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Journals and Conferences
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 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)
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 elements, using just a single core-month.
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)
This paper introduces two new results on Kloosterman sums and their minimal polynomials. We characterise ternary Kloosterman sums modulo 27. We also prove a congruence concerning the minimal polynomial over Q of a Kloosterman sum. This paper also serves as a survey of our recent results on binary Kloosterman sums modulo 16, 32, 64 and 128 with Petr Lisoněk.
In this paper we propose a binary field variant of the JouxLercier medium-sized Function Field Sieve, which results not only in complexities as low as Lqn(1/3, (4/9) ) 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… (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.