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 derive recurrences for counting the number a(n, r) of sequences of length n with Lempel-Ziv complexity r, which has important applications, for instance testing randomness of binary sequences. We also give algorithms to compute these recurrences. We employed these algorithms to compute a(n, r) and expected value, EPn, of number of patterns of a sequence… (More)
Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelberger's theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at a modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of a.
New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum… (More)