Article

# An $L (1/3 + \varepsilon)$ Algorithm for the Discrete Logarithm Problem for Low Degree Curves

04/2007;

Source: arXiv

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**ABSTRACT:**We present an algorithm for solving the discrete logarithm problem in Jacobians of families of plane curves whose degrees in $X$ and $Y$ are low with respect to their genera. The finite base fields $\FF_q$ are arbitrary, but their sizes should not grow too fast compared to the genus. For such families, the group structure and discrete logarithms can be computed in subexponential time of $L_{q^g}(1/3, O(1))$. The runtime bounds rely on heuristics similar to the ones used in the number field sieve or the function field sieve.Journal of Cryptology 01/2009; · 0.84 Impact Factor -
##### Article: An L(1/3) algorithm for ideal class group and regulator computation in certain number fields

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**ABSTRACT:**We analyse the complexity of the computation of the class group structure, regulator, and a system of fundamental units of a certain class of number fields. Our approach differs from Buchmann's, who proved a complexity bound of L(1/2,O(1)) when the discriminant tends to infinity with fixed degree. We achieve a subexponential complexity in O(L(1/3,O(1))) when both the discriminant and the degree of the extension tend to infinity by using techniques due to Enge and Gaudry in the context of algebraic curves over finite fields.12/2009; - [show abstract] [hide abstract]

**ABSTRACT:**We propose an index calculus algorithm for the discrete logarithm problem on general abelian varieties of small dimension. The main difference with the previous approaches is that we do not make use of any embedding into the Jacobian of a well-suited curve. We apply this algorithm to the Weil restriction of elliptic curves and hyperelliptic curves over small degree extension fields. In particular, our attack can solve an elliptic curve discrete logarithm problem defined over Fq3 in heuristic asymptotic running time ; and an elliptic problem over Fq4 or a genus 2 problem over Fq2 in heuristic asymptotic running time .Journal of Symbolic Computation. 01/2009;

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