Gary McGuire

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We consider exceptional APN functions on F2m , which by definition are functions that are APN on infinitely many extensions of F2m . Our main result is that polynomial functions of odd degree are not exceptional, provided the degree is not a Gold number (2 +1) or a Kasami-Welch number (4 −2k +1). We also have partial results on functions of even degree, and(More)
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) = xt is APN (Almost Perfect Nonlinear) over F2n for infinitely many values of n. Equivalently, t is exceptional if the binary cyclic code of length 2n − 1(More)
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)
We introduce two new infinite families of APN functions, one on fields of order 22k for k not divisible by 2, and the other on fields of order 23k 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. © 2007 Elsevier Inc. All rights reserved.
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, we(More)
We study the nonlinearity of the exponential Welch Costas functions, using the Fourier transform on <i>Z</i> <sub>m</sub>. These functions have been proposed for use in nonbinary cryptosystems. High nonlinearity is required to ensure resistance to linear cryptanalysis. We prove some properties of the nonlinearity of these functions, and we suggest a(More)