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ANTICS

Title
Algorithmic Number Theory in Computer Science
Funding
EC | FP7 | SP2 | ERC
Call
ERC-2011-StG_20101014
Contract (GA) number
278537
Start Date
2012/01/01
End Date
2016/12/31
Open Access mandate
no
Special Clause 39
no
Organizations
INRIA
More information
Detailed project information (CORDIS)

 

  • 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...

    Computing arithmetic Kleinian groups

    Page , Aurel (2015)
    Projects: EC | ANTICS (278537)
    International audience; Arithmetic Kleinian groups are arithmetic lattices in PSL_2(C). We present an algorithm which, given such a group Gamma, returns a fundamental domain and a finite presentation for Gamma with a computable isomorphism.

    An algorithm for the principal ideal problem in indefinite quaternion algebras

    Page, Aurel (2014)
    Projects: EC | ANTICS (278537)
    International audience; Deciding whether an ideal of a number field is principal and finding a generator is a fundamental problem with many applications in computational number theory. For indefinite quaternion algebras, the decision problem reduces to that in the underlying number field. Finding a generator is hard, and we present a heuristically subexponential algorithm.

    Generalised Weber Functions

    Enge , Andreas; Morain , François (2014)
    Projects: EC | ANTICS (278537)
    International audience; A generalised Weber function is given by $\w_N(z) = \eta(z/N)/\eta(z)$, where $\eta(z)$ is the Dedekind function and $N$ is any integer; the original function corresponds to $N=2$. We classify the cases where some power $\w_N^e$ evaluated at some quadratic integer generates the ring class field associated to an order of an imaginary quadratic field. We compare the heights of our invariants by giving a general formula for the degree of the modular equation relating $\w_...

    Short addition sequences for theta functions

    Enge, Andreas; Hart, William; Johansson, Fredrik (2016)
    Projects: EC | ANTICS (278537)
    International audience; The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this task using N + o(N) multiplications. Our constructions rely on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For...

    Schertz style class invariants for quartic CM fields

    Enge , Andreas; Streng , Marco (2016)
    Projects: EC | ANTICS (278537)
    A class invariant is a CM value of a modular function that lies in a certain unramified class field. We show that Siegel modular functions over $\mathbb Q$ for $\Gamma^0(N)\subseteq \operatorname {Sp}_4(\mathbb Z)$ yield class invariants under some splitting conditions on $N$. Small class invariants speed up onstructions in explicit class field theory and public-key cryptography. Our results generalise results of Schertz's from elliptic curves to abelian varieties and from classical modular f...

    Vanishing and non-vanishing theta values

    Cohen , Henri; Zagier , Don (2013)
    Projects: EC | ANTICS (278537)
    International audience

    Computing class polynomials for abelian surfaces

    Enge , Andreas; Thomé , Emmanuel (2014)
    Projects: EC | ANTICS (278537)
    International audience; We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating ϑ- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example...

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

    Lubicz, David; Robert, Damien (2014)
    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...
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