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arXiv · 2305.17425

Computing $e$-th roots in number fields

Olivier Bernard·Pierre-Alain Fouque·Andrea Lesavourey
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Abstract

We describe several algorithms for computing $e$-th roots of elements in a number field $K$, where $e$ is an odd prime-power integer. In particular we generalize Couveignes' and Thom\'e's algorithms originally designed to compute square-roots in the Number Field Sieve algorithm for integer factorization. Our algorithms cover most cases of $e$ and $K$ and allow to obtain reasonable timings even for large degree number fields and large exponents $e$. The complexity of our algorithms is better than general root finding algorithms and our implementation compared well in performance to these algorithms implemented in well-known computer algebra softwares. One important application of our algorithms is to compute the saturation phase in the Twisted-PHS algorithm for computing the Ideal-SVP problem over cyclotomic fields in post-quantum cryptography.

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Olivier Bernard, Pierre-Alain Fouque, Andrea Lesavourey. 2023-05-27. Computing $e$-th roots in number fields. https://arxiv.org/abs/2305.17425

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