Designs, Codes and Cryptography最新文献

Cryptanalysis of modern symmetric ciphers may be done by using linear equation systems with multiple right hand sides, which describe the encryption process. The tool was introduced by Raddum and Semaev (Des Codes Cryptogr 49(1):147–160, 2008) where several solving methods were developed. In this work, the probabilities are ascribed to the right hand sides and a statistical attack is then applied. The new approach is a multivariate generalisation of the correlation attack by Siegenthaler (IEEE Trans Comput C 49(1):81–85, 1985). A fast version of the attack is provided too. It may be viewed as an extension of the fast correlation attack in Meier and Staffelbach (J Cryptol 1(3):159–176, 1989), based on exploiting so called parity-checks for linear recurrences. Parity-checks are a particular case of the relations that we introduce in the present work. The notion of a relation is irrelevant to linear recurrences. We show how to apply the method to some LFSR-based stream ciphers including those from the Grain family. The new method generally requires a lower number of the keystream bits to recover the initial states than other techniques reported in the literature.

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters.

Locally repairable codes (LRCs) are linear codes with locality properties for code symbols, which have important applications in distributed storage systems. In this paper, we completely classify all the possible code parameters of optimal ternary LRCs achieving the Singleton-like bound proposed by Gopalan et al. Explicit constructions of optimal ternary LRCs are given for each group of possible code parameters. Moreover, it is also proved that optimal ternary LRCs with maximal minimum distance 6 are unique up to the equivalence of linear codes.

Let (n_k(s)) be the maximal length n such that a quaternary additive ([n,k,n-s]_4)-code exists. We solve a natural asymptotic problem by determining the lim sup (lambda _k) of (n_k(s)/s) for s going to infinity, and the smallest value of s such that (n_k(s)/s=lambda _k.) Our new family of quaternary additive codes has parameters ([4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3cdot 2^{2k-2}]_4) (where (k=l/2) and l is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation with inner code ([3,2,2]_2) meet the Griesmer bound with equality. The proof is in terms of multisets of lines in (PG(l-1,2)).

The circular external difference family and its strong version are of great significance both in theory and in applications. In this paper, we apply the classical cyclotomic construction to the circular external differnece family and exhibit several concrete examples, in particular constructing an infinite family. Furthermore, we prove that all strong circular external differnece families are constructed by patching together several strong external difference families consisting of two subsets, thereby solving the open problem raised by Veitch and Stinson. We also present a new result on the non-existence of a certain type of strong external difference families.

We study the sequences generated by prefer-one rule with different initial vectors. Firstly, we give upper bounds of their periods and for initial vectors with Hamming weight one, we prove that the generated sequences are modified de Bruijn sequences. Moreover, for two of them, we give the truth tables of their feedback functions. We also investigate the feedback functions of prefer-one de Bruijn sequences. For order n prefer-one de Bruijn sequence, we give linear and quadratic terms in its feedback function and prove that the number of degree (n-2) terms has the same parity as n. The statistical result for small n shows that about half of all terms occur in the feedback functions.

We introduce a formal framework to study the multiple unicast problem for a coded network in which the network code is linear over a finite field and fixed. We show that the problem corresponds to an interference alignment problem over a finite field. In this context, we establish an outer bound for the achievable rate region and provide examples of networks where the bound is sharp. We finally give evidence of the crucial role played by the field characteristic in the problem.

Two skew Hadamard matrices are considered SH-equivalent if they are similar by a signed permutation matrix. This paper determines the number of SH-inequivalent skew Hadamard matrices of order 36 for some types. We also study ternary self-dual codes and association schemes constructed from the skew Hadamard matrices of order 36.

For projective Reed–Muller-type codes we give a global duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide a global duality theorem for projective Reed–Muller-type codes over Gorenstein vanishing ideals, generalizing the known case where the vanishing ideal is a complete intersection. We classify self dual Reed–Muller-type codes over Gorenstein ideals using the regularity and a parity check matrix. For projective evaluation codes, we give a duality theorem inspired by that of affine evaluation codes. We show how to compute the regularity index of the r-th generalized Hamming weight function in terms of the standard indicator functions of the set of evaluation points.

Let (mathbb {F}_{q^n}) represent the finite field with (q^n) elements. In this paper, our focus is on determining the number of (mathbb {F}_{q^n})-rational points for two specific objects: an affine Artin–Schreier curve given by the equation (y^q-y = x(x^{q^i}-x)-lambda ), and an Artin–Schreier hypersurface given by the equation (y^q-y=sum _{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-lambda ). Additionally, we establish that the Weil bound is only achieved in these cases when the trace of the element (lambda in mathbb {F}_{q^n}) over the subfield (mathbb {F}_q) is equal to zero.