Algebraic Approaches for the Elliptic Curve Discrete Logarithm Problem over Prime Fields
نویسندگان
چکیده
The elliptic curve discrete logarithm problem is one of the most important problems in cryptography. In recent years, several index calculus algorithms have been introduced for elliptic curves defined over extension fields, but the most important curves in practice, defined over prime fields, have so far appeared immune to these attacks. In this paper we formally generalize previous attacks from binary curves to prime curves. We study the efficiency of our algorithms with computer experiments and we discuss their current and potential impact on elliptic curve standards. Our algorithms are only practical for small parameters at the moment and their asymptotic analysis is limited by our understanding of Gröbner basis algorithms. Nevertheless, they highlight a potential vulnerability on prime curves which our community needs to explore further.
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