Applicable Algebra in Engineering Communication and Computing最新文献

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in (mathbb {K}[x]). The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type (f(alpha )=f(beta )) for (alpha ,beta) in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

This paper presents two modifications for Loidreau’s cryptosystem, a rank metric-based cryptosystem constructed by using Gabidulin codes in the McEliece setting. Recently a polynomial-time key recovery attack was proposed to break this cryptosystem in some cases. To prevent this attack, we propose the use of subcodes to disguise the secret codes in Modification I. In Modification II, we choose a random matrix of low column rank to mix with the secret matrix. Our analysis shows that these two modifications can both resist the existing structural attacks. Furthermore, these modifications have a much more compact representation of public keys compared to Classic McEliece, which has been selected into the fourth round of the NIST-PQC project.

We consider Noetherian operators in the context of symbolic computation. Upon utilizing the theory of holonomic ({mathcal D})-modules, we present a new method for computing Noetherian operators associated to a zero-dimensional ideal. An effective algorithm that consists mainly of linear algebra techniques is proposed for computing them. Moreover, as applications, new computation methods of polynomial ideals are discussed by utilizing the Noetherian operators.

The weight, balancedness and nonlinearity are important properties of Boolean functions, but they can be difficult to determine in general. In this paper, we study how to compute them for two classes of functions where these problems are more tractable. In particular, we study functions of degree three and the so-called “splitting” functions. The latter are functions that can be written as the sum of two functions defined over disjoint sets of variables. We show how, for splitting functions, studying these properties reduces to the study of simpler functions. We provide then a procedure to compute the weight of a cubic Boolean function. We show computationally that, for a cubic Boolean function with limited number of terms, this procedure is on average significantly more efficient than some other methods.

In this paper, using Gallai’s Theorem and the notion of strong resolving graph, we determine the strong metric dimension in annihilating-ideal graph of commutative rings. For reduced rings, an explicit formula is given and for non-reduced rings, under some conditions, strong metric dimension is computed.

Let C be an extremal Type III or IV code and (D_{w}) be the support design of C for weight w. We introduce the numbers, (delta (C)) and s(C), as follows: (delta (C)) is the largest integer t such that, for all weights, (D_{w}) is a t-design; s(C) denotes the largest integer t such that w exists and (D_{w}) is a t-design. Herein, we consider the possible values of (delta (C)) and s(C).

Maximum distance separable (MDS) codes have the highest possible error-correcting capability among codes with the same length and size. Let (gamma ) be nonzero in (mathbb {F}_{2^m}.) We consider all cyclic and ((1+ugamma ))-constacyclic codes of length (2^s) over (mathbb {F}_{2^m}+umathbb {F}_{2^m}) with their Lee distance and investigate all the cases whether the corresponding Gray images are MDS by giving an analogue of the Singleton bound for codes over (mathbb {F}_{2^m}+umathbb {F}_{2^m}) with the Lee distance through Gray map.

We describe software for symbolic computations that we developed in order to find Hamiltonian operators for Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, and verify their compatibility. The computation involves nonlocal (integro-differential) operators, for which specific canonical forms and algorithms have been used.

In this work we define and study binary codes (C_{q,k}) and (overline{C_{q,k}}) obtained from neighborhood designs of Paley-type bipartite graphs P(q, k) and their complements, respectively for q an odd prime. We prove that for some values of q and k the codes ({C}_{q,k}) are self-dual and the codes (overline{C_{q,k}}) are self-orthogonal. Most of these codes tend to be with optimal or near optimal parameters. Next, we extend the codes (C_{q,k}) to get doubly even self dual codes and find that most of these codes are extremal.