Abstract. In this paper, we study several variations of the number field sieve to compute discrete logarithms in finite fields of the form Fpn, with p a medium to large prime. We show that when n is not too large, this yields a Lpn (1/3) algorithm with efficiency similar to that of the regular number field sieve over prime fields. This approach complements the recent results of Joux and Lercier on the function field sieve. Combining both results, we deduce that computing discrete logarithms have heuristic complexity Lpn (1/3) in all finite fields. To illustrate the efficiency of our algorithm, we computed discrete logarithms in a 120-digit finite field F p 3 .
Abstract. In this paper, we study the application of the function field sieve algorithm for computing discrete logarithms over finite fields of the form Fqn when q is a medium-sized prime power. This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations. We show that when q is not too large, a very efficient L(1/3) variation of the function field sieve can be used. Surprisingly, using this algorithm, discrete logarithms computations over some of these fields are even easier than computations in the prime field and characteristic two field cases. We also show that this new algorithm has security implications on some existing cryptosystems, such as torus based cryptography in T30, short signature schemes in characteristic 3 and cryptosystems based on supersingular abelian varieties. On the other hand, cryptosystems involving larger basefields and smaller extension degrees, typically of degree at most 6, such as LUC, XTR or T6 torus cryptography, are not affected.
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We focus on genus 3 hyperelliptic curves. Both geometric and arithmetic aspects are considered. 4. Field of definition 32 4.1. Field of definition and field of moduli: general facts 32 4.2. The hyperelliptic case 33 4.3. The hyperelliptic case with no extra-automorphism 35 4.4. The hyperelliptic case with extra-automorphisms 36 4.5. The hyperelliptic case of genus 3 36 4.6. The case of curves over finite fields 39 5. Conclusion 40 References 40 Appendix A. Stratum equations for the automorphism group D4 42
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