# Tuple lattice sieving

### Shi Bai, Thijs Laarhoven and Damien Stehlé

Abstract: Lattice sieving is asymptotically the fastest approach for
solving the shortest vector problem (SVP) on Euclidean lattices. All
known sieving algorithms for solving SVP require space which
(heuristically) grows as 2^(0.2075n+o(n)), where n is the
lattice dimension. In high dimensions, the memory requirement becomes
a limiting factor for running these algorithms, making them
uncompetitive with enumeration algorithms, despite their superior
asymptotic time complexity. We generalize sieving algorithms to solve
SVP with less memory. We consider reductions of tuples of vectors
rather than pairs of vectors as existing sieve algorithms do. For
triples, we estimate that the space requirement scales as
2^(0.1887n+o(n)). The naive algorithm for this triple sieve
runs in time 2^(0.5661n+o(n)). With appropriate filtering of
pairs, we reduce the time complexity to 2^(0.4812n+o(n))
while keeping the same space complexity. We further analyze the
effects of using larger tuples for reduction, and conjecture how this
provides a continuous tradeoff between the memory-intensive sieving
and the asymptotically slower enumeration.

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