Secure Hyperelliptic Cryptosystems and Their Performances

We investigate the discrete logarithm problem over jacobian varieties of hyperelliptic curves suitable for public-key cryptosystems, and clarify practical advantages of hyperelliptic cryptosystems compared to the elliptic cryptosystems and to RSA. We focus on the curves defined over the ground field of characteristic 2, and we present hyperelliptic cryptosystems from the jacobian associated with curves C : v 2 + v=u 2g+1 of genus g=3 and 11, which are secure against the known attacks. We further discuss the efficiency in implementation of such secure hyperelliptic cryptosystems.

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