On the Quantitative Hardness of CVP

 

For odd integers p >= 1 (and p=\infty), we show that the Closest Vector Problem in the \ell_p norm (CVP_p) over rank n lattices cannot be solved in 2^{(1−\eps)n} time for any constant \eps > 0 unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to “almost all” values of p>=1, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of CVP_2 (i.e., CVP in the Euclidean norm), for which a 2^{n+o(n)}-time algorithm is known. In particular, our result applies for any p=p(n) =/= 2 that approaches 2 as n goes to infinity.

 

Based on joint work with Huck Bennett and Sasha Golovnev.