Christophe Ritzenthaler

The 2-adic CM method for genus 2 curves with application to cryptography

The complex multiplication (CM) method for genus 2 is currently the most efficient way of generating genus 2 hyperelliptic curves defined over large prime fields and suitable for cryptography. Since low class number might be seen as a potential threat, it is of interest to push the method as far as possible. We have thus designed a new algorithm for the construction of CM invariants of genus 2 curves, using 2-adic lifting of an input curve over a small finite field. This provides a numerically stable alternative to the complex analytic method in the first phase of the CM method for genus 2. As an example we compute an irreducible factor of the Igusa class polynomial system for the quartic CM field ℚ (i√(75 + 12√(17))), whose class number is 50. We also introduce a new representation to describe the CM curves: a set of polynomials in (j1,j2,j3) which vanish on the precise set of triples which are the Igusa invariants of curves whose Jacobians have CM by a prescribed field. The new representation provides a speedup in the second phase, which uses Mestres algorithm to construct a genus 2 Jacobian of prime order over a large prime field for use in cryptography.

Distortion maps for supersingular genus two curves

Abstract Distortion maps are a useful tool for pairing based cryptography. Compared with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more complicated, since the full torsion subgroup has rank 2g. In this paper, we prove that distortion maps always exist for supersingular curves of genus g > 1. We also give several examples of curves of genus 2 with explicit distortion maps for embedding degrees 4, 5, 6, and 12.

Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Let k be a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold ( A , a ) over k , which is a Jacobian over , being a Jacobian overxa0 k ; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

Fast addition on non-hyperelliptic genus 3 curves

We present a fast addition algorithm in the Jacobian of a genus 3 non-hyperelliptic curve over a field of any characteristic. When the curve has a rational flex and char(k) > 5, the computational cost for addition is 148M +15SQ+2I and 165M +20SQ+2I for doubling. An appendix focuses on the computation of flexes in all characteristics. For large odd q, we also show that the set of rational points of a nonhyperelliptic curve of genus 3 can not be an arc.

Point Counting on Genus 3 Non Hyperelliptic Curves

We propose an algorithm to compute the Frobenius polynomial of an ordinary non hyperelliptic curve of genus 3 over (mathbb{F}_{2}N). The method is a generalization of Mestre’s AGM-algorithm for hyperelliptic curves and leads to a quasi quadratic time algorithm for point counting.

An Explicit Formula for the Arithmetic--Geometric Mean in Genus 3

The arithmetic–geometric mean algorithm for calculating elliptic integrals of the first type was introduced by Gauss. The analogous algorithm for abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for abelian integrals of genus 3.The arithmetic geometric mean algorithm for calculation of elliptic integrals of the first type was introduced by Gauss. The analog algorithm for Abelian integrals of genus 2 was introduced by Richelot (1837) and Humbert (1901). We present the analogous algorithm for Abelian integrals of genus 3. (This is joint work with Christophe Ritzenthaler, math.AG/0403182.) Friday, May 28 3:15 p.m. Room 383-N http://math.stanford.edu/~vakil/seminar0304/

On some questions of Serre on abelian threefolds

J.-P. Serre asserted a precise form of Torelli Theorem for genus 3 curves, namely, an indecomposable principally polarized abelian threefold is a Jacobian if and only if some specific invariant is a square. We study here a three dimensional family of such threefolds, introduced by Howe, Leprevost and Poonen. By a new formulation, we link their results to Serre’ s assersion. Then, we recover a formula of Klein related to the question for complex threefolds. In this case the invariant is a modular form of weight 18, and the result is proved using theta functions identities.

Jacobians among abelian threefolds: A formula of klein and a question of serre

In this paper we give a criterion when an indecomposable principally polarized abelian threefold (A, a) defined over a field k = ℂ is a Jacobian over k. More precisely, (A, a) is a Jacobian over k if and only if the value of a certain geometric Siegel modular form χ18(A, a) is a square over k. This answers a question of J.-P. Serre.

Complete addition laws on abelian varieties

We prove that under any projective embedding of an abelian variety A of dimension g, a complete system of addition laws has cardinality at least g+1, generalizing of a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in P^2. In contrast with this geometric constraint, we moreover prove that if k is any field with infinite absolute Galois group, then there exists, for every abelian variety A/k, a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or embedding in P^15, respectively, up to a finite number of counterexamples for |k| less or equal to 5.

The Weierstrass Subgroup of a Curve has Maximal Rank

We show that the Weierstrass points of the generic curve of genus over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian.