# Random Self-reducibility of Ideal-SVP via Arakelov Random Walks

3 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an Abelian group, called the Arakelov class group. This fact, well known to number theorists, has so far not been explicitly used in the literature on lattice-based cryptography. Remarkably, the Arakelov class group is a combination of two groups that have already led to significant cryptanalytic advances: the class group and the unit torus. In the present article, we show that the Arakelov class group has more to offer. We start with the development of a new versatile tool: we prove that, subject to the Riemann Hypothesis for Hecke L-functions, certain random walks on the Arakelov class group have a rapid mixing property. We then exploit this result to relate the average-case and the worst-case of the Shortest Vector Problem in ideal lattices. Our reduction appears particularly sharp: for Hermite-SVP in ideal lattices of certain cyclotomic number fields, it loses no more than $\tilde O(\sqrt{n})$ factor on the Hermite approximation factor. Furthermore, we suggest that this rapid-mixing theorem should find other applications in cryptography and in algorithmic number theory.
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Cited literature [49 references]

https://hal.archives-ouvertes.fr/hal-02513308
Contributor : Benjamin Wesolowski Connect in order to contact the contributor
Submitted on : Tuesday, September 22, 2020 - 11:32:40 AM
Last modification on : Saturday, December 4, 2021 - 3:43:37 AM

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2020-297.pdf
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### Citation

Koen de Boer, Leo Ducas, Alice Pellet-Mary, Benjamin Wesolowski. Random Self-reducibility of Ideal-SVP via Arakelov Random Walks. CRYPTO 2020, Aug 2020, Santa Barbara, United States. ⟨10.1007/978-3-030-56880-1_9⟩. ⟨hal-02513308v2⟩

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