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)
The sudoku minimum number of clues problem is the following question: what is the smallest number of clues that a sudoku puzzle can have? For several years it had been conjectured that the answer is 17. We have performed an exhaustive computer search for 16-clue sudoku puzzles, and did not find any, thus proving that the answer is indeed 17. In this article… (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.
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.