Index calculus in class groups of non-hyperelliptic curves of genus three

The original publication is available at www.springerlink.comDescriptionInternational audienceWe study an index calculus algorithm to solve the discrete logarithm problem (DLP) in degree~0 class groups of non-hyperelliptic 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 non-hyperelliptic curves of genus~3 can be solved in an expected time of soft-O(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

Index Calculus in Class Groups of Plane Curves of Small Degree

We present a novel index calculus algorithm for the discrete logarithm problem (DLP) in degree 0 class groups of curves over finite fields. A heuristic analysis of our algorithm indicates that asymptotically for varying q, ``essentially all\u27\u27 instances of the DLP in degree 0 class groups of curves represented by plane models of a fixed degree d over $\mathbb{F}_q$ can be solved in an expected time of $\tilde{O}(q^{2 -2/(d-2)})$. A particular application is that heuristically, ``essentially all\u27\u27 instances of the DLP in degree 0 class groups of non-hyperelliptic curves of genus 3 (represented by plane curves of degree 4) can be solved in an expected time of $\tilde{O}(q)$. We also provide a method to represent ``sufficiently general\u27\u27 (non-hyperelliptic) curves of genus $g \geq 3$ by plane models of degree $g+1$. We conclude that on heuristic grounds the DLP in degree 0 class groups of ``sufficiently general\u27\u27 curves of genus $g \geq 3$ (represented initially by plane models of bounded degree) can be solved in an expected time of $\tilde{O}(q^{2 -2/(g-1)})$

Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

Quantum algorithms for problems in number theory, algebraic geometry, and group theory

Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in Quantum Computation/Information at Kinki Universit

Horizontal isogeny graphs of ordinary abelian varieties and the discrete logarithm problem

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class group, and generating set thereof, induces an isogeny graph on the orbit of the variety for this subgroup. We compute (under the Generalized Riemann Hypothesis) some bounds on the norms of prime ideals generating it, such that the associated graph has good expansion properties. We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and Robert for computing explicit isogenies in genus 2, to prove random self-reducibility of the discrete logarithm problem within the subclasses of principally polarizable ordinary abelian surfaces with fixed endomorphism ring. In addition, we remove the heuristics in the complexity analysis of an algorithm of Galbraith for explicitly computing isogenies between two elliptic curves in the same isogeny class, and extend it to a more general setting including genus 2.Comment: 18 page