of the lattice spanned by A once and then using this short basis to attack allusers. The cost for this decision is the expansion of the matrix A from a random seed during key generationand encapsulation; we discuss this cost more in Subsection 2.1.
Binomial noise. Theoretic treatments of LWE-based encryption typically consider LWE with Gaussian
noise, either rounded Gaussian [94] or discrete Gaussian [28]. As a result, many early implementations also
sampled noise from a discrete Gaussian distribution, which turns out to be either fairly inefficient (see, forexample, [25]) or vulnerable to timing attacks (see, for example, [29, 89, 45]). The performance of the best
known attacks against LWE-based encryption does not depend on the exact distribution of noise, but rather
on the standard deviation (and potentially the entropy). This motivates the use of noise distributions that
we can easily, efficiently, and securely sample from. One example is the centered binomial distribution usedin [10]. Another example is the use of “learning-with-rounding” (LWR), which adds deterministic uniform
noise by dropping bits as in Kyber’s Compressq function. In the design of Kyber we decided to use centeredbinomial noise and thus rely on LWE instead of LWR as the underlying problem.
Allowing decapsulation failures. Another interesting design decision is whether to allow decapsulation
failures (i.e., decryption failures in Kyber.CPAPKE) or choose parameters that not only have a negligible,
but a zero chance of failure. The advantages of zero failure probability are obvious: CCA transforms andsecurity proofs become easier and we can avoid the whole discussion of attacks exploiting decapsulation
failures in Subsection 5.5. The disadvantage of designing LWE-based encryption with zero failures is thatit means either decreasing security against attacks targeting the underlying lattice problem (by significantly
decreasing the noise) or decreasing performance (by compensating for the loss in security via an increase of
the lattice dimension). The decision to allow failure probabilities in all parameter sets of Kyber reflects theintuition that• decapsulation failures are a problem if they appear with non-negligible probability; but• attacks attempting to exploit failures that occur with extremely low probability as in Kyber are amuch smaller threat than, for example, improvements to hybrid attacks [56] targeting schemes withvery low noise.
Different noise values η1 and η2 . Notice that in Algorithm 5, there is additional implicit noise created
via Compressq . This has the effect of adding some (deterministic) noise to the ciphertext, which can beinterpreted as increasing the noise of the error polynomials e1 and e2 . If the decryption error probability is
low enough, then it then makes sense to also increase the noise of the other secret terms (i.e. s, e, r) to be at asimilar level as e1 plus the deterministic noise. We utilize this idea (exclusively) for the Kyber512 parameter
set. Relying on the rounding noise from Compressq to add error is akin to the LWR assumption. But unlikein (Ring/Module)-LWR schemes, where the security completely relies on the noise that’s deterministicallygenerated by rounding, our dependence on the deterministic noise is much smaller. First, it only adds 6
bits of Core-SVP hardness, and second, we are adding noise and rounding together, which presumably hasless algebraic structure than just rounding. In short, without the LWR assumption, our parameter set for
Kyber512 has 112 bits of core-SVP hardness – more specifically, the public keys are protected with 118
bits, and the ciphertexts with 112; with a weak version of the LWR assumption, it has 118-bit security
everywhere.
Additional Hashes. In the CCA transform we hash the (hash of the) public key pk into the pre-key K̄
and into the random coins r (see line 3 of Alg. 8), and we hash the (hash of the) ciphertext into the final
key K. These hashes are not necessary for the security reduction (see Section 4), but they add robustness.
Specifically, the final shared key output by Kyber.CCAKEM depends on the full view of exchanged messages13

(public key and ciphertext), which means that the KEM is contributory and safe to use in authenticated key
exchanges without additional hashing of context. Hashing pk also into the random coins r adds protection
against a certain class of multi-target attacks that attempt to make use of protocol failures. This is discussed
in more detail in Subsection 5.5.
Choice of symmetric primitives. In the design of Kyber we need an extendable output function (XOF),
two hash functions, a pseudorandom function, and a KDF. We decided to rely on only one underlying
primitive for all those functions. This helps to reduce code size in embedded platforms and (for a conservative
choice) reduces concerns that Kyber could be attacked by exploiting weaknesses in one out of several
symmetric primitives. There are only relatively few extendable output functions described in the literature.
The best known ones, which also coined the term XOF, are the SHAKE functions based on Keccak [20] andstandardized in FIPS-202 [81]. This standard conveniently also describes hash functions with the output
lengths we need; furthermore, SHAKE is designed to also work as a PRF. These properties of the FIPS-202
function family made the choice easy, but there are still two decisions that may need explanation:• We could have chosen to instantiate all symmetric primitives with only one function (e.g., SHAKE256) from the FIPS-202 standard. The choice of SHAKE-128 as instantiation of the XOF is actuallyimportant for performance; also we do not need any of the traditional security properties of hashfunctions from SHAKE-128, but rather that the output “looks uniformly random”. In an earlier versionof Kyber we instantiated H, G, and PRF all with SHAKE-256. We decided to change this to different
functions from the FIPS-202 family to avoid any domain-separation discussion. Note that this decision
increases code-size at most marginally: all 4 functions can be obtained by a call to a “Keccak” function
with appropriate arguments (see, for example, [19]).
• We could have decided to use KMAC from NIST Special Publication 800-185 to instantiate the PRF.
We decided against this, because it would increase the numbers of Keccak permutations required in
the generation of the noise polynomials and thus noticeably and unnecessarily decrease performance.
As a modification in round-2, we decided to derive the final key using SHAKE-256 instead of SHA3-256.
This is an advantage for protocols that need keys of more than 256 bits. Instead of first requesting a 256-bit
key from Kyber and then expanding it, they can pass an additional key-length parameter to Kyber andobtain a key of the desired length. This feature is not supported by the NIST API, so in our implementations
we set the keylength to a fixed length of 32 bytes in api.h.
Choice of symmetric primitives in the “90s” variant The 90s variant of Kyber uses symmetricprimitives that are standardized by NIST and accelerated in hardware on a large variety of platforms. These
two criteria narrow the choice to AES and SHA-256, which are, for example, implemented in hardware on
recent Intel, AMD, and ARM processors. A natural choice for the hash function G with 512-bit output isSHA-512 from the same SHA-2 family of hash functions as SHA-256.
Supporting non-incremental hash APIs. In line 3 of Alg. 8 we feed H(pk ) (instead of pk ) into G andin line 5 we feed H(c) (instead of c) into H. Using H(pk ) in the call to G enables a small speedup for
decapsulation as described in Subsection 2.1. However, there is another reason why we first hash pk and c,
namely that it simplifies implementing Kyber with a non-incremental hash API. If Kyber is implemented
in an environment which already offers a library for hashing, but only offers calls of the form h = H(m), thenproducing a hash of the form h = H(m1 km2 ) would first require copying m1 and m2 into one consecutive
area of memory. This would require unnecessary copies and, more importantly, additional stack space. Such
non-incremental hash APIs are not uncommon: one example is the API of NaCl [18].
Return value for decapsulation failure. Traditionally the FO transform returns ⊥ (i.e., a special failure
symbol) when decapsulation fails. We use a variant that instead sets the resulting shared key to a pseudorandom value computed as the hash of a secret z and the ciphertext c. This variant of the FO transformwas proven secure in [55]. In practice it has the advantage that implementations of Kyber’s decapsulation
are safe to use even if higher level protocols fail to check the return value. As a consequence of this implicit
rejection approach, our implementations of decapsulation always return 0.

14

2

Performance analysis

In this section we consider implementational aspects of Kyber and report performance results of two implementations: the ANSI C reference implementation requested by NIST and an implementation optimized using
AVX2 vector instructions. included in the submission package under Additional_Implementations/avx2/.
We remark that the optimized implementation in ANSI C in subdirectory Optimized_Implementation/, asrequested by the Call for Proposals, is a copy of the reference implementation.
The big picture of Kyber performance. Thanks to the extremely efficient NTT-based multiplication andsampling of A in NTT domain, the performance of Kyber is largely determined by the performance of thesymmetric primitives. This is illustrated, for example, by the fact that the AVX2-optimized implementationof the 90s variant of Kyber is almost twice as fast as Kyber with symmetric primitives based on Keccak. Thisdifference is going to be even larger on systems with hardware-accelerated SHA-256. It is also illustrated
by the fact that for optimized implementations, decapsulation is faster than encapsulation, even though
the former contains a re-encapsulation necessary for the FO transform. The main reason for this is that
decapsulation does not need to compute H(pk ), which more than compensates for the extra NTT-dominatedre-encapsulation.
The fact that (sequential) hashing is very slow in relation to polynomial arithmetic on recent Intelprocessors also means that the additional hashes discussed in Section 1.5 have a very significant impact on
performance: they account for almost half of the encapsulation cycles of our AVX2-optimized implementation
on Intel processors.2 As also mentioned in Section 1.5, hashing the public key and the ciphertext is not
required for CCA security, but is instead done to make the function more robust and directly usable inapplications that require the shared key to depend on the entire transcript. Our rationale is that because
the basic operations comprising Kyber are extremely fast, we can afford to pay a time penalty and makethe default version of Kyber as robust and misuse-resilient as possible. Nevertheless, in light of the factthat this hashing has a profound effect on the absolute performance of the scheme, it is an interesting future
research direction to see whether there are applications that indeed require hashing the entire ciphertext
and/or public key. If it’s enough to only a hash a small part of them, then the running time of Kyber can
be cut in half (on many processors) without sacrificing any robustness.

2.1

Implementation considerations and tradeoffs

Implementing the NTT. Many different tradeoffs are possible when implementing the number-theoretictransform. The most important ones are between code size (which becomes mainly relevant on embedded
processors) and speed. The two implementations of Kyber included in the submission package have a
dedicated forward NTT (from normal to bitreversed order) and inverse NTT (from bitreversed to normal
order). Also, both implementations use precomputed tables of powers of ζ. What is particularly interesting
about using the NTT on embedded platforms is that the multiplication of two elements of Rq can becomputed without any additional temporary storage. What is particularly interesting about using the NTT
on large processors is that it is extremely efficiently vectorizable. Since 2013, the most efficient approach tocompute the NTT on 64-bit Intel processors was to represent coefficients as double-precision floating-pointvalues [51, 10]. In our AVX2-optimized implementation of Kyber, we show that carefully optimizing the
NTT using AVX2 integer instructions results in much better performance. Specifically, on Intel HaswellCPUs one forward NTT in Kyber takes only about 320 cycles; an inverse NTT takes only about 290 cycles.
Keccak. The second speed-critical component inside Kyber are the symmetric primitives, i.e., SHA3-256,
SHA3-512, SHAKE-128, and SHAKE-256, all based on the Keccak permutation. SHA3 has the reputation to
not be the fastest hash function in software (see, for example, [66]). To some extent this is compensated by thefact that most calls to Keccak are parallel and thus very efficiently vectorizable. Our AVX2 implementation
makes use of this fact. Also, ARM recently announced that future ARMv8 processors will have hardware
support for SHA3 [50], so there is a good chance that at least on some architectures, software performanceof SHA3 will not be an issue in the future.
2 The reason that the use of Keccak as an XOF to expand the public matrix A from the seed ρ, and as a PRF to generatesecrets and noise, has a smaller impact is because these Keccak invocations are efficiently parallelizable.

15

AES and SHA-2. To illustrate what performance Kyber can achieve with hardware-accelerated symmetricprimitives we include the 90s variant using AES, SHA256 and SHA512 instead of symmetric primitives basedon Keccak. This variant is interesting only if at least AES is accelerated in hardware, because constant-time
software implementations (required for the use of AES as a PRF) are not faster than parallel Keccak.
Hardware-RNGs for key generation. During key generation, the generation of s and e is performed
using SHAKE-256 (and AES in the 90s variant). However, this is not required. The choice of RNG during
key generation is a local decision that any user and platform can make independently. In particular on
platforms with fast hardware AES one can adapt the AES-based PRF from the 90s variant also for theotherwise Keccak-based Kyber. We considered using this in our AVX2 implementation, but using thisoptimization means that testvectors would not match between our two implementations. This is not an issue
in actual deployments, where randombytes is not deterministic.
Caching of ephemeral keys. Applications that are even more conscious of key-generation time can decideto cache ephemeral keys for some time. This is enabled by the fact that Kyber is IND-CCA2 secure.
Tradeoffs between secret-key size and speed. It is possible to use different tradeoffs between secret-key
size and decapsulation speed. If secret-key size is critical, it is of course possible to not store H(pk) andalso to not store the public key as part of the secret key but instead recompute it during decapsulation.
Furthermore, not keeping the secret key in NTT domain makes it possible to compress each coefficient toonly 3 bits, which means that a Kyber768 secret key can be compressed into just 288 bytes. This has the
additional advantage that it makes key recovery via cold-boot attacks [52] somewhat harder [3]. Finally,as all randomness in key generation is generated from two 32-byte seeds, it is also possible to only store
these seeds and re-run key generation during decapsulation. When opting for such 32-byte secret keys, there-encapsulation step of decapsulation can save the expansion of the matrix A, is it is already expanded (intransposed form) in key generation.
In the other direction, if secret-key size does not matter very much and decapsulation speed is critical,
one might decide to store the expanded matrix A as part of the secret key and avoid recomputation fromthe seed ρ during the re-encapsulation part of decapsulation.
Both implementations included in the submission package use the secret-key format described in Algorithm 7, i.e., with polynomials in NTT domain, including the public key and H(pk), but not including thematrix A.
Local storage format of static public keys. A user who is frequently encapsulating messages to thesame public key can speed up encapsulation by locally storing an expanded public key containing the matrix
A and H(pk ). This saves the cost of expanding the matrix A from the seed ρ and the cost of hashing pk inevery encapsulation.

2.2

Performance of Kyber on Intel Haswell CPUs

Table 2 reports Intel Haswell performance results of the reference implementation and an AVX2-optimized
implementation of Kyber and the 90s variant of Kyber together with the sizes of keys and ciphertexts. All
benchmarks were obtained on one core of an Intel Core i7-4770K (Haswell) processor clocked at 3492 MHz (asreported by /proc/cpuinfo) with TurboBoost and hyperthreading disabled. The benchmarking machine has
32 GB of RAM and is running Debian GNU/Linux with Linux kernel version 4.19.0. Both implementations
were compiled with gcc version 8.3.0 and the compiler flags as indicated in the Makefiles included in thesubmission package. All cycle counts reported are the median of the cycle counts of 10 000 executions of therespective function. The implementations are not optimized for memory usage, but generally Kyber has
only very modest memory requirements. This means that in particular our implementations do not need toallocate any memory on the heap.

2.3

Performance of Kyber on ARM Cortex-M4 CPUs

Table 3 reports cycle counts and RAM usage of a C implementation (“clean”) and an optimized implementation (“m4”) of Kyber on an ARM Cortex-M4. The clean C implementation is a slightly modified versionof our C reference implementation from the PQClean project [58]. The optimized implementation is based16

Table 2: Key and ciphertext sizes and cycle counts for all paramter sets of Kyber. Cycle counts wereobtained on one core of an Intel Core i7-4770K (Haswell); “ref” refers to the C reference implementation,“AVX2” to the implementation using AVX2 vector instructions; sk stands for secret key, pk for public key,and ct for ciphertext. In parenthesis are approximate values when including key generation in decapsulation
to avoid having to store expanded secret keys. In this scenario, we only store the initial seed d in line 1 ofAlgorithm 4. The approximate cycle counts for this scenario are computed as the sum of cycle counts forstandard decapsulation and key generation minus the number of cycles require to generate the matrix A fromthe public seed ρ. Note that this is a very conservative estimate; actual implementations of the approach canalso save, for example, sampling the 32 bytes of randomness. See also the discussion on “tradeoffs between
secret-key size and speed” in Subsection 2.1.

Kyber512
Sizes (in Bytes)sk:1632 (or 32)
pk:
800
ct:
768
Kyber512-90s

Haswell Cycles (ref )gen:
122684enc:
154524
dec: 187960 (or ≈ 288912)

Haswell Cycles (AVX2)
gen:
33856
enc:
45200
dec: 34572 (or ≈ 59088)

Sizes (in Bytes)sk:1632 (or 32)
pk:
800
ct:
768

Haswell Cycles (ref )gen:
213156enc:
249084
dec: 277612 (or ≈ 405268)

Haswell Cycles (AVX2)
gen:
21880
enc:
28592
dec: 20980 (or ≈ 38752)

Sizes (in Bytes)
sk:
2400 (or 32)
pk:
1184
ct:
1088
Kyber768-90s

Haswell Cycles (ref )gen:199408enc:
235260dec: 274900 (or ≈ 425492)

Haswell Cycles (AVX2)
gen:
52732
enc:
67624
dec: 53156 (or ≈ 82220)

Sizes (in Bytes)
sk:
2400 (or 32)
pk:
1184
ct:
1088

Haswell Cycles (ref )gen:389760enc:
432764
dec: 473984 (or ≈ 671864)

Haswell Cycles (AVX2)
gen:
30460
enc:
40140
dec: 30108 (or ≈ 51512)

Sizes (in Bytes)sk:3168 (or 32)
pk:
1568
ct:
1568
Kyber1024-90s

Haswell Cycles (ref )gen:307148enc:
346648dec: 396584 (or ≈ 617848)

Haswell Cycles (AVX2)
gen:
73544
enc:
97324
dec: 79128 (or ≈ 115332)

Sizes (in Bytes)sk:3168 (or 32)
pk:
1568
ct:
1568

Haswell Cycles (ref )gen:636380enc:
672644
dec: 724144 (or ≈ 1009448)

Haswell Cycles (AVX2)
gen:
43212
enc:
56556
dec: 44328 (or ≈ 71180)

Kyber768

Kyber1024

17

Table 3: Cycle counts and RAM usage (in Bytes) for Kyber on a Cortex-M4 (STM32F407Discovery). The“clean” implementation is a slightly modified C reference implementation from the PQClean project, the “m4”implementation is optimized for the Cortex-M4. All benchmarks are obtained using the pqm4 framework.

Kyber512
gen:
enc:
dec:

M4 Cycles (clean)
655595865256961648

M4 RAM (clean)
6020
8668
9444

M4 Cycles (m4)463068561518519237

M4 RAM (m4)
2844
2484
2508

M4 Cycles (clean)
108789713737441491214

M4 RAM (clean)
10052
13212
14308

M4 Cycles (m4)756224915676853001

M4 RAM (m4)
3292
2980
3004

M4 Cycles (clean)
169631420575222199958

M4 RAM (clean)
15180
18844
20420

M4 Cycles (m4)12133031407769
1326409

M4 RAM (m4)
3804
3492
3516

Kyber768
gen:
enc:
dec:
Kyber1024
gen:
enc:
dec:

on [26] with the improvements from [9] and the round-3 tweaks applied. All benchmarks are obtained using
the pqm4 framework [59] on an STM32F407 Discovery board. The two implementations are also includedin the submission package. They do not include standalone main functionality, but instead are assumed to
be used within pqm4. We do not report benchmarks of the 90s version of Kyber from the Cortex-M4 here,
because pqm4 does not yet include a constant-time implementation of AES.

3

Known Answer Test values

All KAT values are included in subdirectories of the directory KAT of the submission package. Specifically, theKAT values of Kyber512 are in the subdirectory KAT/kyber512; the KAT values of the Kyber512-90s arein the subdirectory KAT/kyber512-90s; the KAT values of Kyber768 are in the subdirectory KAT/kyber768;
the KAT values of the Kyber768-90s are in the subdirectory KAT/kyber768-90s; the KAT values of Kyber1024 are in the subdirectory KAT/kyber1024; and the KAT values of the Kyber1024-90s are in thesubdirectory KAT/kyber1024-90s. Each of those directories contains the KAT values as generated by the
PQCgenKAT_kem program provided by NIST. Specifically, those files are:
• KAT/kyber512/PQCkemKAT_1632.req,• KAT/kyber512/PQCkemKAT_1632.rsp,• KAT/kyber512-90s/PQCkemKAT_1632.req,• KAT/kyber512-90s/PQCkemKAT_1632.rsp,• KAT/kyber768/PQCkemKAT_2400.req,• KAT/kyber768/PQCkemKAT_2400.rsp,• KAT/kyber768-90s/PQCkemKAT_2400.req,18• KAT/kyber768-90s/PQCkemKAT_2400.rsp,• KAT/kyber1024/PQCkemKAT_3168.req,• KAT/kyber1024/PQCkemKAT_3168.rsp,• KAT/kyber1024-90s/PQCkemKAT_3168.req, and• KAT/kyber1024-90s/PQCkemKAT_3168.rsp,

4

Expected security strength

4.1

Security definition

Kyber.CCAKEM (or short, Kyber) is an IND-CCA2-secure key encapsulation mechanism, i.e., it fulfills
the security definition stated in Section 4.A.2 of the Call for Proposals.

4.2

Rationale of our security estimates

Our estimates of the security strength for the three different parameter sets of Kyber—and consequently
the classification into security levels as defined in Section 4.A.5 of the Call for Proposals—are based on thecost estimates of attacks against the underlying module-learning-with-errors (MLWE) problem as detailedin Subsection 5.1.
To justify this rationale, we will in the following give two reductions from MLWE: a tight reduction in
the random-oracle model (ROM) in Theorem 2 and a non-tight reduction in the quantum-random-oracle
model (QROM) in Theorem 3. With those reductions at hand, there remain two avenues of attack thatwould break Kyber without solving the underlying MLWE problem, namely
1. breaking one of the assumptions of the reductions, in particular attacking the symmetric primitives
used in Kyber; or
2. exploiting the non-tightness of the QROM reduction.
We briefly discuss 1.) in Subsection 5.4. The discussion of 2.) requires considering two separate issues,
namely• a (quadratic) non-tightness in the decryption-failure probability of Kyber.CPAPKE, and• a (quadratic) non-tightness between the advantage of the MLWE attacker and the quantum attacker
against Kyber.
In Subsection 5.5 we discuss quantum attacks exploiting decryption failures and in the presentation of the
non-tight QROM reduction we explain why the non-tightness between quantum attacks against MLWE
and quantum attacks against Kyber is unlikely to matter in practice. More specifically, we show how toeliminate this non-tightness if we allow the reasonable, but non-standard, assumption that Kyber.CPAPKE
ciphertexts are pseudorandom, even if all randomness is generated pseudorandomly from a hash of theencrypted message.

4.3

Security Assumption

The hard problem underlying the security of our schemes is Module-LWE [27, 67]. It consists in distinguishing
uniform samples (ai , bi ) ← Rqk × Rq from samples (ai , bi ) ∈ Rqk × Rq where ai ← Rqk is uniform andbi = aTi s + ei with s ← Bηk common to all samples and ei ← Bη fresh for every sample. More precisely, for
an algorithm A, we define Advmlwe
m,k,η (A) =Pr b0 = 1 :A ← Rqm×k ; (s, e) ← βηk × βηm ;b = As + e; b0 ← A(A, b)− Pr b0 = 1 :19

A ← Rqm×k ; b ← Rqm ; b0 ← A(A, b).

4.3.1

Tight reduction from MLWE in the ROM

We first note that Kyber.CPAPKE is tightly IND-CPA secure under the Module-LWE hardness assumption.
Theorem 1. Suppose XOF and G are random oracles. For any adversary A, there exist adversaries B andmlwe
C with roughly the same running time as that of A such that Advcpa
Kyber.CPAPKE (A) ≤ 2 · Advk+1,k,η (B) +Advprf
PRF (C).
The proof of this theorem is easily obtained by noting that, under the MLWE assumption, public-key
and ciphertext are pseudo-random.
Kyber.CCAKEM is obtained via a slightly tweaked Fujisaki-Okamoto transform [55, 47] applied to
Kyber.CPAPKE. The following concrete security statement proves Kyber.CCAKEM’s IND-CCA2-security
when the hash functions G and H are modeled as random oracles. It is obtained by combining the genericbounds from [55] with Theorem 1 (and optimizing the constants appearing in the bound).
Theorem 2. Suppose XOF, H, and G are random oracles. For any classical adversary A that makes at mostqRO many queries to random oracles XOF, H and G, there exist adversaries B and C of roughly the samerunning time as that of A such thatprf
mlwe
AdvccaKyber.CCAKEM (A) ≤ 2Advk+1,k,η (B) + AdvPRF (C) + 4qRO δ.

Note that the security bound is tight. The negligible additive term 4qRO δ stems from Kyber.CPAPKE’s
decryption-failure probability δ.
4.3.2

Non-tight reduction from MLWE in the QROM

As for security in the quantum random oracle model (QROM), [55, 97] proved that Kyber.CCAKEM is INDCCA2 secure in the QROM, provided that Kyber.CPAPKE is IND-CPA secure. A slightly tighter reduction
can be obtained by requiring the base scheme Kyber.CPAPKE to be pseudo-random. Pseudo-randomness[97] requires that, for every message m, a (randomly generated) ciphertext (c1 , c2 ) ← Kyber.CPAPKE.Enc(pk , m)is computationally indistinguishable from a random ciphertext of the form (Compressq (u, du ), Compressq (v, dv )),for uniform (u, v). (We also require the property of “statistical disjointness” [97] which is trivially fulfilled for
Kyber.CPAPKE.) The proof of Kyber.CPAPKE’s IND-CPA security indeed shows that Kyber.CPAPKE istightly pseudo-random under the Module-LWE hardness assumption.
Theorem 3. Suppose XOF, H, and G are random oracles. For any quantum adversary A that makes atmost qRO many queries to quantum random oracles XOF, H and G, there exist quantum adversaries B and
C of roughly the same running time as that of A such thatqprf
2
Advmlwe
Advcca
(A)
≤
4q
·
RO
k+1,k,η (B) + AdvPRF (C) + 8qRO δ.
Kyber.CCAKEM
Unfortunately, the above security bound is non-tight and therefore can only serve as an asymptotic
indication of Kyber.CCAKEM’s CCA-security in the quantum random oracle model.
Tight reduction under non-standard assumption. We can use [55, 97] to derive a tight security
bound in the QROM from a non-standard security assumption, namely that a deterministic version of
Kyber.CPAPKE, called DKyber.CPAPKE, is pseudo-random in the QROM. Deterministic Kyber.CPAPKE
is defined as Kyber.CPAPKE, but the random coins r used in encryption are derived deterministicallyfrom the message m as r := G(m). Pseudo-randomness for deterministic encryption states that an encryption (c1 , c2 ) of a randomly chosen message is computationally indistinguishable from a random ciphertext (Compressq (u, du ), Compressq (v, dv )), for uniform (u, v). In the classical ROM, pseudo-randomness of
DKyber.CPAPKE is tightly
q equivalent to MLWE but in the QROM the reduction is non-tight (and is thereason for the term qRO ·Advmlwe
k+1,k,η (B) in Theorem 3). Concretely, we obtain the following bound:

pr
prf
mlwe
2
AdvccaKyber.CCAKEM (A) ≤ 2Advk+1,k,η (B) + AdvDKyber.CPAPKE (C) + AdvPRF (D) + 8qRO δ.

We remark that we are not aware of any quantum attack on deterministic Kyber.CPAPKE that performs
better than breaking the MLWE problem.
20

Table 4: Classical and quantum hardness of the different proposed parameter sets of Kyber together withthe claimed security level as defined in Section 4.A.5 of the Call for Proposals.
Kyber512 Kyber768 Kyber1024
NIST Security level

1

3

5

Core-SVP methodology, Primal attack only (Sec. 5.1.4)
lattice attack dim. d
999
1419
1885
BKZ-blocksize β
406
626
878
core-SVP classical hardness
118
183
256
core-SVP quantum hardness
107
166
232
Refined estimate for classical attacks using [40, 36, 5] (Sec. 5.2)
lattice attack dim. d
1025
1467
1918
BKZ-blocksize β
413
637
894
sieving dimension β 0 = β − d4f
375
586
829
log2 (gates)
151.5
215.1
287.3
log2 (memory in bits)
93.8
138.5
189.7

4.4

Estimated security strength

Table 4 lists the security levels according to the definition in Section 4.A.5 of the Call for Proposals for thedifferent parameter sets of Kyber. Our claims are based on the cost estimates of the best known attacks
against the MLWE problem underlying Kyber as detailed in Subsection 5.1. Specifically we list the classicaland the quantum core-SVP hardness and use those to derive security levels.
All parameter sets of Kyber have some probability of decryption failure. These failures are a security
concern (see Section 5.5) and so the probabilities with which they occur need to be small. But because in
the classical random oracle model, the decryption failure probability is information-theoretic, we do not see
a need for it to decrease with the security parameter. In particular, decryption failure for our level 3 and 5
parameter sets is less than 2−160 , which means that if 280 instances of Kyber were run every second fromnow until our sun becomes a white dwarf, the odds still heavily favor there never being a decryption failure.
We therefore exclude these attacks from our claims regarding the NIST security estimates.
The impact of the deterministic noise caused by Compressq on Kyber512. Each coefficient of e1(and e2 ) in Algorithm 5 is distributed as a binomial distribution with parameter η2 = 2, which has varianceη2 /2 = 1. The parameter du = 10 implies that the Compressq function maps elements modulo q to a set ofsize 210 . When this set is mapped back to Zq via the Decompressq function, the difference between every twoelements in the latter set is either 3 or 4. This implies that the error created by the Compressq function foreach coefficient is either uniform over {−1, 0, 1}, {−1, 0, 1, 2}, or {−2, −1, 0, 1}. It therefore has variance atleast as large as the uniform distribution over the set {−1, 0, 1}, which is 2/3. This makes the total varianceof each coefficient of e1 plus the deterministic error at least 1 + 2/3 = 5/3. The other secret and noise terms
have binomial distributions with parameter η1 = 3 for a variance of η1 /2 = 3/2 < 5/3. When accounting forthe errors added by Compressq , we therefore calculate the hardness of Kyber512 assuming that it is as hard
as Module-LWE where the variance of each secret/error coefficient is 3/2.
We now make the above discussion more precise. We define the Module-LWER problem as the problem
of distinguishing samples (ai , Compressq (bi , d)) for random (ai , bi ) ← Rqk × Rq from samples where ai ← Rqk
and bi = aTi s + ei with s ← Bηk1 common to all samples and ei ← Bη2 fresh for every sample.3 Using the3 This problem generalizes both Module-LWE and Module-LWR. When e = 0, this problem is Module-LWR, whereas wheni21

same terminology as in the beginning of Section 4.3, we define the advantage of an algorithm A breaking
Module-LWER as Advmlwer
m,k,η1 ,η2 ,d (A) =A ← Rqm×k ; (s, e) ← βηk1 × βηm2 ;A ← Rqm×k ; b ← Rqm ;0.
Pr b0 = 1 :
−
Pr
b
=
1
:
b = As + e; b0 ← A(A, Compressq (b, d))b0 ← A(A, Compressq (b, d))
In addition to the parameters in the Module-LWE problem, the Module-LWER problem has the additionalparameter d which defines the range (i.e., {0, . . . , 2d − 1}) of the Compressq function. Also, the η parameteris split into η1 , which defines the distribution of s, and η2 , which defines the distribution of e. The intuition
for having η1 6= η2 is that the Compressq function creates another noise vector e0 which gets added to e (wesdiscuss this in more detail below), and so the short target vector crucial for the lattice attack is. If
e + e0we would like to have this vector be balanced, we should set η1 > η2 .
We can then obtain the analogues of Theorems 1 and 2 with the term 2Advmlwe
k+1,k,η (B) being replaced bymlwer
0
Advmlwek,k,η1 (B) + Advk+1,k,η1 ,η2 ,du (B ),(6)where the Module-LWE assumption is used in the key generation (as before) and the Module-LWER assumption is used in the encryption. We now assess the security of the Module-LWER problem. First,
when η1 ≥ η2 , any algorithm B for Module-LWE implies an algorithm A for Module-LWER such thatmlwe
Advmlwerm,k,η1 ,η2 ,d (A) ≤ Advm,k,η2 (B). When η1 > η2 , as in the Kyber512 parameter set, the preceding inequality ignores the fact that having a larger s in the Module-LWER instance makes the short target lattice
vector longer. And when d < log q, the inequality similarly ignores the fact that the adversary A only getsto see (A, t = Compressq (As + e, d)) rather than (A, As + e). If we use this rough inequality, then for our
Kyber512 parameters we would have the Core-SVP classical hardness of the Module-LWER problem be112. So the logarithm of the required number of gates stated in the Kyber512 column of Table 4 would
decrease by 6 to approximately 145.
We argue, however, that based on the current state of knowledge of lattice attacks, the Module-LWER
problem with Kyber512 parameters is somewhat harder. In particular, we contend that it is at least as
hard as the Module-LWE problem where s, e ← η1 . This implies that (6)≤ 2Advmlwe
k+1,k,η1 (B). If we thenanalyze the hardness of Module-LWE with these parameters, we get the numbers in Table 4.
The quantity that determines the efficacy of the attacks  (in terms of the Core-SVP measurement) againststhe Module-LWE problem is the norm of the vector
where As + e = b over the ring Rq . In order toecreate such an equation out of a Module-LWER instance, we map t = Compressq (As+e, d) to Rq by applying
Decompressq (t, d), which allows us to write As + e + e0 = t, where s ← βηk1 , e ← βηm2 and e0 is the additional
error term caused by the compression. Without knowledge of s and e, such a mapping is the best that one
can use to minimize the norm of e0 . To finish our argument, it would suffice to show that the vector e + e0
has larger norm than a vector whose coefficients are drawn from the distribution βη1 . For this, we use theheuristic assumption that e0 is independent of s and e and each of its coefficients has the distribution
Decompressq Compressq (x, d), d − x mod± q; x ← Zq .4(7)
When d = du = 10 and q = 3329 are as in the Kyber512 parameter set, then the distribution in (7) is:
257
2048
1024
, Pr[e0 ∈ {−1, 1}] =, Pr[e0 = 0] =
.
(8)
3329
3329
3329
Using the above distribution on the coefficients of e0 and the fact that each coefficient of e comes fromthe distribution βη2 , we can exactly compute the distribution of the squared norm of e + e0 . In Table
5, we compare some bounds of this distribution with that of a 512-dimensional vector whose coefficientsare all chosen from βη1 . It can be seen from the table that the norm of the vector generated as e + e0
is noticeably larger, at all probabilities, than the one generated from βη1 . In fact, because the varianceof each coefficient of e is 1 and of the distribution in (8) (when the probabilities of the ± coefficients are
Pr[e0 ∈ {−2, 2}] =the compression coefficient d is greater than log q, this is the Module-LWE problem.
4 This assumption (with e = 0) is also used in the security and decryption error analysis of (Module)-LWR problems.

22

p
2

max α s.t. PrX=e+e0 ;e←βη512 ,e0 ←(8) [kXk < α] < p2
max α s.t. PrX←βη512 [kXk2 < α] < p1

2−1 2−30 2−60 2−90 2−120984 679 561 480
419
767 529 436 373
325

Table 5: Probability bounds on kXk2 when X is distributed as e + e0 or as a binomial. For example, when
X = e + e0 , then Pr[kXk2 < 561] < 2−60 while Pr[kXk2 < 562] ≥ 2−60 .
balanced) is approximately 0.9, the distribution of e + e0 behaves very similarly to a Gaussian with varianceapproximately 1.9. It is therefore reasonable to assume that this error distribution makes the problem harderthan if the distribution were β3 , which behaves similarly to a continuous Gaussian of variance 1.5. In short,by also accounting for the rounding error e0 , we gain an additional 6 bits of security with respect to thecurrently-best attacks.
The impact of MAXDEPTH. The best known quantum speedups for the sieving algorithm, which we
consider in our cost analysis (see Subsection 5.1.1), are only mildly affected by limiting the depth of a quantum
circuit, because it uses Grover search on sets of small size (compared to searching through the whole keyspaceof AES). For the core-SVP-hardness operation estimates to match the quantum gate cost of breaking AES at
the respective security levels, a quantum computer would need to support a maximum depth of 70–80. When
limiting the maximum depth to smaller values, or when considering classical attacks, the core-SVP-hardness
estimates are smaller than the gate counts for attacks against AES. Recent study [5] on the concrete cost ofsieving suggest that the quantum speed-ups of these algorithms are tenuous, independently of the value of
MAXDEPTH.

4.5

Additional security properties

4.5.1

Forward secrecy.

Kyber has a very efficient key-generation procedure (see also Section 2) and is therefore particularly well
suited for applications that use frequent key generations to achieve forward secrecy.
4.5.2

Side-channel attacks.

Timing attacks. Neither straight-forward reference implementations nor optimized implementations ofKyber use any secret-dependent branches or table lookups5 .This means that typical implementations ofKyber are free from the two most notorious sources of timing leakage. Another possible source of timing
leakage are non-constant-time multipliers like the UMULL instruction on ARM Cortex-M3 processors, which
multiplies two 32-bit integers to obtain a 64-bit result. However, multiplications in Kyber have only 16-bit
inputs, and most non-constant-time multipliers show timing variation only for larger inputs. For example, on
ARM Cortex-M3 processors the obvious way to implement multiplications in Kyber is through the constanttime MUL instruction, which multiplies two 32-bit integers, but returns only the bottom 32-bits of the result.
What remains as a source of timing leakage are modular reductions, which are sometimes implemented via
conditional statements. However, timing leakage in modular reductions is easily avoided by using (faster)Montgomery [79] and Barrett reductions [16] as illustrated in our reference and AVX2 implementations.
We note that the 90s variant is only really attractive to use if AES hardware support is available. If
hardware support is not available, then table-based implementations of AES are notorious for leaking secrets
trough cache timing. In our C reference implementation of the 90s variant we use a constant-time bitsliced
implementation of AES, which is based on code from BearSSL [91].
Differential attacks. We expect that any implementation of Kyber without dedicated protection against
differential power or electromagnetic radiation (EM) attacks will be vulnerable to such attacks. This is truefor essentially any implementation of a cryptographic scheme that uses long-term (non-ephemeral) keys.
Deployment scenarios of Kyber in which an attacker is assumed to have the power to mount such an attack
5 Note that the rejection sampling in generating the matrix A does not involve any secret data.

23

require specially protected—typically masked—implementations. In [83], Oder, Schneider, Pöppelmann, andGüneysu present such a masked implementation of Ring-LWE decryption with a CCA transform very similar
to the one used in Kyber. The implementation targets Cortex-M4F microcontrollers; the conclusion of thework is that protecting the decryption (decapsulation) step against first-order DPA incurs an overhead of
about about a factor of 5.5. The techniques presented in that paper also apply to Kyber and we expectthat the overhead for protecting Kyber against differential attacks is in the same ballpark.
Template attacks. Protections against differential attacks do not help if an attacker is able to recover even
ephemeral secrets from a single power or EM trace. At CHES 2017, Primas, Pessl, and Mangard presented
such a single-trace attack against an implementation of Ring-LWE on a Cortex-M4F microcontroller [92].
The attacker model in this attack is rather strong: it is the typical setting of template attacks, which assumes
an attacker who is able to generate template traces on known inputs on a device with leakage very similar
to the actual target device. In [92], the authors used the same device for generating target traces and in theattack. The attack was facilitated (maybe even enabled) by the fact that the implementation under attack
used variable-time modular reductions. Consequently, the paper states that “One of the first measures tostrengthen an implementation against SPA attacks is to ensure a constant runtime and control flow”. This is
the case for all implementations of Kyber. The attack from [92] would thus certainly not straight-forwardly
apply to implementations of Kyber, but more research is required to investigate whether also constant-time
implementations of Kyber (and other lattice-based schemes) succumb to template attacks, and what thecost of suitable countermeasures is.
4.5.3

Multi-target attacks

Our security analysis makes no formal claims about security bounds in the multi-target setting. However,in the design of Kyber we made two decisions that aim at improving security against attackers targeting
multiple users:• We adopt the “against-all-authority” approach of re-generating the matrix A for each public key from
NewHope [10]. This protects against an attacker attempting to break many keys at the cost of breaking
one key.
• In the CCA transform (see Alg. 8) we hash the public key into the pre-key K̄ and the coins r. Making
the coins r dependent of the public key protects against precomputation attacks that attempt to break
one out of many keys. For details, see Subsection 5.5.
4.5.4

Misuse resilience

The first, and most important, line of defense against misuse is the decision to make IND-CCA2 security
non-optional. As discussed in Subsection 1.5, it would have been possible to achieve slightly shorter publickeys and ciphertexts, and faster decapsulation, in a CPA-secure variant of Kyber. Using IND-CCA2 security
by default makes it safe to use Kyber with static keys and as a consequence also to re-use ephemeral keys
for some time. What is not safe, is to reuse the same randomness in encapsulation, but that randomness isalso not exposed to the outside by the API. The CCA transform has a second effect in terms of robustness:
it protects against a broken implementation of the noise sampling. A rather peculiar aspect of LWE-based
cryptography is that it will pass typical functional tests even if one communication partner does not add any
noise (or by accident samples all-zero noise). The deterministic generation of noise via SHAKE-256 during
encapsulation and the re-encryption step during decapsulation will reveal such an implementation mistake
immediately.
An additional line of defense against misuse is to hash the public-key into the “pre-key” K̄ and thus
make sure that the KEM is contributory. Only few protocols require a KEM to be contributory and those
protocols can always turn a non-contributory KEM into a contributory one by hashing the public key into
the final key. Making this hash part of the KEM design in Kyber ensures that nothing will go wrong on
the protocol level if implementers omit the hash there.
A similar statement holds for additionally hashing the ciphertext into the final key. Several protocols
need to ensure that the key depends on the complete view of exchanged protocol messages. This is thecase, for example, for the authenticated-key-exchange protocols described in the Kyber paper [24, Sec. 5].
24

Hashing the full protocol view (public key and ciphertext) into the final key already as part of the KEM
makes it unnecessary (although of course still safe) to take care of these hashes on the higher protocol layer.

5

Analysis with respect to known attacks

5.1

Attacks against the underlying MLWE problem

MLWE as LWE. The best known attacks against the underlying MLWE problem in Kyber do not makeuse of the structure in the lattice. We therefore analyze the hardness of the MLWE problem as an LWE
problem. We briefly discuss the current state of the art in algebraic attacks, i.e., attacks that exploit thestructure of module lattices (or ideal lattices) at the end of this subsection.
5.1.1

Attacks against LWE

Many algorithms exist for solving LWE (for a survey see [7]), but many of those are irrelevant for ourparameter set. In particular, because there are onlym = (k + 1)n
LWE samples available to the attacker, we can rule out BKW type of attacks [60] and linearization attacks [13]. This essentially leaves us with two BKZ [98, 32] attacks, usually referred to as primal and dual
attacks that we will recall in Subsections 5.1.2 and 5.1.3.
The algorithm BKZ proceeds by reducing a lattice basis using an SVP oracle in a smaller dimension b.
It is known [53] that the number of calls to that oracle remains polynomial, yet concretely evaluating thenumber of calls is rather painful, and this is subject to new heuristic ideas [32, 31, 11].
We start with an analysis that ignores this polynomial factor, i.e. only considers the cost of a single callto an SVP oracle in dimension b. We will also use for now a very simple cost estimate for the hardness of
SVP. This core-SVP hardness methodology was introduced in [10, Sec. 6], as a simple way of estimating
security. In the light of the cryptanalytic progress that happened during the NIST evaluation Rounds 1 and
2, we consider that this method remains informative, but too coarse to produce accurate security estimates,
especially the for security against classical attackers. This will be addressed by the new Section 5.2, added
for the Round 3.
Enumeration vs. sieving. There are two algorithmic approaches for the SVP oracle in BKZ: enumerationand sieving algorithms. These two classes of algorithms have very different performance characteristics and,in particular for sieving, it is hard to predict how practical performance scales from lattice dimensions thathave been successfully tackled to larger dimensions that are relevant in attacks against cryptosystems like
Kyber. The starting point of such an analysis is the fact that enumeration algorithms have super-exponential
running time, while sieving algorithms have only exponential running time. Experimental evidence fromtypical implementations of BKZ [49, 32, 38] shows that enumeration algorithms are more efficient in “small”
dimensions, so one question is at what dimension sieving becomes more efficient. At the beginning of the
NIST PQC project, the best known sieving techniques were slower in practice for accessible dimensions of
up to b ≈ 130.
The work [40] showed (in the classical setting) that sieving techniques can be sped up in practice forexact-SVP, becoming less than an order of magnitude slower than enumeration already in dimension 60 to 80.
One reason for the improvement is the “dimensions-for-free” technique, allowing to solve SVP in dimension bby sieving in a sligthly smaller dimension b0 = b − d4f . Further algorithmic and implementation efforts havefinally managed to get sieving to outperform enumeration in practice [4], with a cross-over around dimension
80.
The hardness estimation is complicated by the fact that sieving algorithms are much more memory
intensive than enumeration algorithms. Specifically, sieving algorithms have exponential complexity not
only in time, but also in memory, while enumeration algorithms require only small amounts of memory.
In practice, the cost of access to memory increases with the size of memory, which typically only becomes
noticeable once the memory requirement exceeds fast local memory (RAM). There is no study, yet, thatinvestigates the algorithmic optimization and practical performance of sieving using slow background storage.
25

We follow the approach of [10, Sec. 6] to obtain a conservative lower bound on the performance of bothsieving and enumeration for the dimensions that are relevant for the cryptanalysis of Kyber. This approachworks in the RAM model, i.e., it assumes that access into even exponentially large memory is free.
A lot of recent work has pushed the efficiency of the original lattice sieving algorithms [82, 78], improving
p b+o(b)≈ 20.292b+o(b) using localitythe heuristic complexity from (4/3)b+o(b) ≈ 20.415b+o(b) down to 3/2
sensitive hashing (LSH) techniques [63, 17]. Without the dimensions-for-free technique, the hidden subexponential factors are typically much greater than 1 in practice [77, 40].
Most of the sieving algorithms have been shown [65, 62] to benefit from Grover’s quantum search algorithm, bringing the complexity down to 20.265b+o(b) . However, a concrete analysis shows that, in practice,
the quantum speed-up remains tenuous [5].
For our first analysis, we will use 20.292b as the classical and 20.265b and the quantum cost estimate ofboth the primal and dual attacks with block size (dimension) b. We recall those two attacks in the following.
5.1.2

Primal attack.

The primal attack consists of constructing a unique-SVP instance from the LWE problem and solving it using
BKZ. We examine how large the block dimension b is required to be for BKZ to find the unique solution. Given
the matrix LWE instance (A, b = As+e) one builds the lattice Λ = {x ∈ Zm+kn+1 : (A|Im |−b)x =√0 mod q}
of dimension d = m+kn+1, volume q m , and with a unique-SVP solution v = (s, e, 1) of norm λ ≈ ς kn + m,
where ς is the standard deviation (i.e., the square root of the variance) of the individual secret / error
coefficients. Note that the number of used samples m may be chosen between 0 and (k + 1)n in our case andwe numerically optimize this choice.
Success condition. We model the behavior of BKZ using the geometric series assumption (which is known
to be optimistic from the attacker’s point of view), that finds a basis whose Gram-Schmidt norms are given
by kb?i k = δ d−2i−1 · Vol(Λ)1/d , where δ = ((πb)1/b · b/2πe)1/2(b−1) [31, 7]. The unique short vector v will be
detected if the projection of v onto the vector space√ spanned by the last b Gram-Schmidt vectors is shorter
than b?d−b . Its projected norm is expected to be ς b, that is, the attack is successful if and only if√(9)ς b ≤ δ 2b−d−1 · q m/d .
We note that this analysis introduced in [10] differs and is more conservative than prior works, whichwere typically based on the hardness of unique-SVP estimates of [48]. The validity of the new analysis has
been confirmed by further analysis and experiments in [6].
5.1.3

Dual attack

The dual attack consists of finding a short vector in the dual lattice w ∈ Λ0 = {(x, y) ∈ Zm × Zkn : At x =y mod q}. Assume we have found a vector (x, y) of length ` and compute z = vt · b = vt As + vt e =wt s + vt e mod q, which is distributed as a Gaussian of standard deviation `ς if (A, b) is indeed an LWE
sample (otherwise it is uniform mod q). Those two distributions have maximal variation distance boundedby  = 4 exp(−2π 2 τ 2 ), where τ = `ς/q, that is, given such a vector of length ` one has an advantage  against
decision-LWE.
The length ` of a vector given by the BKZ algorithm is given by ` = kb0 k. Knowing that Λ0 has dimension
d = m + kn and volume q kn we get ` = δ d−1 q kn/d . Therefore, obtaining an -distinguisher requires running
BKZ with block dimension b, where− 2π 2 τ 2 ≥ ln(/4).
(10)
Note that small advantages  are not relevant since the agreed key is hashed: an attacker needs an advantage
of at least 1/2 to significantly decrease the search space of the agreed key. He must therefore amplify his
success probability by building about 1/2 many such short vectors. Because the sieving algorithms provides
20.2075b vectors, the attack must be repeated at least R times whereR = max(1, 1/(20.2075b 2 )).
This makes the conservative assumption that all the vectors provided by the Sieve algorithm are as short as
the shortest one.
26

5.1.4Core-SVP hardness of Kyber

In Table 4 we list the classical and quantum core-SVP-hardness of the three parameter sets of Kyber. The
lower bounds of the cost of the primal and dual attack were computed following the approach outlined above
using the analysis script Kyber.py that is available online at https://github.com/pq-crystals/securityestimates.

5.2

Beyond core-SVP hardness

At the time the core-SVP hardness measure was introduced by [10], the best implementations of sieving [17,77] had performance significantly worse than the 2.292b CPU cycles proposed as a conservative estimate bythis methodology. This was due to substantial polynomial or even sub-exponential overheads hidden in thecomplexity analysis of 2.292b+o(b) given in [17]. Before this [10] proposed the core-SVP approach, security
estimates of lattice schemes were typically based on the cost of SVP via enumeration given in [32, 7], leading
to much more aggressive parameters. Beyond the cost of SVP-calls, this methodology also introduced adifferent prediction of when BKZ solves LWE, which was later confirmed [6] and refined [36].
While doubts were expressed to whether sieving would ever outperform the super-exponential, yet practically smaller, costs of enumeration [32] for relevant cryptographic dimensions, significant progress on sieving
algorithms [40, 4] has brought down the cross-over point down to dimension about b = 80. In fact, thecurrent SVP records are now held by algorithms that employ sieving6 . This progress mandates a revisionand refinement of Kyber security estimates, especially regading classical attacks. In particular, while it waspretty clear from experiments than the costs hidden in the o(b) before those improvement were positive bothin practice and asymptotically, the dimensions-for-free technique of [40] offers a sub-exponential speed-up,
making it a priori unclear whether the total o(b) term is positive or negative, both asymptotically andconcretely.
In summary, while the core-SVP methodology introduced 5 years ago has pushed designers to be moreconservative than previously, it now appears this estimation technique is too coarse to produce accurate
security estimates. In the following, we give a refined analysis based on the latest developments describedin the literature. We complement this with a discussion of all the approximations, simplifications andforeseeable developments that remain to be explored.
We also note that the choice of the gate count metric was recently discussed in the NIST PQC forum
mailing list, and in the case of the algorithm of [17], we do not believe this metric to be very realistic. However,it appears that alternative metrics unavoidably involve physical and technological constants (speed of light,
density of information, energy efficiency of gates and data transfers). There seems to be no clear consensus
on what some those constants would be with current technology, let alone future technologies. These othermetrics would also greatly increase the complications involved in the already delicate tuning and analysis ofsieving.
In the following subsection we therefore discuss the current understanding of attacks against Kyber inthe gate-count metric. We focus the discussion on the concrete case of Kyber512. This preliminary analysis
gives a cost of 2151 gates, which is a 28 factor margin over the targeted security of the 2143 gates required for
attacks against AES. Our discussion of the ‘known unknowns’ conclude that this number could be affectedby a factor of up to 216 in either direction. While there is a risk to see the security claim drop below the 2143
bar in the gate count metric, one should consider the choice of the gate count metric itself as a substantial
margin. We do not think that even a drop as large as 216 would be catastrophic, in particular given themassive memory requirements that are ignored in the gate-count metric. Having listed and documented thesources of uncertainties, we hope that many of them can be tackled in the months to come, to narrow the“confidence interval”.
5.2.1

A tentative gate-count estimate accounting for recent progress

For concreteness, we focus this discussion to the case of Kyber512. Let us
Pstart by defining the progressivity
overhead C = 1/(1 − 2−.292 ) = 5.46, that is the limit of ratio between i≤b 2.292i+o(i) and 2.292b+o(b) as b
grows.
6 https://www.latticechallenge.org/svp-challenge/27

Primal Attack Only. Our first point is that, while the core-SVP hardness methodology suggest thatthe dual attack is slightly cheaper than the primal one, it is in fact significantly more expensive. Indeed,
the analysis of the dual attack of [10] (repeated above) assumes that one gets exponentially many vectors
from sieving
in the first block that will be as short as the shortest one. In fact, most of them will have aplength 4/3 larger. Furthermore, the assumption that we obtain exponentially many such short vectors
is incompatible with some of the latest sieving improvements, specifically the dimensions-for-free techinquefrom [40]. We note that the analysis from [2], which does not assume those extra short vectors, indeed does
predict much larger cost for the dual attack than the primal.
BKZ simulation. The analysis of the BKZ success condition from [10, 6] is based on the so called
geometric-series assumption, an assumption that has several inaccuracies; in particular it misses a “tail”
phenomenon [99]. We instead rely on the simulator provided as part of the leaky-LWE-estimator of [36].
This simulator uses progressive-BKZ [11, 4], which we believe gives somewhat better performance than fixedblock BKZ. It predicts a median success when reaching blocksize b = 413. Success grows by a factor of 1.373
from b = 412 to b = 413, which is larger than the cost factor of 20.292 ≈ 1.224: aiming at lower blocksize andretrying is not a worthy strategy (it grows even faster for b < 412). The overall dimension of the lattice isn = 1025.
The cost of progressive BKZ with sieving up to blocksize b is essentially C · (n − b) ≈ 3340 times the costof sieving for SVP in dimension b.
b ln(4/3), which is
Dimensions for free. According to [40], the number of “dimensions for free” is d4f = ln(b/(2πe)))d4f = 37.3 (or 38, rounding upward). That is, each SVP oracle call in dimension b = 413 will require runningsieving in dimension b0 = b − d4f = 375.
We note that [4, Fig. 5] obtains a few more dimensions for free in practice than [40] with two tricks:“on-the-fly lifting” and “pump-down sieves”. In practice, this saves a bit of memory (2.5 ), and has no visible
effect on time. The trick at hand is really fit for CPUs (as it uses more floating-point arithmetic), andprobably less interesting for implementations in dedicated circuits (which favor XOR-popcounts).

Gate cost of sieving. Recent work [5] has proposed an analysis of the cost of sieving with a classical andquantum circuit (with RAM access to data). More specifically, they focus on the analysis of a “AllPairSearch”
function, account for exact volumes of spherical caps and wedges, compute explicit gate counts for the
innermost loop operations (XOR-popcounts, inner products), and automatically tune parameters to obtainconcrete classical and quantum costs. Regarding the best classical algorithm, they conclude with a cost of
about 2137.4 gates for AllPairSearch in dimension 375.7
The work in [5] is motivated by the quantum/classical speed-up, therefore it does not consider the required
number of calls to AllPairSearch. Naive sieving requires a polynomial number of calls to this routine, however
this number of calls appears rather small in practice using progressive sieving [40, 64], and we will assume
that it needs to be called only once per dimension during progressive sieving, for a cost of C · 2137.4 gates8 .
Final gate count. Overall, we conclude with a gate count of
G = (1025 − 413) · C 2 · 2137.4 = 2151.5 .
We can also estimate the memory requirement following the analysis of [5], maybe working under theassumption that each coordinate of a lattice vector used during the sieving can be represented within a byte.
Automatizing the above calculations,9 we obtain Table 4, summarizing the refined estimates for all threeparameter sets.
We also note that a similar refined count of quantum gates seems essentially irrelevant for our security
claim: the work of [5] concluded that quantum speed-up of sieving are rather tenuous, while the quantum
security target for each category is significantly lower than the classical target.
7 This was interpolated from the datafile cost-estimate-list_decoding-classical.csv from the version of May 2020 of [5],available at https://eprint.iacr.org/eprint-bin/getfile.pl?entry=2019/1161&version=20200520:144757&file=1161.pdf .
This datafile can be extracted from the pdf via the linux tool pdfdetach. Another version of that datafile is more easily
accessible at https://github.com/jschanck/eprint-2019-1161/blob/main/data/cost-estimate-list_decoding-classical.
csv, and differs from the one we used by less than a bit.
8 This is using the fact inside BKZ, doing a single pump per SVP instance seems sufficient [2], while an isolated SVP instancerequires an extra progressive loop, called “work-out” on top of the “pump”. That is, an isolated SVP instance should have an
extra C factor.
9 Scripts available at https://github.com/lducas/leaky-LWE-Estimator/tree/NIST-round3

28

5.3

Approximations, overheads, and foreseeable improvements

We now propose a list of research direction to refine the best attack or its cost estimation, together with an
educated guess of their potential concrete impact. We hope that this list of open problems may motivate
the community towards a more informed consensus on the actual cost of lattice attacks, that would apply
not only to Kyber, but to all lattice-based candidates in the NIST PQC standardization project.
Q1: Idealized Angles and Lengths. A first approximation made in the concrete analysis of [5] is thatall vectors have the same length, and therefore that a pair of vectors leads to a reduction precisely if they
have an angle below π/3. However, both heuristically and experimentally, this is not exactly true. Indeed,
the length of the sum kx ± yk may be smaller than max(kxk, kyk) despite an angle slightly larger than π/3
when kxk and kyk differ a bit. Furthermore, implementations such as [4] even relax the reduction condition,and perform a reduction z ← x ± y where z is neither x nor y, but the current worst vector of the database.
This approximation and relaxation may explain the discrepancy of about a factor of 8 between thenumber of vectors required in the analysis [5] and the experiments up to dimension b = 120 [4]. However,these reducing pairs at higher angles are a bit harder to detect with near-neighbors-search techniques.
Expected influence on memory estimate: A factor of 2−3 .
Expected influence on gate-count estimate: A factor between 2−3 and 2−1 .
Q2: Idealized near-neighbors search. The analysis of [5] assumes that the BDGL [17] sieve behaves as
if the buckets are perfectly uniformly distributed over the sphere; however the actual algorithm [17] mustresort to some structure in the way these buckets are distributed. The original analysis shows that thisstructure
does not affect the success probability of finding each pair by a factor more than subexponential√
2Õ( b) [17]. However, this asymptotic analysis may not be that tight, and indeed, preliminary experimentssuggest that this success-probability loss may not be so large in practice.
Furthermore, depending on the parametrization, this algorithm has overheads compared to the idealization of [5]: minimizing memory requirement induces a 2O(b/ log(b)) overhead on time, which can in principle
be traded for an asymptotically similar memory overhead.
Expected influence on gate-count estimate: A factor of up to 25 . (part of which tradeable for morememory)Q3: Saturation, collisions, and dimensions for free. Another question that affects the concrete performance of the sieving is the delicate choice of when to stop it. In practice the latest implementations[40, 4]pchose to stop the sieving when a fraction s = 1/2 of the expected vectors of length 4/3 · GH has beenfound.10 This means that the success probability of actually solving SVP with dimensions for free can decrease by a factor s (or maybe a bit less because inppractice the sieving outputs about 6 times more vectors,
the remaining of which are a sligthly longer than 4/3 GH, and also have some probability of lifting to theshortest vector).
On the other hand, stopping the sieving with a larger saturation value s also means that we observe
more collisions: a pair does reduces, but result more often in an already known vector, making it useless.
This must be compensated by enlarging the database, and/or running more reduction rounds. Preliminary
experiments showed that the frequency of collision can be significantly larger than s; this may be explainedby repeated reduction of the same pair, and the fact that the database tends to not be uniform over the ball,and biased toward shorter vectors.
Expected influence on gate-count estimate: A factor between 2−1 and 22 .
Q4: Number of calls to AllPairSearch. The number of calls to AllPairSearch at each step of Progressive
Sieving was assumed to be 1. This could be tested experimentally. It is not unthinkable that this is in fact
a fractional number less than 1, as, in progressive-sieving, the previous calls in lower dimension may alreadyprovide a close to target set of vectors. Preliminary experiments suggest 1/3 or 1/2 in very small dimensions;these experiments have been carried without NNS to avoid interference, and deserve to be extended.
Expected influence on gate-count estimate: A factor between 2−2 and 22 .
Q5: Routing and Congestion (from RAM to circuit). As far as we understand, the sieving circuit
studied in [5] still considers RAM access, which, even in the gate count model, and without considering
speed-of-light delays easily underestimates cost. For example, once it has been determined in which bucket
10 Here, GH denote the expected length of the shortest vector according to the Gaussian Heuristic.

29

a vectors belongs, there must be some routing logic to send it there; if there are 2x addresses, then the sent
bits must traverse at least O(x) gates. This gets quite more involved as several cores may want to senddata to the same bucket at the same time, inducing overheads for queuing logic and/or packet drops. Extralogistic costs are also required in the bucket itself, and if the targeted bucket size is small, variations in theactual numbers of vectors that reach a bucket may induce further overheads or packet drops.
Such logistic costs can be partially mitigated by re-tuning the algorithm: if, once bucketed, data is used
often enough, these overheads will become negligible. However, making buckets larger negatively impacts
algorithmic performance in the RAM model, and despite mitigation, some loss seems unavoidable. We notethat the current parametrization from [5] suggests to target an average bucket size of 32 in the relevant
dimensions, leading to each bit of data being used in 32 many XOR-popcounts. We do not think that thisparametrization makes logistic overheads negligible.
Expected influence on gate-count estimate: A factor between 22 and 28 .
Q6: Beyond the gate-count metric. While we mostly stick to the gate-count metric in this discussion,
we note that the previous research question Q5 might be naturally paired with a similar study in othermetrics, maybe having physical and technological constants as parameters. In particular, we would like tomention an ongoing implementation of Sieving with GPU [41], for which optimal performances are reachedfor quite large bucket-size (say 215 ), and this despite using a single computation server (i.e., one is limited
by RAM to GPU-RAM bandwidth, and not network bandwidth).
Q7: Refined BKZ strategies. We note that there are ways to refine a bit the (progressive) BKZ strategy
used in the simulator of [36]. Indeed, a standard idea [68] is to run a single SVP call in a dimension that is
considerably larger than the BKZ blocksize after the BKZ preprocessing, in the hope to partially amortizethe (n − b) · C factor. While this has been used for sieving in practice [4] on LWE challenges, a precise
analysis of the gains in large dimensions is still missing. We note that this is a time-memory trade-off, as
this last SVP call will require more memory than the BKZ pre-processing.
Another folklore idea which remains to be documented and quantified, is to try to swap the “heads” and“tails” of the BKZ profile [99], by instead running dual-BKZ as the pre-processing.
One could also tweak the progressive steps of the blocksize in progressive-BKZ, but our preliminary
experiments with the simulations of [36] was not suggesting that significant gains are to be expected there.
Expected influence on gate-count estimate: A factor between 2−8 and 2−2 .
Q8: Module-BKZ in practice. It is known that sieving in cyclotomic/cyclic ideal lattices can benefit
from speed-ups and memory gain, namely a factor d can be saved on memory, and a factor between O(d)and O(d2 / log d) can be gained depending on whether Near Neighboor Search techniques are used or not.
Namely, the symmetries of those lattices allows to get of full orbit of d vectors of the same length for the
price of 1.
In principle, the same gains apply to module lattices of dimension b = kd, growing with the module
dimension d. It was mentioned on the NIST pqc-forum11 that a module variant of BKZ could exists, andtherefore that its internal SVP oracle could benefit from symmetries. This is confirmed by the theoreticalwork of [80].
However, there are numerous issues that are likely to prevent substantial gain from this approach. The
first one is that one is now very constrained in terms of blocksize: to run module-BKZ with blocksize b on
Kyber, one must rely on a sub-module structure of dimension d = gcd(b, 256). This may force one to over orundershoot the optimal block-size for the attack, and leaves less flexibility, especially for progressivity bothat the sieving and BKZ levels. This is even more constraining in combination with the dimension for free,where, as far we can see, one would require both the sieving and the lift context to be aligned with the module
structure: d = gcd(b, d4f , 256). It is also not clear whether, for the same block-size, module-BKZ would givea basis perfectly as good as a BKZ, given that the basis is now constrained by the module structure. One
will also need to adjust Kannan embedding to the module structure, which increases the overall dimensionof the lattice by d rather than by 1.
Expected influence on gate-count estimate: A factor between 2−4 and at most 20 .
Summary. Combining all the discussion above, it appears that further refinements of the analysis of various
aspect of sieving, as well as some foreseeable algorithmic improvements, the estimates may move by a factor
11 https://groups.google.com/a/list.nist.gov/g/pqc-forum/c/JXN9NWGt9Ys/m/37tgAlU7DAAJ

30

somewhere between 2−16 and 214 . Again, we hope that at least some of these research directions will soon be
documented and elucidated, narrowing down this confidence interval. We recall that they essentially affect
all lattice-based candidates in similar way (except maybe Q8).
5.3.1

Algebraic attacks.

While the best known attacks against the MLWE instance underlying Kyber do not make use of thestructure in the lattice, we still discuss the current state of the art of such attacks. Most noticeably, several
recent works propose new quantum algorithms against Ideal-SVP [44, 30, 21, 34, 35], i.e., solving the shortest
vector problem in ideal lattices. The work of [35] mentions obstacles towards a quantum attack on Ring-LWE
from their new techniques, but nevertheless suggests using Module-LWE, as it plausibly creates even moreobstacles. In [1], Albrecht and Deo establish a reduction from MLWE to RLWE, whose implication is thata polynomial-time algorithm against RLWE with certain parameters would translate to a polynomial-timealgorithm against MLWE. In practical terms, however, this attack has a significant slow-down (and this is
not just due to the proof) as the dimension of the module increases. This does suggest that increasing thedimension of the module may make the scheme more secure in concrete terms. In particular, going throughthis reduction to attack Kyber768 would lead to an RLWE problem with quite large modulus and error(q 0 = q 3 , ς 0 > q 2 ς), and therefore require the attacker to consider more than 1 sample: the underlying lattice
remains a module with a rank strictly larger than 2.

5.4

Attacks against symmetric primitives

All symmetric building blocks of Kyber are instantiated with functions derived from Keccak [20]. In thedeterministic expansion of A from ρ we essentially need SHAKE-128 to produce output that “looks uniformly
random” and does not create any backdoors in the underlying lattice problem. In the noise generation we
require that concatenating a secret and a public input and feeding this concatenation to SHAKE-256 as
input results in a secure pseudorandom function. Breaking any of these properties of SHAKE would be amajor breakthrough in the cryptanalysis of SHAKE, which would require replacing SHAKE inside Kyber
by another XOF.
The security proofs model SHAKE-128, SHA3-256, and SHA3-512 as random oracles, i.e., they are subject
to the standard limitations of proofs in the (quantum-)random-oracle model. Turning these limitations into
an attack exploiting the instantiation of XOF, H, or G with SHAKE and SHA3 would again constitute amajor breakthrough in the understanding of either Keccak or random-oracle proofs in general.

5.5

Attacks exploiting decryption failures

In Theorems 2 and 3 we see that decryption failure probability plays a role in the attacker’s advantage: in2the classical context in the term 4qRO δ and in the quantum context in the term 8qRO
δ, where qRO is thenumber of queries to the (classical or quantum) random oracle.
Attacks exploiting failures. This term in the attacker’s advantage is not merely a proof artifact, it canbe explained by the following attack: An attacker searches through many different values of m (see line 1 ofAlg. 8) until he finds one that produces random coins r (line 3 of Alg. 8) that lead to a decapsulation failure,
which will give the attacker information about the secret key. In the quantum setting the search through2
different values of m is accelerated by Grover’s algorithm, which explains the square in the term qRO. With64
this attack in mind note that with 2 ciphertexts (cmp. Section 4.A.2 of the Call for Proposals), there is achance of 2−100 of a decapsulation failure in Kyber768 without any particular effort by the attacker.
To understand what exactly this means for attacks against Kyber, we need to address the following twoquestions:
1. How hard is it for an attacker to trigger a Kyber decapsulation failure?
2. How hard is it for an attacker equipped with a ciphertext triggering a failure to mount a successful
attack against Kyber?

31

Regarding the first question, the naive approach of an attacker is to try random ciphertexts, which has
a success probability of qd δ, where qd is the number of decapsulation queries. In the classical randomoracle model, the cost of an attack exploiting failures will also never get lower than that, as it matches theinformation-theoretic success probability.
Failure boosting using Grover.
A quantum attacker can try to use Grover search to precompute values of m that have a slightly higher
chance to produce a failure. The efficacy of Grover search is limited by the fact that an attacker cannot
determine offline whether a given value of m, or more specifically, the derived values r (line 9 of Alg. 5) ande1 (line 13 of Alg. 5), produce a decapsulation failure. The reason is that the probability of decapsulation
failures largely depends on the products sT e1 and eT r and the attacker does not know (the signs of) the
coefficients of s and e.
Original Kyber analysis. The original Kyber submission document gave an analysis of a particular
strategy for using Grover’s algorithm to search for values of m that produce e1 and r with above-averagenorm. Intuitively, the larger these values are, the bigger the probability of a decryption failure. The
gain achieved through such an approach is, however, limited due to the fact that the distribution of ahigh-dimensional Gaussian is tightly concentrated around its expected value, while that of a 1-dimensional
Gaussian is not as tightly concentrated around its mean. We present this original analysis below, focusing
on Kyber768, which has failure probability of 2−164 , for concreteness:
The polynomial pairp(e1 , r) can be seen as a vector in Z1536 distributed as a discrete Gaussian withstandard deviation σ = η1 /2 = 1 (because η1 = η2 ). By standard tail bounds on discrete Gaussians [14],we know that an m-dimensional vector v drawn from a discrete Gaussian of standard deviation σ will satisfy√
2m
Pr[kvk > κσ m] < κm · e 2 (1−κ ) ,(11)

for any κ > 1.
√
So for example, the probability of finding a vector which is of length 1.33 · σ 1536 is already as small as
2−220 . Even if Grover’s algorithm reduces the search space and increases the probability to 2−110 , finding
such a vector merely increases the chances of getting a decryption error; and the probability increase isgoverned by the tail-bounds for 1-dimensional Gaussians12 . For any vector v, if z is chosen according to a
Gaussian with standard deviation σ, then for any κ,
2Pr[hz, vi| > κσkvk] ≤ 2e−κ /2 .

(12)

If originally, the above probability is set so that decryption errors occur with probability ≈ 2−160 , thenκ ≈ 15.13 If the adversary is then able to increase kvk by a factor of 1.33 (by being able to find larger(e1 , r)), then we can decrease κ by a factor of 1.33 to ≈ 11.25 in (12), which would still give us a probabilityof a decryption error of less than 2−90 . However, finding such a large v would take at least 2110 time, which
would make the whole attack cost at least 2200 .
Of course one can try to find a slightly smaller v in the first step so that the entire attack takes less time.
If Grover’s algorithm really saves a square-root factor, then the optimal value is ≈ 1.05 for κ in (11), which
would allow us to lower κ by a factor of 1.05 in (12) to 15/1.05 ≈ 14.28, and would still give a total time tofind one decryption error ≈ 2−150 . This makes the attack completely impractical.
Different approaches since 2017. Since the original analysis of failure boosting against Kyber, multiple
works have considered multiple aspects of this topic [37, 42]. The latest work on this topic is [22], considers
using Grover’s algorithm to produce a strategic set of decryption query points which results in a higher
chance of triggering a decryption failure when the number of decryption queries is limited, as in the NIST
Call for Proposals. The overall running time is, however, not less than in the above attack, which didn’t
restrict itself with a limit on the number of decapsulation queries.
12 The decryption noise is generated as an inner product of two vectors, and the distribution of this inner product closely
resembles the Gaussian distribution.
13 The above formula only roughly approximates how the decryption error is calculated where z corresponds to the secret key(s, e). We should also point out that a part of the decryption error in Kyber is caused by the rounding function Compress,
which the adversary has no control over. Therefore this attack will be even less practical than what we describe.

32

From one to multiple failures. The original Kyber submission document also discussed that an attacker
may need more than a single failure to mount a key-recovery attack. For example, the work of [46] required
up to 4000 failures, which may suggest that the number of online queries required for a successful attack
against Kyber increases by a factor of ≈ 212 . However, recent work [43] proposed an adaptive strategy,
which uses previous failures to significantly lower the cost of the next one. This directional failure boosting
technique reduces the cost of the full attack to barely more than the cost of triggering the first failure. Thismeans that the answer to the second question in the beginning of this subsection is that one should be on
the safe side and make sure that it’s hard to trigger even one failure.
Multitarget attacks using failures. Despite the limited gain, an attacker could consider using Grover’s
algorithm to precompute values of m that produce r and e1 with large norm and then use this precomputed
set of values of m against many users. This multi-target attack is prevented by hashing the public key pk
into the random coins r and thereby into r and e1 (line 3 of Alg. 8).

6Advantages and limitations

6.1

Advantages

In addition to the very competitive speeds, small parameters, and being based on a well-studied problem,the unique advantages of Kyber are:Ease of implementation: Optimized implementations only have to focus on a fast dimension-256 NTT
and a fast Keccak permutation. This will give very competitive performance for all parameter sets ofKyber.
Scalability: Switching from one Kyber parameter set to another only requires changing the matrix
dimension (i.e., a #define in most C implementations), the noise sampling, and the rounding of theciphertext via different parameters to the Compressq function.
We will now give a brief comparison of Kyber to other types of post-quantum schemes (that we areaware of) and, more importantly, to other manners in which lattice-based schemes could be instantiated.

6.2

Comparison to SIDH

An interesting alternative to lattice-based KEMs is supersingular-isogeny Diffie-Hellman (SIDH) [57]. The
obvious advantage of SIDH is the sizes of public keys and ciphertexts that—with suitable compression [33]—are about a factor of 3 smaller than Kyber’s public keys and ciphertexts. The downside of SIDH is that it
is more than 2 orders of magnitude slower than Kyber. The scheme is also rather new, which makes it hard
to make definitive comparisons. In the coming years, both implementation speeds and (quantum) attacks
against SIDH can improve which may result in faster schemes and/or larger parameters.

6.3

Comparison to code-based KEMs

When considering code-based KEMs, one needs to distinguish the “classical” McEliece and Niederreiterschemes based on binary Goppa codes, and schemes with a less conservative (but more efficient) choiceof code. A KEM based on binary Goppa codes can reasonably claim to be a very conservative choice of
post-quantum primitive; however, its deployment will, in many scenarios, be hampered by massive publickey size and key-generation time. Less conservative choices, like quasi-cyclic medium-density parity-check(QC-MDPC) codes, are a closer competition in terms of performance but suffer from the fact that for
efficient parameters at high security levels they do not achieve (provably) negligible failure probability,
which precludes their use in CCA-secure KEMs.

6.4

Comparison to other lattice-based schemes

There are certain design choices that one can make when designing lattice-based schemes, some of which canhave significant effects on the efficiency of the resulting scheme and on the underlying security assumption.
33

Below we list the most important ones and explain the advantages / disadvantages of them versus what wechose for Kyber.
6.4.1

Schemes that build a KEM directly

The Kyber KEM is constructed by encrypting a random message using the LPR encryption [72] (with“bit-dropping”). Another approach one could take is directly building a KEM using the slightly different
ideas described in [39, 86]. The advantage of the constructions in [39, 86] over our approach is that if one
were to construct a CPA-secure KEM transmitting a b-bit key, then the ciphertext would be b bits shorter,
which is about a 3% saving for typical parameters [69]. If, however, one wishes to construct a CCA-secure
KEM like Kyber, then this advantage disappears since transformations from CPA-secure KEMs to CCAsecure ones implicitly go through a CPA-secure encryption scheme, which will result in adding b bits to theKEM. This is why, in Kyber, we simply use the LPR encryption scheme (instead of the CPA-secure key
encapsulation) to define Kyber.CPAPKE, and then use this as a building block to construct the IND-CCA2secure KEM Kyber.CCAKEM. Since there is virtually no difference between the two approaches, we will not
draw a distinction between schemes constructed in either manner throughout the rest of this section.
6.4.2

LWE based schemes

If one does not want to use any algebraic structure in the LWE problem (i.e. if one takes the MLWE problem
over the ring Z), then there are two possibilities for constructing encryption or key-exchange schemes. The
first approach makes the public key and the secret key very large (on the order of Megabytes), while keeping
the ciphertext at essentially the same size as in Kyber. This type of scheme is the [88] version of the original
Regev scheme from [94]. Because of the very large public-key size, this scheme would be extremely inefficient
as a key exchange. A scheme more amenable to key exchange is [23], whose public key and ciphertext sizes
are both approximately 11 KB each, which is approximately 10 times larger than in Kyber. The running
time of each party is also larger by a factor of at least 10. In short, LWE-based schemes do not have any ring
structure but are an order of magnitude slower and larger than Kyber. They are good back-up schemes incase algebraic structure in lattice schemes could somehow be devastatingly exploited by attackers.
6.4.3

Ring-LWE based schemes

The other extreme in the LWE design space are Ring-LWE (RLWE) schemes based on [72] (e.g., [10]).
RLWE is a special case of the MLWE problem where the width of the matrix A over the ring R is always 1(and typically, its height would be 2 for a PKE or KEM scheme). Varying the hardness of an RLWE scheme
therefore requires to change the dimension of the ring, whereas in Kyber, the ring is always the same andthe dimension of the module is being varied. As we mentioned above, one advantage of the approach we
chose for Kyber is that we only need to have one good implementation for operations over the ring; varying
the dimension of the module simply involves doing more (or fewer) of the same ring operations. Changing
the ring, on the other hand, would require completely re-implementing all the operations.
Another advantage of working with a constant-degree “small” ring is that it enables more fine-grained
tradeoffs between performance and security. The simplest and most efficient way of implementing RLWE is
to work over rings Z[X]/(X n + 1) where n is a power of 2. Since n is the only parameter that determines
the efficiency and security of RLWE schemes, limiting it to powers of 2 may require overshooting the needed
security bound. For example, the dimension of Kyber768 is not reachable. One could of course work directly
modulo a polynomial of any desired degree (with the main restriction being that it has to be irreducible
over Z), but then the security would decrease slightly due the geometry of non-power-of-2 number fields(see [75, 76]).
The one advantage of RLWE over Kyber is that if A is a k × k matrix, then extracting it from a seedrequires k times more XOF output than for a 1 × 1 matrix.
6.4.4

NTRU

When compared to Kyber, NTRU [54] has all the advantages and disadvantages of RLWE, but in addition
has two further negative points against it. First NTRU key generation is considerably more expensive

34

than in RLWE when the ring does not support NTT. The reason is that NTRU key generation requires
polynomial division, whereas RLWE key generation requires only multiplication (if the ring supports NTT,
then division is not much slower than multiplication). The second possible downside of NTRU is that thegeometry of its underlying lattice leads to attacks that do not exist against RLWE or MLWE schemes [61].
While this property does not seem to aid in attacks against the small parameters that are used for defining
NTRU cryptosystems, it may point to a possible weakness that could be further exploited. The one possible
advantage of using NTRU is a small performance advantage during encryption (encapsulation), but given the
disadvantages we do not consider this a good tradeoff. Furthermore, it is not possible to define an efficientversion of “Module-NTRU” that would allow for the advantages of Kyber described above in Section 6.1.
6.4.5

Different Polynomial Rings

One could consider using Kyber with a ring that is not Z[X]/(X n + 1). An argument that could be madefor using different rings is that the rings currently used in Kyber have algebraic properties (e.g., subrings,
large Galois groups, etc.) which may be exploited in attacks. We choose to work with Z[X]/(X n + 1) for
the following reasons:• From a performance perspective, there is no serious competition; the NTT-based multiplication supported by the parameters we chose for Kyber is at the same time very memory efficient and fasterthan any other algorithm for multiplication in polynomial rings.
• Lattice-based schemes using the ring Z[X]/(X n + 1) have been studied since at least [70]. When thenoise vectors are chosen as specified in [72], there have been no improved attacks against RLWE (or
MLWE) that use the underlying algebraic structure [87]. Furthermore, being based on MLWE, thealgebraic structure of Kyber is very different from that which was exploited in the attacks against
ideal lattices in [21, 34, 35]14 – we emphasize that the lattice problems underlying the hardness of
Kyber are not ideal lattices.
• Some of the additional algebraic structure of Z[X]/(X n + 1) is actually helpful against certain possible
attack vectors. As a simple example, it can be proven that when X n + 1 (almost) fully splits modulo
q, there do not exist polynomials in the ring that have small norm and many zeros in the NTT
representation—the existence of such polynomials for any q would weaken the security of MLWE.
Along the same lines, it can also be shown that X n + 1 does not have any factors (modulo q) ofsmall degree and small norm. For example, all (of which there are exponentially-many in n) factors√of degree n/2 have `2 -norm at least q. This prevents attacks that map the MLWE instance to alower-dimensional sub-ring without increasing the errors by too much.
• Finally, Z[X]/(X n + 1) is one of the most widely studied, and best understood, rings (along with othercyclotomic rings) in algebraic number theory. The fact that no attacks have been found against its use
for cryptosystems like Kyber makes it a much more conservative choice than some ring that is harderto analyze and may show weaknesses only after many more years of study.
6.4.6

Deterministic Noise.

Instead of adding noise e, e1 , and e2 , one can add “deterministic” noise by simply dropping bits. This is thebasis behind the “Learning with Rounding” (LWR) problem [15], which for certain parameters is as hard as
the LWE problem. We believe that asymptotically this is a sound approach, but the number of bits thatcan be dropped before significant decryption error is introduced is not very large (≈ 2 or 3). Relying on justthis assumption adds algebraic structure and may allow for a possibility of slightly improved attacks against
the scheme. For this reason, and the fact that generating noise is not a particularly costly operation, we did
not choose to potentially weaken the scheme to save a little time or have slightly shorter outputs.
14 Also, like the attacks against NTRU, these do not apply for the small parameters used public key encryption schemes.

35

7

Brief discussion of relevant results since Nov. 2017

In this section we briefly reference and comment on results relevant to Kyber that were published after theoriginal NIST PQC submission deadline in November 2017.
“small Kyber”. The idea of decreasing the modulus q from q = 7681 to q = 3329 for Kyber was alreadypresent in [100]; the paper refers to the resulting parameter set as “small-Kyber”. Other parameters chosen
in small-Kyber are different than what we describe in this round-2 update; in particular, small-Kyber does
not eliminate the public-key compression. More importantly, the polynomial multiplication using the NTT
algorithm is different in small-Kyber. Our multiplication algorithm uses the NTT algorithm to computethe decomposition of polynomials ai ∈ Zq [X]/(X 256 + 1) as(ai mod X 2 − r1 , . . . , ai mod X 2 − r128 ),and multiplication is performed using pointwise products modulo X 2 − rj . The algorithm in [100], on theother hand, divides the ai into two polynomials of degree 128 and performs a “full-splitting” NTT (i.e. modulo
X − ri0 ) on each of them. This method, which computes two (smaller) NTTs and some additional steps toreassemble the polynomials, has slightly worse performance than the original Kyber and considerably worse
than Kyber with our round-2 tweaks; though we believe that applying our new tweaks would speed up thealgorithm in [100] as well.
Fault attacks. In [93], Ravi, Roy, Bhasin, Chattopadhyay, and Mukhopadhyay describe a fault attack
against implementations of NewHope, Kyber, Frodo, and Dilithium on an ARM Cortex-M4. Specifically,
these attacks target the instructions loading the nonce in the PRF; skipping this load through a clock glitch
reuses the same nonce multiple times, which leads to efficiently breakable instances of the schemes. We
could have decided to modify the specification of Kyber to generate randomness in a different way. Indeed,
generating all randomness at once through one call to the PRF would thwart this specific attack and maybe
aid future fault-attack protected implementations. However, such a change to the specification would reduceparallelism and thus cost performance on many platforms. More importantly however, the benefits of sucha change are still completely unclear. It is not at all surprising that an implementation without any faultattack countermeasures, such as the one targeted in [93], succumbs to some fault attack. It is not clear
if the attack described in [93] is the most powerful attack and it is also unclear what serious fault-attack
countermeasures for any of the targeted schemes have to look like to thwart any realistic attack.
Improvements to sieving. Various papers have improved our understanding of attacks against latticebased cryptosystems and in particular the performance of sieving algorithms. Instead of listing those papers
here, we updated the discussion in Sections 5.1 and 5.2.
Attacks exploiting decapsulation failures. Since the beginning of the NIST PQC project, several papers
investigate chosen-ciphertext attacks exploiting decapsulation failures. Instead of listing those papers here,
we updated the discussion in Section 5.5.

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