Finite Fields and Their Applications

In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a ...

Let F p n be a finite field with p n elements. Ness and Helleseth in [29] first studied a class of functions over F p n with the form f ( x ) = u x p n − 3 2 + x p n − 2 , u ∈ F p n ⁎, which is called the Ness-Helleseth function. The f ( x ) has ...

It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P 4 ( F q 2 ) has at most d ( q 5 + q 2 ) + q 3 + 1 points in common with a threefold of degree d defined over F q 2. He proved the conjecture for d = 2. In this ...

In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open ...

A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F q 2 is maximal if the number N q 2 of its F q 2-rational points attains the Hasse-Weil upper bound, that is N q 2 = q 2 + 2 g q + 1 where g is the genus ...

Let ε > 0 be a fixed small constant, F p be the finite field of p elements for prime p. We consider additive and multiplicative problems in F p that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let ...

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the ( 2 k + 1 )-th moment to the zeroeth ...

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such ...

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f = y n + ∑ i = 0 n − 1 a i ( x ) y i ∈ F q [ x ] [ y ] with i.i.d. coefficients a i taking values in the set { a ( x ) ∈ F ...

In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the automorphism group of maximal order of a smooth cubic ...

Let q = 2 m. In this paper, we investigate permutation pentanomials over F q 2 of the form f ( x ) = x t + x r 1 ( q − 1 ) + t + x r 2 ( q − 1 ) + t + x r 3 ( q − 1 ) + t + x r 4 ( q − 1 ) + t with gcd ( x r 4 + x r 3 + x r 2 + x r 1 + 1 , x t + ...

An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an ...

n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n-to-1 mappings over F q ⁎. Then we characterize a new ...

Permutation polynomials with low c-differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect c-nonlinear (PcN) and almost perfect c-nonlinear (APcN) polynomials over finite ...