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We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer t ≥ 3 is said to be exceptional if f (x) = x t is APN (Almost Perfect Nonlinear) over F 2 n for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length 2 n −… (More)
We consider exceptional APN functions on F 2 m , which by definition are functions that are APN on infinitely many extensions of F 2 m. Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold number (2 k + 1) or a Kasami-Welch number (4 k − 2 k + 1). We also have partial results on functions of even… (More)
We introduce two new infinite families of APN functions, one on fields of order 2 2k for k not divisible by 2, and the other on fields of order 2 3k for k not divisible by 3. The polynomials in the first family have between three and k + 2 terms, the second family's polynomials have three terms.
We apply our new hitting set enumeration algorithm to solve the sudoku minimum number of clues problem, which is the following question: What is the smallest number of clues (givens) that a sudoku puzzle may have? It was conjectured that the answer is 17. We have performed an exhaustive search for a 16-clue sudoku puzzle, and we did not find one, thereby… (More)
In this paper we propose a binary field variant of the Joux-Lercier medium-sized Function Field Sieve, which results not only in complexities as low as Lqn (1/3, 2/3) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one elements. To illustrate the efficiency of the method,… (More)
In this paper we show how some recent ideas regarding the discrete logarithm problem (DLP) in finite fields of small characteristic may be applied to compute logarithms in some very large fields extremely efficiently. In particular, we demonstrate a practical DLP break in the finite field of 2 6120 elements, using just a single core-month.
We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.