Rajendra Kumar

Divesh Aggarwal, Yanlin Chen, Rajendra Kumar et al.

$ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}} \newcommand{\eps}{\varepsilon} $We show a number of reductions between the Shortest Vector Problem and the Closest Vector Problem over lattices in different $\ell_p$ norms ($\SVP_p$ and $\CVP_p$ respectively). Specifically, we present the following $2^{\eps m}$-time reductions for $1 \leq p \leq q \leq \infty$, which all increase the rank $n$ and dimension $m$ of the input lattice by at most one: $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p})γ$-approximate $\SVP_q$ to $γ$-approximate $\SVP_p$; $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p}) γ$-approximate $\CVP_p$ to $γ$-approximate $\CVP_q$; and $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1+1/p})$-$\CVP_q$ to $(1+\eps)$-unique $\SVP_p$ (which in turn trivially reduces to $(1+\eps)$-approximate $\SVP_p$). The last reduction is interesting even in the case $p = q$. In particular, this special case subsumes much prior work adapting $2^{O(m)}$-time $\SVP_p$ algorithms to solve $O(1)$-approximate $\CVP_p$. In the (important) special case when $p = q$, $1 \leq p \leq 2$, and the $\SVP_p$ oracle is exact, we show a stronger reduction, from $O(1/\eps^{1/p})\text{-}\CVP_p$ to (exact) $\SVP_p$ in $2^{\eps m}$ time. For example, taking $\eps = \log m/m$ and $p = 2$ gives a slight improvement over Kannan's celebrated polynomial-time reduction from $\sqrt{m}\text{-}\CVP_2$ to $\SVP_2$. We also note that the last two reductions can be combined to give a reduction from approximate-$\CVP_p$ to $\SVP_q$ for any $p$ and $q$, regardless of whether $p \leq q$ or $p > q$. Our techniques combine those from the recent breakthrough work of Eisenbrand and Venzin (which showed how to adapt the current fastest known algorithm for these problems in the $\ell_2$ norm to all $\ell_p$ norms) together with sparsification-based techniques.

Divesh Aggarwal, Yanlin Chen, Rajendra Kumar et al.

The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. $\bullet$ A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer $4\leq q\leq \sqrt{n}$, our algorithm takes $q^{13n+o(n)}$ time and requires $poly(n)\cdot q^{16n/q^2}$ memory. This tradeoff which ranges from enumeration ($q=\sqrt{n}$) to sieving ($q$ constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. $\bullet$ A quantum algorithm for SVP that runs in time $2^{0.950n+o(n)}$ and requires $2^{0.5n+o(n)}$ classical memory and poly(n) qubits. In Quantum Random Access Memory (QRAM) model this algorithm takes only $2^{0.835n+o(n)}$ time and requires a QRAM of size $2^{0.293n+o(n)}$, poly(n) qubits and $2^{0.5n}$ classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [ADRS15] that has a time and space complexity $2^{n+o(n)}$. $\bullet$ A classical algorithm for SVP that runs in time $2^{1.669n+o(n)}$ time and $2^{0.5n+o(n)}$ space. This improves over an algorithm of [CCL18] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number which is $2^{0.402n}$. We conjecture that for most lattices this quantity is a $2^{o(n)}$. Assuming that this is the case, our classical algorithm runs in time $2^{1.292n+o(n)}$, our quantum algorithm runs in time $2^{0.750n+o(n)}$ and our quantum algorithm in QRAM model runs in time $2^{0.667n+o(n)}$.