On the geometry of cyclic lattices

TL;DR

This paper explores the geometric properties of cyclic lattices, showing they are more common among well-rounded lattices and establishing the equivalence of SVP and SIVP in certain cases, with implications for cryptography.

Contribution

It provides a counting estimate for well-rounded cyclic lattices and demonstrates SVP and SIVP equivalence on a significant subset, extending results to generalized permutation-invariant lattices.

Findings

Well-rounded cyclic lattices are more prevalent than expected.

SVP and SIVP are equivalent on a positive proportion of certain cyclic lattices.

Results extend to lattices invariant under permutation group actions.

Abstract

Cyclic lattices are sublattices of $\mathbb Z^N$ that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded cyclic lattices, indicating that well-rounded lattices are more common among cyclic lattices than generically. We also show that SVP is equivalent to SIVP on a positive proportion of Minkowskian well-rounded cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of such lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of $\mathbb Z^N$ closed under the action of subgroups of the permutation group $S_N$, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any $N$-cycle.