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

Koen de Boer, Léo Ducas, Alice Pellet-Mary, and Benjamin Wesolowski

##### 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 a $\tilde O(\sqrt n)$ 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.

Note: more details about Gentry's reduction

Available format(s)
Category
Public-key cryptography
Publication info
A minor revision of an IACR publication in CRYPTO 2020
Keywords
Ideal LatticesRandom WalkWorst-Case HardnessArakelovL-functions
Contact author(s)
K de Boer @ cwi nl
ducas @ cwi nl
alice pelletmary @ kuleuven be
benjamin wesolowski @ math u-bordeaux fr
History
2020-09-08: last of 2 revisions
See all versions
Short URL
https://ia.cr/2020/297

CC BY

BibTeX

@misc{cryptoeprint:2020/297,
author = {Koen de Boer and Léo Ducas and Alice Pellet-Mary and Benjamin Wesolowski},
title = {Random Self-reducibility of Ideal-SVP via Arakelov Random Walks},
howpublished = {Cryptology ePrint Archive, Paper 2020/297},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/297}},
url = {https://eprint.iacr.org/2020/297}
}

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