A generalized slide reduction that handles every block size now gives the fastest provable algorithms for δ-approximate SVP in the cryptographic range n^{1/2+ε} ≤ δ ≤ n^{O(1)}.
Just Take the Average! An Embarrassingly Simple $2^n$-Time Algorithm for SVP (and CVP)
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abstract
We show a $2^{n+o(n)}$-time (and space) algorithm for the Shortest Vector Problem on lattices (SVP) that works by repeatedly running an embarrassingly simple "pair and average" sieving-like procedure on a list of lattice vectors. This matches the running time (and space) of the current fastest known algorithm, due to Aggarwal, Dadush, Regev, and Stephens-Davidowitz (ADRS, in STOC, 2015), with a far simpler algorithm. Our algorithm is in fact a modification of the ADRS algorithm, with a certain careful rejection sampling step removed. The correctness of our algorithm follows from a more general "meta-theorem," showing that such rejection sampling steps are unnecessary for a certain class of algorithms and use cases. In particular, this also applies to the related $2^{n + o(n)}$-time algorithm for the Closest Vector Problem (CVP), due to Aggarwal, Dadush, and Stephens-Davidowitz (ADS, in FOCS, 2015), yielding a similar embarrassingly simple algorithm for $\gamma$-approximate CVP for any $\gamma = 1+2^{-o(n/\log n)}$. (We can also remove the rejection sampling procedure from the $2^{n+o(n)}$-time ADS algorithm for exact CVP, but the resulting algorithm is still quite complicated.)
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Slide Reduction, Revisited---Filling the Gaps in SVP Approximation
A generalized slide reduction that handles every block size now gives the fastest provable algorithms for δ-approximate SVP in the cryptographic range n^{1/2+ε} ≤ δ ≤ n^{O(1)}.