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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 $\theta$.
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Submitted on : Thursday, June 2, 2016 - 8:20:14 PM
Last modification on : Friday, July 8, 2022 - 10:08:34 AM
Long-term archiving on: : Saturday, September 3, 2016 - 11:40:16 AM


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Hugo Labrande. Computing Jacobi's $\theta$ in quasi-linear time. Mathematics of Computation, 2016, ⟨10.1090/mcom/3245⟩. ⟨hal-01227699v2⟩



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