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We investigate the discrete logarithm problem over jacobians of hyperelliptic curves suitable for public-key cryptosystems. We focus on the case when the definition field has small characteristic 2, 3, 5 and 7, then we present hyperelliptic cryptosystems that resist against all known attacks. We further implement our designed hyperelliptic cryptosystems over finite fields F 2n in software on Alpha and Pentium-II computers. Our results indicate that if we choose curves carefully, hyperelliptic cryptosystems do have practical performance.

Design of Hyperelliptic Cryptosystems in Small Characteristic and a Software Implementation over F2 n