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, (4/9) 1/3) for computing arbitrary logarithms, but also in an heuristic polynomial time algorithm for finding the discrete logarithms of degree one and two elements when the field has a subfield of an appropriate size. To illustrate the efficiency of the method, we have successfully solved the DLP in the finite fields with 2 1971 and 2 3164 elements, setting a record for binary fields.

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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 degree, and functions that have degree 2 k + 1.

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 we describe our method and the actual search. As a part of this project we developed a novel way for enumerating hitting sets. The hitting set problem is computationally hard; it is one of Karp's 21 classic NP-complete problems. A standard backtracking algorithm for finding hitting sets would not be fast enough to search for a 16-clue sudoku puzzle exhaustively, even at today's supercomputer speeds. To make an exhaustive search possible, we designed an algorithm that allowed us to efficiently enumerate hitting sets of a suitable size.Solving the sudoku minimum number of clues problem serves as a way to introduce our algorithm for enumerating hitting sets. However, this algorithm has uses beyond combinatorics. To begin with, it is in principle applicable to any instance of the hitting set problem, be it the decision version (determining if hitting sets of a given size exist) or the optimization version (finding a smallest hitting set). Such situations occur in bioinformatics (gene expression analysis [6]), software testing as well as computer networks [7], and when finding optimal drug combinations [8,9]; the last of these papers lists protein network discovery, metabolic network analysis and gene ontology as further areas in which hitting set problems naturally arise. Besides, our algorithm can be applied to the hypergraph transversal problem, which appears in artificial intelligence and database theory [10]. Lastly, we note that the vertex cover problem from graph theory is a special case of the hitting set problem, and that the set cover problem is in fact equivalent to the hitting set problem. 2 The former occurs in computational biology, see [11], while the latter finds application in reducing interference in cellular networks [12].1 In a 7-clue sudoku puzzle, the two missing digits can be interchanged in any solution to yield another solution. 2 We did not find any papers about the set cover problem that were useful to us in improving our hitting set algorithm.

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