Finding Shortest Lattice Vectors in the Presence of Gaps

Abstract

The \(\lambda _i\)-gap \(\lambda _i/\lambda _1\) among the successive minima of a lattice especially its \(\lambda _2\)-gap often provides useful information for analyzing the security of lattice-based cryptographic schemes. In this paper, we mainly study the efficiency of shortest vector problem (SVP) algorithms for lattices with \(\lambda _i\)-gap. First, we prove new upper bounds for the packing density of this type of lattices. Based on these results, we discuss the efficiency of the ListSieve-Birthday algorithm proposed by Pujol and Stehle for SVP, and obtain the conclusion that the complexity will decrease obviously as the \(\lambda _i\)-gap increases. Particularly, ListSieve-Birthday becomes faster than the current best deterministic (Voronoi cell-based) algorithm for SVP, as long as \(\lambda _2\)-gap is larger than 1.78. When \(\lambda _2\)-gap is up to 28, the time complexity is \(2^{0.9992n+o(n)}\), and the coefficient factor of \(n\) is approximately to 0.802 if \(\lambda _2\)-gap is large enough. Moreover, we provide an SVP approximation algorithm modified by the ListSieve-Birthday algorithm. This algorithm terminates sieve process earlier and relaxes the birthday search, and hence decreases the time complexity significantly.