Irreducible Compositions of Polynomials over Finite Fields

The paper studies constructions of irreducible polynomials over finite fields using polynomial composition method

Complexity of Computing Quadratic Nonresidues

This note provides new methods for constructing quadratic nonresidues in finite fields of characteristic p. It will be shown that there is an effective deterministic polynomial time algorithm for constructing quadratic nonresidues in finite fields.Comment: References and Improvement

Sequences of irreducible polynomials without prescribed coefficients over finite fields of even characteristic

In this paper we deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the $Q$-transform employed previously by Varshamov and Meyn for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper we construct infinite sequences of irreducible polynomials of non-decreasing degree starting from any irreducible polynomial.Comment: 6 pages; expanded proofs; examples added; minor fixe

Some Algebraic Properties of a Subclass of Finite Normal Form Games

We study the problem of computing all Nash equilibria of a subclass of finite normal form games. With algebraic characterization of the games, we present a method for computing all its Nash equilibria. Further, we present a method for deciding membership to the class of games with its related results. An appendix, containing an example to show working of each of the presented methods, concludes the work.Comment: 18 Page

Subsets of Fq[x]\mathbb{F}_q[x] free of 3-term geometric progressions

Several recent papers have considered the Ramsey-theoretic problem of how large a subset of integers can be without containing any 3-term geometric progressions. This problem has also recently been generalized to number fields, determining bounds on the greatest possible density of ideals avoiding geometric progressions. We study the analogous problem over $\mathbb{F}_q[x]$, first constructing a set greedily which avoids these progressions and calculating its density, and then considering bounds on the upper density of subsets of $\mathbb{F}_q[x]$ which avoid 3-term geometric progressions. This new setting gives us a parameter $q$ to vary and study how our bounds converge to 1 as it changes, and finite characteristic introduces some extra combinatorial structure that increases the tractibility of common questions in this area.Comment: 13 pages. 1 figure, 3 table

Formal Analysis of Galois Field Arithmetics - Parallel Verification and Reverse Engineering

Galois field (GF) arithmetic circuits find numerous applications in communications, signal processing, and security engineering. Formal verification techniques of GF circuits are scarce and limited to circuits with known bit positions of the primary inputs and outputs. They also require knowledge of the irreducible polynomial $P(x)$, which affects final hardware implementation. This paper presents a computer algebra technique that performs verification and reverse engineering of GF($2^m$) multipliers directly from the gate-level implementation. The approach is based on extracting a unique irreducible polynomial in a parallel fashion and proceeds in three steps: 1) determine the bit position of the output bits; 2) determine the bit position of the input bits; and 3) extract the irreducible polynomial used in the design. We demonstrate that this method is able to reverse engineer GF($2^m$) multipliers in \textit{m} threads. Experiments performed on synthesized \textit{Mastrovito} and \textit{Montgomery} multipliers with different $P(x)$, including NIST-recommended polynomials, demonstrate high efficiency of the proposed method.Comment: To appear in IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. (TCAD'18); extended version. arXiv admin note: text overlap with arXiv:1611.0510

Enumerating all the Irreducible Polynomials over Finite Field

In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$ is the size of the output.} time cost per element. Our algorithm is based on an improved algorithm for enumerating all the Lyndon words of length $n$ in linear delay time and the known reduction of Lyndon words to irreducible polynomials

Network-Error Correcting Codes using Small Fields

Existing construction algorithms of block network-error correcting codes require a rather large field size, which grows with the size of the network and the number of sinks, and thereby can be prohibitive in large networks. In this work, we give an algorithm which, starting from a given network-error correcting code, can obtain another network code using a small field, with the same error correcting capability as the original code. An algorithm for designing network codes using small field sizes proposed recently by Ebrahimi and Fragouli can be seen as a special case of our algorithm. The major step in our algorithm is to find a least degree irreducible polynomial which is coprime to another large degree polynomial. We utilize the algebraic properties of finite fields to implement this step so that it becomes much faster than the brute-force method. As a result the algorithm given by Ebrahimi and Fragouli is also quickened.Comment: Minor changes from previous versio

More Classes of Complete Permutation Polynomials over \F_q

In this paper, by using a powerful criterion for permutation polynomials given by Zieve, we give several classes of complete permutation monomials over \F_{q^r}. In addition, we present a class of complete permutation multinomials, which is a generalization of recent work.Comment: 17 page

A note on the multiple-recursive matrix method for generating pseudorandom vectors

The multiple-recursive matrix method for generating pseudorandom vectors was introduced by Niederreiter (Linear Algebra Appl. 192 (1993), 301-328). We propose an algorithm for finding an efficient primitive multiple-recursive matrix method. Moreover, for improving the linear complexity, we introduce a tweak on the contents of the primitive multiple-recursive matrix method.Comment: 14 page