Applicable Algebra in Engineering Communication and Computing最新文献

An almost perfect non-linear (APN) function over (mathbb {F}_{2^n}) is called exceptional APN if it remains APN over infinitely many extensions of (mathbb {F}_{2^n}). Exceptional APN functions have attracted attention of many researchers in the last decades. While the classification of exceptional APN monomials has been done by Hernando and McGuire, it has been conjectured by Aubry, McGuire and Rodier that up to equivalence, the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. Since then, many partial results have been on classifying non-exceptional APN polynomials. In this paper, for the classification of the exceptional property of APN functions, we introduce a new method that uses techniques from curves over finite fields. Then, we apply the method with Eisenstein’s irreducibility criterion and Kummer’s theorem to obtain new non-exceptional APN functions.

Low-Rank Parity-Check (LRPC) codes are a class of rank metric codes that have many applications specifically in network coding and cryptography. Recently, LRPC codes have been extended to Galois rings which are a specific case of finite rings. In this paper, we first define LRPC codes over finite commutative local rings, which are bricks of finite rings, with an efficient decoder. We improve the theoretical bound of the failure probability of the decoder. Then, we extend the work to arbitrary finite commutative rings. Certain conditions are generally used to ensure the success of the decoder. Over finite fields, one of these conditions is to choose a prime number as the extension degree of the Galois field. We have shown that one can construct LRPC codes without this condition on the degree of Galois extension.

In complex real-world networks, the relation among vertices (people) changes over time. Even with millions of vertices, adding new vertices or deleting a few previous ones can drastically change the network’s dynamics. The Iterated Local Transitivity model is a deterministic model based on the principle of transitivity and local interaction among people. The same has been extended to signed social networks. Let (Sigma) be a signed graph with underlying graph (G = (V, E)) and a function (sigma :Erightarrow {+,-}) assigning signs to the edges. We determine the relation between the characteristic polynomials of signed graph (Sigma) and the signed graph obtained from (Sigma) by adding (deleting) vertices or by adding (deleting) edges. Consequently, we present a recurrence relation for a characteristic polynomial of the Iterated Local Transitivity model for signed graphs.

Abstract

As an important class of cyclic codes, BCH codes are widely employed in satellite communications, DVDs, CD, DAT etc. In this paper, we determine the dimension of BCH codes of length (frac{q^{2m}-1}{q+1}) over the finite fields ({mathbb {F}}_q) . We settle a conjecture about the largest q-cyclotomic coset leader modulo n which was proposed by Wu et al. We also get the second largest q-cyclotomic coset leader modulo n if m is odd. Moreover, we investigate the parameters of ({mathcal {C}}_{(n,q,delta _i)}) ( ({mathcal {C}}_{(n,q,delta _i)}^perp) ) for (i=1,2) and ({mathcal {C}}_{(n,q,delta )}) ( ({mathcal {C}}_{(n,q,delta )}^perp) ) for (2le delta le q-1) . What’s more, we obtain many (almost) optimal codes.

Linear codes are widely studied due to their important applications in authentication codes, association schemes and strongly regular graphs, etc. In this paper, a class of at most three-weight linear codes is constructed by selecting a new defining set, then the parameters and weight distributions of codes are determined by exponential sums. Results show that almost all the linear codes we constructed are minimal and we also describe the access structures of the secret sharing schemes based on their dual. Especially, the new binary code is a two-weight projective code and based on which a strongly regular graph with new parameters is designed.

As a class of linear codes, cyclic codes are widely used in communication systems, consumer electronics and data storage systems due to their favorable properties. In this paper, we construct two classes of optimal p-ary cyclic codes with parameters ([p^m-1, p^m-frac{3m}{2}-2, 4]) by analyzing the solutions of certain polynomials over finite fields. Furthermore, we propose an efficient method to determine the optimality of the 7-ary cyclic code (mathcal {C}_{(0,1,e)}) and present three classes of optimal codes with parameters ([7^m-1,7^m-2m-2,4]). Additionally, we provide the weight distribution of one class of their duals.

At present, GLV/GLS scalar multiplication mainly focuses on elliptic curves in Weierstrass form, attempting to find and construct more and more efficiently computable endomorphism. In this paper, we investigate the application of the GLV/GLS scalar multiplication technique to Jacobi Quartic curves. Firstly, we present a concrete construction of efficiently computable endomorphisms for this type of curves over prime fields by exploiting birational equivalence between curves, and obtain a 2-dimensional GLV method. Secondly, we consider the quadratic twists of elliptic curves. By using birational equivalence and Frobenius mapping between curves, we present methods to construct efficiently computable endomorphisms for this type of curves over the quadratic extension field, and obtain a 2-dimensional GLS method. Finally, we obtain a 4-dimensional GLV method on elliptic curves with j-invariant 0 or 1728 by using higher degree twists. The experimental results show that the speedups of the 2-dimensional GLV method and 4-dimensional GLV method compared to 5-NAF method exceed (37.2%) and (109.4%) for Jacobi Quartic curves, respectively. At the same time, when utilizing one of the proposed methods, the scalar multiplication on Jacobi Quartic curves is consistently more efficient than on elliptic curves in Weierstrass form.