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CORR
2004
Springer

Ramanujan Graphs and the Random Reducibility of Discrete Log on Isogenous Elliptic Curves

8 years 11 months ago
Ramanujan Graphs and the Random Reducibility of Discrete Log on Isogenous Elliptic Curves
Cryptographic applications using an elliptic curve over a finite field filter curves for suitability using their order as the primary criterion: e.g. checking that their order has a large prime divisor before accepting it. It is therefore natural to ask whether the discrete log problem (dlog) has the same difficulty for all curves with the same order; if so it would justify the above practice. We prove that this is essentially true by showing random reducibility of dlog among such curves, assuming the Generalized Riemann Hypothesis (GRH). Our reduction proof works for curves with (nearly) the same endomorphism rings, but it is unclear if such a reduction exists in general. This suggests that in addition to the order, the conductor of its endomorphism ring may play a role. The random self-reducibility for dlog over finite fields is well known; the non-trivial part here is that one must relate non-isomorphic algebraic groups of two isogenous curves. We construct certain expander graphs ...
David Jao, Stephen D. Miller, Ramarathnam Venkates
Added 17 Dec 2010
Updated 17 Dec 2010
Type Journal
Year 2004
Where CORR
Authors David Jao, Stephen D. Miller, Ramarathnam Venkatesan
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