Computing Jacobi's $\theta$ in quasi-linear time

Hugo Labrande 1, 2
1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z, \tau)$, for $z$, $\tau$ verifying certain conditions, with precision $P$ in $O(M(P) \sqrt{P})$ bit operations, where $M(P)$ denotes the number of operations needed to multiply two complex $P$-bit numbers. We generalize an algorithm which computes specific values of the $\theta$ function (the theta-constants) in asymptotically faster time; this gives us an algorithm to compute $\theta(z, \tau)$ with precision $P$ in $O(M(P) \log P)$ bit operations, for any $\tau\in F$ and $z$ reduced using the quasi-periodicity of $\theta$.
Type de document :
Article dans une revue
Mathematics of Computation, American Mathematical Society, 2016
Contributeur : Hugo Labrande <>
Soumis le : jeudi 2 juin 2016 - 20:20:14
Dernière modification le : mardi 13 décembre 2016 - 15:45:09
Document(s) archivé(s) le : samedi 3 septembre 2016 - 11:40:16


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  • HAL Id : hal-01227699, version 2
  • ARXIV : 1511.04248



Hugo Labrande. Computing Jacobi's $\theta$ in quasi-linear time. Mathematics of Computation, American Mathematical Society, 2016. <hal-01227699v2>



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