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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]