Cryptology ePrint Archive: Report 2011/098

Computing Discrete Logarithms in the Jacobian of High-Genus Hyperelliptic Curves over Even Characteristic Finite Fields

M. D. Velichka and M. J. Jacobson, Jr. and A. Stein

Abstract: We describe improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of high-genus hyperelliptic curves defi ned over even characteristic fields. Our first improvement is to incorporate several ideas for the low-genus case by Gaudry and Theriault, including the large prime variant and using a smaller factor base, into the large-genus algorithm of Enge and Gaudry. We extend the analysis in [24] to our new algorithm, allowing us to predict accurately the number of random walk steps required to find all relations, and to select optimal degree bounds for the factor base. Our second improvement is the adaptation of sieving techniques from Flassenberg and Paulus, and Jacobson to our setting. The new algorithms are applied to concrete problem instances arising from the Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating signifi cant improvements in practice.

Category / Keywords: foundations / hyperelliptic curves, discrete logarithm problem, sieving, Weil descent

Publication Info: Submitted to Mathematics of Computation

Date: received 25 Feb 2011

Contact author: jacobs at cpsc ucalgary ca

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Version: 20110228:224542 (All versions of this report)

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