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Index calculus in class groups of nonhyperelliptic curves of genus three, in "Journal of Cryptology", The original publication is available at www.springerlink.com
, 2007
"... We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of nonhyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of nonhyperelliptic curves of ..."
Abstract

Cited by 26 (4 self)
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We study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree 0 class groups of nonhyperelliptic curves of genus 3 over finite fields. We present a heuristic analysis of the algorithm which indicates that the DLP in degree 0 class groups of nonhyperelliptic curves of genus 3 can be solved in an expected time of Õ(q). This heuristic result relies on one heuristic assumption which is studied experimentally. We also present experimental data which show that a variant of the algorithm is faster than the Rho method even for small group sizes, and we address practical limitations of the algorithm.
Appendix to the preceding paper: A rank 3 generalization of the Conjecture of
, 2005
"... The Conjecture of Shimura and Taniyama is a special case of a general philosophy according to which a motive of a certain type should correspond to a special type of automorphic forms on a reductive group. Now familiar extensions of the conjecture include (i) essentially the same statement for ellip ..."
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The Conjecture of Shimura and Taniyama is a special case of a general philosophy according to which a motive of a certain type should correspond to a special type of automorphic forms on a reductive group. Now familiar extensions of the conjecture include (i) essentially the same statement for elliptic curves over totally real fields and (ii) the statement that for each irreducible abelian surface over Q there is a genus 2 Siegel modular holomorphic cusp form of weight 2 with the same degree 4 Lfunction. Our purpose here is to give a new rank 3 variant of the ShimuraTaniyama conjecture in which elliptic curves are replaced by a naive analogue, the Picard curves. These curves have recently been much studied over finite fields, but their arithmetic study in characteristic zero seems not yet to have attracted much interest. This paper is an Appendix to [Ti] which treats at length the case (ii) above. Motives. Let K be a number field and let M be an irreducible motive defined over K with coefficients in Q. We recall that for l a rational prime, this means that the ladic realization Ml of M is a Qldirect summand of a Tatetwist
Index
, 2006
"... calculus in class groups of nonhyperelliptic curves of genus three ..."
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