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4 documents, page 1 of 1

Computing isogenies between Abelian Varieties

Lubicz , David; Robert , Damien (2012)
Projects: EC | ANTICS (278537)
We describe an efficient algorithm for the computation of separable isogenies between abelian varieties represented in the coordinate system given by algebraic theta functions. Let $A$ be an abelian variety of dimension $g$ defined over a field of odd characteristic. Our algorithm decomposes in two principal steps. First, given a theta null point for $A$ and a subgroup $K$ isotropic for the Weil pairing, we explain how to compute the theta null point corresponding to the quotient abelian vari...

Arithmetic on Abelian and Kummer Varieties

Lubicz , David; Robert , Damien (2016)
Projects: EC | ANTICS (278537)
International audience; A Kummer variety is the quotient of an abelian variety by the automorphism $(-1)$ acting on it. Kummer varieties can be seen as a higher dimensional generalisation of the $x$-coordinate representation of a point of an elliptic curve given by its Weierstrass model. Although there is no group law on the set of points of a Kummer variety, there remains enough arithmetic to enable the computation of exponentiations via a Montgomery ladder based on differential additions. I...

A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties

Lubicz , David; Robert , Damien (2015)
Projects: EC | ANTICS (278537)
International audience; In this paper, we use the theory of theta functions to generalize to all abelian varieties the usual Miller's algorithm to compute a function associated to a principal divisor. We also explain how to use the Frobenius morphism on abelian varieties defined over a finite field in order to shorten the loop of the Weil and Tate pairings algorithms. This extend preceding results about ate and twisted ate pairings to all abelian varieties. Then building upon the two precedin...

Computing separable isogenies in quasi-optimal time

Lubicz , David; Robert , Damien (2015)
Projects: EC | ANTICS (278537)
International audience; Let $A$ be an abelian variety of dimension $g$ together with a principal polarization $\phi: A \rightarrow \hat{A}$ defined over a field $k$. Let $\ell$ be an odd integer prime to the characteristic of $k$ and let $K$ be a subgroup of $A[\ell]$ which is maximal isotropic for the Riemann form associated to $\phi$. We suppose that $K$ is defined over $k$ and let $B=A/K$ be the quotient abelian variety together with a polarization compatible with $\phi$. Then $B$, as a po...
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