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A Rigorous Proof Of The Waterloo Algorithm For The Discrete Logarithm Problem
"... In this paper we are concerned with the Waterloo variant of the index calculus method for the discrete logarithm problem in F 2 n . We provide a rigorous proof for the heuristic arguments for the running time of the Waterloo algorithm. This implies in studying the behavior of pairs of coprime smooth ..."
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In this paper we are concerned with the Waterloo variant of the index calculus method for the discrete logarithm problem in F 2 n . We provide a rigorous proof for the heuristic arguments for the running time of the Waterloo algorithm. This implies in studying the behavior of pairs of coprime smooth polynomials over finite fields. Our proof involves a double saddle point method, and it is in nature similar to the one of Odlyzko for the rigorous analysis of the basic index calculus.
SEQUENCES OF CONSECUTIVE SMOOTH POLYNOMIALS over a finite field
, 2006
"... Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n. ..."
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Given ε>0, we show that there are infinitely many sequences of consecutive εnsmooth polynomials over a finite field. The number of polynomials in each sequence is approximately ln ln ln n.
Weil Descent Attack for ArtinSchreier Curves
"... In this paper, we show how the method introduced by Gaudry, Hess and Smart can be extended to a family of algebraic curves using ArtinSchreier extensions. This family also extends the number of hyperelliptic curves in characteristic 2 vulnarable to the Weil decent attack obtained by Galbraith. We a ..."
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In this paper, we show how the method introduced by Gaudry, Hess and Smart can be extended to a family of algebraic curves using ArtinSchreier extensions. This family also extends the number of hyperelliptic curves in characteristic 2 vulnarable to the Weil decent attack obtained by Galbraith. We also show that the genus of the resulting curve will be one of two easily computable values. 1
Weil descent attack for Kummer extensions
"... In this paper, we show how the Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic. This attack applies to a family of hyperelliptic and superelliptic curves over quadratic field extensions, as well as two famil ..."
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In this paper, we show how the Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic. This attack applies to a family of hyperelliptic and superelliptic curves over quadratic field extensions, as well as two families of hyperelliptic curves defined over cubic extensions. We also show that those are the only families of nonsingular curves defining Kummer extensions for which this method will work. 1