On the Quantitative Hardness of CVP

$\newcommand{\eps}{\varepsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1}$For odd integers $p \geq 1$ (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 $p \geq 1$, 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 $p = p(n) \neq 2$ that approaches $2$ as $n \to \infty$. We also show a similar SETH-hardness result for \SVP_\infty; hardness of approximating \CVP_p to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for \CVP_p and \CVPP_p for any $1 \leq p < \infty$ under different assumptions

COUNTING LATTICE VECTORS

Abstract. We consider the problem of counting the number of lattice vectors of a given length and prove several results regarding its computational complexity. We show that the problem is ♯Pcomplete resolving an open problem. Furthermore, we show that the problem is at least as hard as integer factorization even for lattices of bounded rank or lattices generated by vectors of bounded norm. Next, we discuss a deterministic algorithm for counting the number of lattice vectors of length d in time 2 O(rs+log d) , where r is the rank of the lattice, s is the number of bits that encode the basis of the lattice. The algorithm is based on the theory of modular forms

Counting Lattice Vectors

We consider the problem of counting the number of lattice vectors of a given length. We show that problem is #P-complete resolving an open problem. Furthermore, we show that the problem is at least as hard as integer factorization even for lattices of bounded rank or lattices generated by vectors of bounded norm. Next, we discuss a deterministic algorithm for counting the number of lattice vectors of length d in time .Z0(rs+109d) , where 1- is the rank of the lattice, s is the number of bits that encode the basis of the lattice. The algorithm is based on the theory of modular forms