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UDC 004.453.2 DOI 10.17223/2226308X/14/39
TOWARDS THE SECURITY OF McEliece's CRYPTOSYSTEM BASED ON HERMITIAN SUBFIELD SUBCODES1
G. P. Nagy, S. El Khalfaoui
The purpose of this paper is to provide a comprehensive security analysis for the parameter selection process, which involves the computational cost of the information set decoding algorithm using the parameters of subfield subcodes of 1-point Hermitian codes.
Keywords: code-based cryptography, McEliece Cryptosystem, Hermitian subfield subcodes, Schur square dimension.
1. Introduction
Recently, there has been a big amount of research addressed to quantum computers that use quantum mechanical techniques to solve hard computational problems in mathematics [1]. The existence of these powerful machines threaten many of the public-key cryptosystem that are widely in use [2]. McEliece [3] introduced the first code-based public-key cryptosystem in 1978. The crucial issues in cryptography today is to reduce the key size and improve the security level of the McEliece cryptosystem, which is a promising cryptographic scheme for the post-quantum era [4]. Error correcting codes, used in code-based cryptographic protocols, must have efficient decoding algorithms. A rich class of such codes is the family algebraic-geometric (AG) codes, their subcodes and subfield subcodes. This includes the generalized Reed — Solomon codes, the alternant codes, the binary Goppa codes and BCH codes. See [5] for a survey on the decoding of AG codes.
The authors of [6-8] provided polynomial-time attacks against the McEliece cryptosystem that relies either on AG codes or on their subcodes. In general, evaluation codes do not behave like random codes which demonstrate the quite range of attacks proposed against the McEliece cryptosystem based on AG codes. The approach given in [6, 8] is inspired by the so-called filtration attacks that rely on computing the Schur product that make AG codes distinguishable form random ones. Wieschebrink [9] used this observation to provide an attack against McEliece scheme based on subcodes of GRS codes [10]. Many attacks have been founded on this argument, and have employed a
1Project no. 2018-1.2.1-NKP-2018-00004 has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the 2018-1.2.1-NKP funding scheme.
combination of powerful techniques such as the filtration method, an error-correcting pair (ECP) or an error-correcting array (ECA), that lead to a key recovery attack or a blind reconstruction of a decoding algorithm [6, 8, 11]. These vulnerabilities are based on two operations: Schur product and s-closure. In some cases, the Schur filtration method can expand the latter to develop an efficient decoding algorithm.
The purpose of this paper is to provide a comprehensive security analysis for the parameter selection process, which involves the computational cost of the information set decoding (ISD) algorithm using Hermitian subfield subcode parameters. Our approach focuses on the optimal parameters that improve the key size for a given security level. Furthermore, due to practical considerations, the key size of several parameter selections is compared to that of the classical McEliece cryptosystem submitted to NIST [4] for the same security level. Besides, we identify the Hermitian subfield subcodes parameters that achieve a Schur square dimension roughly equal to that of random codes. This technique is employed in the so-called distinguisher attack, that allows the attacker to determine the Schur square dimension of the code used as a public key.
2. Preliminaries
Let q be a prime power. A q-ary linear code of length n is a linear subspace C ^ F^. The dimension of C is denoted by k, C is usually given by its n x k generator matrix G, or its n x (n — k) parity check matrix H:
C = {Gx : x e Fkq} = {y e F^" : HTy = 0}.
The minimal distance of C is d(C) = min{wt(x) : x e C \ {0}}, where wt(x) denotes the Hamming weight of the vector x. If 2t < d(C), then for each y e F^, there is at most one pair x, e e F^ of vectors such that x e C, wt(e) ^ t and y = x + e. Define the map DCt : F^" ^CU {*} by y M x if the decomposition y = x + e exists, and y M * otherwise. We call DC,t a nearest neighbor decoding of C, correcting up to t errors. In general, known nearest neighbor decoding algorithms have exponential time complexity in the size of the input G,t,y. The seminal result by Berlekamp, McEliece and van Tilborg [12] shows that the decoding problem is NP-complete even for the binary case q = 2. The simplest general decoding technique is called information set decoding (ISD), with goes back to an old algorithm of Prange [13]. This algorithm has time complexity
Gprange(n, k,t) = Q/ t ^Coauss(n, k,q),
where Gcauss(n, k, q) is the time complexity of the Gauss —Jordan elimination of a kxn matrix over Fq. There are many improvements of Prange's algorithms, but all known variants have the same asymptotic behavior, see [14] and the references therein.
Let us fix the parameters n, k,t and q. The McEliece cryptographic scheme [3], or in general code-based cryptosystems has a [n, k, t]q linear code C as public key, and an efficient decoding algorithm DC,t as private key. Usually, C is given by a generator matrix
G =
4 G0
in systematic form, that is, the key size is
k(n — k)[log2 (q)].
The plain text message m e Fq is encrypted to c = Gm + e e F^, where e is a random element of weight t in F™. The security of the scheme relies on two facts:
To prevent a message recovery attack, the parameters n,k,t and q must be chosen such that the time complexity of ISD exceeds a given level L of security. This level is usually measured in bits, and corresponds to the time complexity of breaking an L-bits symmetric-key block cipher, like AES. Since it is not the purpose of this paper to give a detailed cryptanalysis of symmetric-key block ciphers, we interpret this condition as
:)/(n-k)> •
2) To prevent a key recovery attack, it should not be possible to give an efficient decoding algorithm for C. At this point, we not only assume the knowledge of G, but also the technique that was used to construct C from a given family of codes. In the classic McEliece proposal, G = PG\S, where Gi is the generator matrix of the binary Goppa code r(L,g). Here, g = g(X) is a polynomial over Fq with no repeated roots, L is an ordered n-tuple of distinct elements of Fq that are no roots of g, P is a random n x n permutation matrix, and S is a random k x k invertible matrix; L,g, P, S are parts of the private key; they are supposed to be kept secret for a long period of time.
The second requirement implies that the public key C must be indistinguishable from a random subspace of F^. In general, distinguishing attacks do not necessarily lead to message or key recovery attacks. However, often they do, and cryptosystems must resist to distinguishing attacks.
3. The Schur product distinguisher
We briefly introduce some notions on attack techniques that allow us to describe some important results stated in [6].
Definition 1. Given two elements a = (ai,..., an) and b = (bi, . . ., bn) in F^, the Schur product is the component-wise
a * b = (aibi, • • •, anbn)
product on F^^. For two linear subspaces A, B C F^ their Schur product is the linear subspace
A * B = SpanFq{a * b : a E A and b E B}.
If B = A, then A * A is denoted as A*2, and we define A*1 by induction for any positive integer t.
One of the main results in [15] is that when the length n is such that n > k(k + 1)/2, the dimension of the square of the random code C is exactly k(k + 1)/2, with probability tending to 1 as n — k(k + 1)/2 approaches infinity. More precisely, we define F(n,k) as the family of linear codes of length n and dimension k. Let n : N —> N be such that n(k) > k(k + 1)/2. Then there exits a constant y E R such that, for all large enough k,
* C = __i. 1 > I _ 2 ,(n(k)-k(k+ i)/2)
Pr (dimC * C = 2+1)) ^ 1 - 2Y(
2
where C is chosen uniformly at random from F(n(k), k). This observation serves as a useful distinguisher between random linear subspaces and those with a rich algebraic structure. Definition 2. Let C be an [n, k]q linear code. We say that C is s-good, if
dim(C * C) = dim(Cx * C±) = n.
4. Algebraic-geometric codes: constructions and parameters
Algebraic-geometric codes are linear error-correcting codes constructed from algebraic curves over finite fields. They are defined by evaluating functions or by using residues of differentials. Their parameters can be derived from well-known theorems of algebraic geometry. Our notation and terminology on algebraic plane curves over finite fields, their function fields, divisors and Riemann —Roch spaces are standard, see for instance [16].
Let X be an algebraic curve, i.e., an affine or projective variety of dimension one, which is absolutely irreducible and nonsingular, and whose defining equations are (homogeneous) polynomials with coefficients in Fq. Let g = g(X) be the genus of X, Fq(X) denotes the function field of X. A divisor D of X is a formal sum D = niPi + ... + nkPk, where ni,..., nk e Z and P1,..., Pk are places of Fq(X). If n1..., nk ^ 0, then D y 0. If D, E are two divisors and D — E y 0, then D y E. For a non-zero function f in the function field Fq(X) and a place P, vP(f) stands for the order of f at P. If vP(f) > 0, then P is a zero of f, while if vP(f) < 0, then P is a pole of f with multiplicity —vP(f). The principal divisor of a non-zero function f is Div(f) = vP(f )P.
P
For a divisor D, the associated Riemann —Roch space L(D) is the vector space
L(D) = {f e Fq(X) \ {0} : Div(f) y —D} U {0}.
The dimension €(D) of L(D) is given by the Riemann-Roch Theorem [16, Theorem 1.1.15]:
£(D) = £(W — D) + deg D — g + 1,
where W is a canonical divisor. We denote the set of differentials on X by Q. The differential space of the dividor D is
Q(D) = {dh e Q : Div(dh) y A} U {0}.
In the following, Pi, P2,..., Pn are pairwise distinct places on X, and D is the divisor D = P1 + ... + Pn. Let G be another divisor with support disjoint from D. We define two types of AG codes, the functional and the differential codes, respectively:
Gl(D, G) = {(f (Pi),..., f (Pn)) : f e L(G)} , Cn (D,G) = {(resp-L (w),..., respn (w)) : w e Q(G — D)} .
These codes are dual to each other, and Gq(D, G) = CL(D, K + D — G) for an well-chosen canonical divisor K. The Riemann — Roch theorem enables us to estimate the dimension and the minimum distance of AG codes:
deg(G) — g + 1, 0 ^ deg(G) ^ 2g — 2, = deg(G) — g + 1, 2g — 2 ^ deg(G) ^ n, ^ deg(G) — g + 1, n ^ deg(G) ^ n + 2g — 2.
The minimum distance of a functional code is at least its designed minimum distance
= n — deg(G).
AG codes have polynomial time decoding algorithms, that can correct up to t = — g)/2 errors [5]. However, they are vulnerable to Schur filtration attacks. In particular, AG codes are far from being s-good. The following proposition is derived from [17, Theorem 6].
Proposition 1. Let G and G' be two divisors on the curve X both with disjoint support with the divisor D and such that deg G ^ 2g + 1 and deg G' ^ 2g. Then
Cl(D, G) * CL(D, G') = CL(D, G + G').
In particular,
dim(CL(D, G)*2) ^ 2 dim(CL(D, G)) + g — 1.
Let C be a linear subspace of the functional code CL(D,G). The Schur filtration attack constructs an effective decoding algorithm using a system of linear subspaces
Wij = {z e F^ : z * C" ^ C*j}.
Here, i, j ^ 1 may be arbitrary. Clearly, if i < j, then C*(j-i) ^ Wi)j-.
5. Subfield subcodes of 1-point Hermitian codes
Reed — Solomon codes form a well-known subclass of AG codes. In this section, we present the construction of Hermitian codes, another subclass of interest. Let Hq be a Hermitian curve over a finite field Fq2. In affine coordinates, Hq is given by the equation
Hq : Yq + Y = Xq+1.
It is a non-singular curve, and its genus is g = q(q — 1)/2 by the genus formula. Hq has one point Px = (0:1:0) at infinity, and q3 affine rational points P1,...,Pq3. This makes the class of Hermitian curves interesting since they attain the maximal number of rational points for Hasse —Weil bound [18]. Such curves are called Fq2-maximal. Xing and Stichtenoth [19] showed that for fixed q, the genus of a Fq2-maximal curve is ^ q(q — 1)/2, and equality holds if and only if X is isomorphic to Hq.
Definition 3. Let s be a positive integer. The Fq2-linear code CL(D,sPx) of length n = q3 is called a Hermitian 1-point code.
In general, it is a hard computational problem to determine a bases of the Riemann — Roch space L(D). For 1-point divisors D = sPx of the Hermitian curve, such a basis is given in [18, Theorem 10.4]. The dual of CL(D, sPx) is a 1-point Hermitian code too, with parameter
= n + 2g — 2 — s = q(q2 — q — 1) — s.
Clearly, 1-point Hermitian codes form an increasing series of linear subspaces of F^.
Let m be a positive integer and r = qm. Let C be a linear [n,k,t] code over Fr. The Fr/Fq subfield subcode of C is defined as
C|Fq = Cn F™
The true dimension k* of C|Fq is hard to determine, but the bound
k* ^ n — m(n — k)
is straightforward. Any algorithm that can decode up to t errors of C can be used to correct up to t errors of the subfield subcode C|Fq. In this paper, we examine the class
Cq (s) = CL(D,sPx)|Fq
of Hermitian subfield subcodes, and propose their usage in code-based cryptosystems. In [20], we determined the true dimension of Cq(s) for some specific values of s. In [21], we conducted an experimental study to analyze the true dimension of Cq(s) for q ^ 16, and concluded that the datasets can be best approximated by the extreme value distribution.
Here, our focus is on the resistance of Cq(s) to the Schur distinguishing attack. We determine the parameters such that the key size is significantly smaller than in the classic McEliece scheme. Notice that the Schur filtration technique may be used for key recovery attacks on subfield subcodes of AG codes, as well, provided the degree m of the field extension, and the genus of the underlying algebraic curve are small. In our case, m =2 is small, but the genus is the largest possible with fixed field Fq2 and maximal length n.
Proposition 2. Let q be a prime power and let Cq(s) = CL(D, sP»)|Fq be a 1-point Hermitian subfield subcode. There are positive integers aq, bq such that Cq(s) is s-good if and only if aq ^ s ^ bq.
We conducted numerical experiments to determine the values aq,bq for q ^ 16, see Table 1. The results motivate the following conjecture.
Open problem 1. Let q be a prime power and let Cq(s) = CL(D, sP^)|Fq be a 1-point Hermitian subfield subcode. Then dim(Cq(s)x * Cq(s)x) = q3 if and only if s ^ q3 — q — 1.
Table 1
Interval bounds for s-goodness
q 4 5 7 8 9 11 13 16
aq bq 45 59 72 119 192 335 315 503 400 719 720 1319 1176 2183 2 295 4 079
6. Comparative of Hermitian subfield subcodes to McEliece cryptosystem:
key size and security level
The National Institute of Standards and Technology (NIST) has recently begun a selection process to standardize asymmetric cryptosystems resistant to quantum computer attacks [4]. Code-based cryptosystems are promising candidates for NIST selection.
In this section, we analyse the computational cost of solving the ISD problem for various sets of parameters relevant to post-quantum cryptography. To do so, we consider classical McEliece cryptosystem variants built on Goppa codes. The parameters for cryptosystems reported in [22] are designed to be comparable to the computational cost required to break AES-128 (Category 1), AES-192 (Category 3), and AES-256 (Category 5). The Tables 2 and 3 summarize the code parameters of Classical McEliece cryptosystem submitted to NIST round 2-code-based cryptosystems, and those of 1-point Hermitian subfield subcodes Cq(s) (code length n, dimension k, and error-capability t), as well as the computational cost of Prange's ISD algorithm, expressed as log2 (bit operations) with the public key size.
Table 2
Classic McEliece cryptosystem
Classic McEliece n k t Prange complexity Key size (bit)
Category 1 3 488 2 720 64 142.78 2088 960
Category 3 4 608 3 360 96 184.89 4193 280
Category 5 6 688 5 024 128 262.35 8 359 936
6 960 5413 119 263.44 8373911
8192 6 528 128 300.14 10 862 592
Table 3
McEliece cryptosystem based on s-good 1-point Hermitian subfield subcodes
Code Type n k t Prange complexity Key size (bit)
Category 1 Cn(1174) 1331 927 78 142.33 1 123 524
Category 3 Ci3(2039) 2197 1735 79 185.89 3 206 280
C16(3 980) 4 096 3 634 58 187.40 6 715 632
Category 5 Cis(1861) 2197 1398 168 263.01 4 468 008
Ci6(3 874) 4 096 3 422 111 300.65 9 225 712
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