ENGE, Andreas; GAUDRY, Pierrick

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ENGE, Andreas
Algorithmic number theory for cryptology [TANC]

GAUDRY, Pierrick
Curves, Algebra, Computer Arithmetic, and so On [CACAO]

Langue

Communication dans un congrès avec actes

Ce document a été publié dans

Eurocrypt 2007, 2007-05-20, Barcelona. 2007, vol. 4515, p. 379-393

Springer

Résumé en anglais

The discrete logarithm problem in Jacobians of curves of high genus $g$ over finite fields $\FF_q$ is known to be computable with subexponential complexity $L_{q^g}(1/2, O(1))$. We present an algorithm for a family of plane curves whose degrees in $X$ and $Y$ are low with respect to the curve genus, and suitably unbalanced. The finite base fields are arbitrary, but their sizes should not grow too fast compared to the genus. For this family, the group structure can be computed in subexponential time of $L_{q^g}(1/3, O(1))$, and a discrete logarithm computation takes subexponential time of $L_{q^g}(1/3+\varepsilon, o(1))$ for any positive~$\varepsilon$. These runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve algorithms.< Réduire

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Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251