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Let F ( x ) = x 2 2 k + 2 k + 1 be a power function over the finite field GF ( 2 n ) , where k is a positive integer and n = 4 k . It is known by a recent result of Bracken and Leander that F ( x ) is a highly nonlinear differentially 4-uniform function. Building upon this work, in this paper we determine the differential spectrum of F ( x ) . [ABSTRACT FROM AUTHOR]

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37] constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities. [ABSTRACT FROM AUTHOR]

For a function field k over a finite field with F q as the field of constants, and a finite abelian group G whose exponent divides q − 1 , we study the distribution of zeta zeroes for a random G -extension of k , ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution. [ABSTRACT FROM AUTHOR]

Abstract: Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [10], Ding et al. (2011) [6], Wang et al. (2012) [15], Xiong (2012) [16,17]. In this paper we solve one more special case. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which in turn can be solved by using the Jacobi sums directly. [Copyright &y& Elsevier]

Abstract: Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) , Ding et al. (2011) , Wang et al. (2011) . In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method. [Copyright &y& Elsevier]

Abstract: We prove the existence of the pair correlation measure associated to torsion points on the real locus of an elliptic curve E and provide an explicit formula for the limiting pair correlation function. [Copyright &y& Elsevier]

Abstract: We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in . We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate–Shafarevich group and a large Tate–Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1–2) (2005) 139–154 (2006)], this shows that the mean value of the 2-rank of the large Tate–Shafarevich group for square-free positive integers n less than X is , as . [Copyright &y& Elsevier]

Abstract: We prove that the pair correlation of the sequence of rational numbers in the unit interval with prime denominators is Poissonian. The result is also true for rational numbers with denominators having exactly two distinct prime factors. [Copyright &y& Elsevier]

Abstract: We investigate local spacing problems along curves via smooth maps. Moreover, we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over , and of rational points on the unit circle. [Copyright &y& Elsevier]

In this paper we determine all elliptic curves En:y2=x3-n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers. [Copyright &y& Elsevier]

Abstract: For the finite field of q elements (q odd) and a quadratic non-residue (that is, a non-square) , we define the distance function on the upper half plane . For two sets with , and a non-trivial additive character ψ on , we give the following estimate which (assuming that ) is non-trivial if as . [Copyright &y& Elsevier]