In this paper, we present a characterization of a semi-multiplicative analogue of planar functions over finite fields. When the field is a prime field, these functions are equivalent to a variant of a doubly-periodic Costas array and so we call these functions Costas. We prove an equivalent conjecture of Golomb and Moreno that any Costas polynomial over a prime field is a monomial. Moreover, we...

In the last twenty years several methods for computing primary decompositions of ideals in multivariate polynomial rings over fields (Seidenberg (1974), Lazard (1985), Kredel (1987), Eisenbud et al. (1992)), the integers (Seidenberg, 1978), factorially closed principal ideal domains (Ayoub (1982), Gianni et al. (1988)) and more general rings (Seidenberg, 1984) have been proposed. A related prob...

In 1999 Silverman [21] introduced a family of binary finite fields which are composite extensions of F2 and on which arithmetic operations can be performed more quickly than on prime extensions of F2 of the same size. We present here a fast approach to elliptic curve cryptography using a distinguished subset of the set of Silverman fields F2N = Fhn . This approach leads to a theoretical computa...

Some nonconstant polynomials with a finite string of prime values are known; in this paper, some polynomials of this kind are described, starting from Euler’s example (1772) P(x) = x2+x+41: other quadratic polynomials with prime values were studied, and their properties were related to properties of quadratic fields; in this paper, some quadratic polynomials with prime values are described and ...

General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.

In this paper we describe how to efficiently implement pairing calculation on supersingular genus 2 curves over prime fields. We find that, contrary to the results reported in [8], pairing calculation on supersingular genus 2 curves over prime fields is efficient and a viable candidate for practical implementation. We also show how to eliminate divisions in an efficient manner when computing th...

In this talk I will present estimates on incomplete character sums in finite fields, with special emphasize on the non-prime case. Some of the results are of the same strength as Burgess celebrated theorem for prime fields. The improvements are mainly based on arguments from arithmetic combinatorics providing new bounds on multiplicative energy and an improved amplification strategy. In particu...