Designs, Codes and Cryptography最新文献

GIFT-64 is a block cipher that has received a lot of attention from the community since its proposal in 2017. The attack on the highest number of rounds is a differential related-key attack on 26 rounds. We studied this attack, in particular with respect to some recent generic frameworks for improving key recovery, and we realised that this framework, combined with an efficient parallel key guessing of interesting subsets of the key and a consequent list merging applied to the partial solutions, can improve the complexity of the attack. We propose two different trade-offs, as a result of the improved key-recovery. We believe that the techniques are quite generic and that it is possible to apply them to improve other differential attacks.

Cyclic codes, as a significant subclass of linear codes, can be constructed and analyzed using algebraic methods. Due to its cyclic nature, they have efficient encoding and decoding algorithms. To date, cyclic codes have found applications in various domains, including consumer electronics, data storage systems, and communication systems. In this paper, we investigate the full automorphism groups of binary cyclic codes. A matrix presentation technique of cyclic codewords is introduced, which subsequently serves well for presenting binary cyclic codes of long lengths. These constructions are significantly useful in facilitating the determination of the full automorphism groups of binary cyclic codes of specified lengths.

A locally recoverable code of locality r over (mathbb {F}_{q}) is a code where every coordinate of a codeword can be recovered using the values of at most r other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length (n=mell ) and index (ell ) are linear codes that are invariant under cyclic shifts by (ell ) places. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of (mathbb {F}_q). Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let (alpha ) be a generator of (mathbb F_{3^m}setminus {0}), where m is a positive integer. Denote by (mathcal {C}_{(i_1,i_2,cdots , i_t)}) the cyclic code with generator polynomial (m_{alpha ^{i_1}}(x)m_{alpha ^{i_2}}(x)cdots m_{alpha ^{i_t}}(x)), where ({{m}_{alpha ^{i}}}(x)) is the minimal polynomial of ({{alpha }^{i}}) over ({{mathbb {F}}_{3}}). In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal ternary cyclic codes (mathcal {C}_{(0,1,e)}) and (mathcal {C}_{(1,e,s)}) with parameters ([3^m-1,3^m-frac{3m}{2}-2,4]), where (s=frac{3^m-1}{2}). In addition, by determining the solutions of certain equations and analyzing the irreducible factors of certain polynomials over (mathbb F_{3^m}), we present four classes of optimal ternary cyclic codes (mathcal {C}_{(2,e)}) and (mathcal {C}_{(1,e)}) with parameters ([3^m-1,3^m-2m-1,4]). We show that our new optimal cyclic codes are not covered by known ones.

In this paper, we promote Trojan message attacks against Merkle–Damgård hash functions and their concatenation combiner in quantum settings for the first time. Two main quantum scenarios are considered, involving the scenarios where a substantial amount of cheap quantum random access memory (qRAM) is available and where qRAM is limited and expensive to access. We first discuss the construction of diamond structures and analyze the corresponding time complexity in both of these quantum scenarios. Secondly, we propose quantum versions of the generic Trojan message attacks on Merkle–Damgård hash functions as well as their improved versions by combining with diamond structures and expandable messages, and then determine their cost. Finally, we propose Trojan message attack against Merkle–Damgård hash concatenation combiner in quantum setting. The results show that Trojan message attacks can be improved significantly with quantum computers under both scenarios, so the security of hash constructions in classical setting requires careful re-evaluation before being deployed to the post-quantum cryptography schemes.

Combinatorial neural (CN) codes are binary codes introduced firstly by Curto et al. for asymmetric channel, and then are further studied by Cotardo and Ravagnani under the metric (delta _r) (called asymmetric discrepancy) which measures the differentiation of codewords in CN codes. When (r>1), CN codes are different from the usual error-correcting codes in symmetric channel ((r=1)). In this paper, we focus on the optimality of some CN codes with (r>1). An upper bound for the size of CN codes with (delta _r=r+1) is deduced, by discussing the relationship between such CN codes and error-detecting codes for asymmetric channels, which is shown to be tight in this case. We also propose an improved Plotkin bound for CN codes. Notably, by applying symmetric designs related with Hadamard matrices, we not only generalize one former construction of optimal CN codes by bent functions obtained by Zhang et al. (IEEE Trans Inf Theory 69:5440–5448, 2023), but also obtain seven classes of new optimal CN codes meeting the improved Plotkin bound.

We study the complexity of the Code Equivalence Problem on linear error-correcting codes by relating its variants to isomorphism problems on other discrete structures—graphs, lattices, and matroids. Our main results are a fine-grained reduction from the Graph Isomorphism Problem to the Linear Code Equivalence Problem over any field (mathbb {F}), and a reduction from the Linear Code Equivalence Problem over any field (mathbb {F}_p) of prime, polynomially bounded order p to the Lattice Isomorphism Problem. Both of these reductions are simple and natural. We also give reductions between variants of the Code Equivalence Problem, and study the relationship between isomorphism problems on codes and linear matroids.

Projective Reed–Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field (mathbb {F}_q). We consider the relationship between projective Reed–Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when q is sufficiently large, the dimension of the hull of a projective Reed–Muller code is 1 less than the dimension of the code. We determine the dimension of the hull for a wider range of parameters and describe how this leads to a new proof of a recent result of Ruano and San-José.

Set systems with strongly restricted intersections, called (alpha )-intersecting families for a vector (alpha ), were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and eventown. Given a binary vector (alpha =(a_1, ldots , a_k)), a collection ({mathcal {F}}) of subsets over an n element set is an (alpha )-intersecting family modulo 2 if for each (i=1,2,ldots ,k), all i-wise intersections of distinct members in ({mathcal {F}}) have sizes with the same parity as (a_i). Let (f_alpha (n)) denote the maximum size of such a family. In this paper, we study the asymptotic behavior of (f_alpha (n)) when n goes to infinity. We show that if t is the maximum integer such that (a_t=1) and (2tle k), then (f_alpha (n)sim (t! n)^{1/t}). More importantly, we show that for any constant (c>0), as the length k goes larger, (f_alpha (n)) is upper bounded by (O (n^c)) for almost all (alpha ). Equivalently, no matter what k is, there are only finitely many (alpha ) satisfying (f_alpha (n)=Omega (n^c)). This answers an open problem raised by Johnston and O’Neill in 2023. All of our results can be generalized to modulo p setting for any prime p smoothly.

In this work we present results on the classification of (mathbb {F}_{q^n})-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD codes (mathcal {C}=langle X^{q^t}, F(X), G(X) rangle subseteq mathcal {L}_{n,q}) of exceptional type, i.e. such that (mathcal {C}) is MRD over infinitely many extensions of the base field. These results partially address a conjecture of Bartoli, Zini and Zullo in 2023.