Applicable Algebra in Engineering Communication and Computing最新文献

The LG cryptosystem is a public-key encryption scheme in the rank metric using the recent family of (mathbf{mathrm {lambda }}-)Gabidulin codes and introduced in 2019 by Lau and Tan. In this paper, we present a cryptanalysis showing that the security of several parameters of the scheme have been overestimated. We also show the existence of some weak keys allowing an attacker to find in polynomial time an alternative private key.

Type II ({mathbb {Z}}_4)-codes are a class of self-dual ({mathbb {Z}}_4)-codes with Euclidean weights divisible by eight. A Type II ({mathbb {Z}}_4)-code of length n is extremal if its minimum Euclidean weight is equal to (8leftlfloor frac{n}{24}rightrfloor +8.) A small number of such codes is known for lengths greater than or equal to 48. Based on the doubling method, in this paper we develop a method to construct new extremal Type II ({mathbb {Z}}_4)-codes starting from a free extremal Type II ({mathbb {Z}}_4)-code of length 48, 56 or 64. Using this method, we construct extremal Type II ({mathbb {Z}}_4)-codes of length 64 and type (4^{31}2^2). Extremal Type II ({mathbb {Z}}_4)-codes of length 64 of this type were not known before. Moreover, the residue codes of the constructed extremal ({mathbb {Z}}_4)-codes are best known [64, 31] binary codes and the supports of the minimum weight codewords of the residue code and the torsion code of one of these codes yields self-orthogonal 1-design.

Given an algebraic germ of a plane curve at the origin, in terms of a bivariate polynomial, we analyze the complexity of computing an irreducible decomposition up to any given truncation order. With a suitable representation of the irreducible components, and whenever the characteristic of the ground field is zero or larger than the degree of the germ, we design a new algorithm that involves a nearly linear number of arithmetic operations in the ground field plus a small amount of irreducible univariate polynomial factorizations.

In this paper, two classes of permutation pentanomials over finite fields (mathbb {F}_{p^{2m}}) are investigated by transforming the permutation property of polynomials to verifying that some low-degree equations has no solutions in the unit circle. Furthermore, based on the study of the algebraic curves for fractional polynomials, several classes of permutation pentanomials and hexanomials over (mathbb {F}_{5^{2m}}) are discovered. Additionally, we obtain some new permutation pentanomials, quadrinomials and octonomials over (mathbb {F}_{5^{2m}}) from known permutation polynomials in the unit circle.

In this paper, we define the complex-type k-Padovan numbers and then give the relationships between the (left( 1,k-1right))-bonacci numbers, the k -Padovan numbers and the complex-type k-Padovan numbers by matrix method. In addition, we study the complex-type k-Padovan sequence modulo m and then we show that for some m the periods of the complex type k-Padovan and k-Padovan sequences modulo m are related. Furthermore, we extend the complex-type k-Padovan sequences to groups. Finally, we obtain the periods of the complex-type 4, 5, 6-Padovan sequences in the semidihedral group (SD_{2^{m}}), (left( mge 4right)) with respect to the generating pairs (left( x,yright)) and (left( y,xright)).

Let (mathbb{F}_q) be a finite field of characteristic p. In this paper we prove that the c-Boomerang Uniformity, (c ne 0), for all permutation monomials (x^d), where (d > 1) and (p not mid d), is bounded by (left{ begin{array}{ll} d^2, & c^2 ne 1, d cdot (d - 1), & c = - 1, d cdot (d - 2), & c = 1 end{array} right} .) Further, we utilize this bound to estimate the c-boomerang uniformity of a large class of generalized triangular dynamical systems, a polynomial-based approach to describe cryptographic permutations of (mathbb{F}_{q}^{n}), including the well-known substitution–permutation network.