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

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 $\theta$.
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Article dans une revue
Mathematics of Computation, American Mathematical Society, 2016

https://hal.inria.fr/hal-01227699
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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theta.pdf
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• HAL Id : hal-01227699, version 2
• ARXIV : 1511.04248

### Citation

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

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