935 research outputs found
Computing Elliptic Curve Discrete Logarithms with Improved Baby-step Giant-step Algorithm
The negation map can be used to speed up the computation of elliptic curve discrete logarithms using either the baby-step giant-step algorithm (BSGS) or Pollard rho. Montgomery\u27s simultaneous modular inversion can also be used to speed up Pollard rho when running many walks in parallel. We generalize these ideas and exploit the fact that for any two elliptic curve points and , we can efficiently get when we compute . We apply these ideas to speed up the baby-step giant-step algorithm. Compared to the previous methods, the new methods can achieve a significant speedup for computing elliptic curve discrete logarithms in small groups or small intervals.
Another contribution of our paper is to give an analysis of the average-case running time of Bernstein and Lange\u27s ``grumpy giants and a baby\u27\u27 algorithm, and also to consider this algorithm in the case of groups with efficient inversion.
Our conclusion is that, in the fully-optimised context, both the interleaved BSGS and grumpy-giants algorithms have superior average-case running time compared with Pollard rho. Furthermore, for the discrete logarithm problem in an interval, the interleaved BSGS algorithm is considerably faster than the Pollard kangaroo or Gaudry-Schost methods
Recent progress on the elliptic curve discrete logarithm problem
International audienceWe survey recent work on the elliptic curve discrete logarithm problem. In particular we review index calculus algorithms using summation polynomials, and claims about their complexity
Xavier University Newswire
https://www.exhibit.xavier.edu/student_newspaper/1525/thumbnail.jp
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