Lattice Problems Beyond Polynomial Time

Abstract

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also show a novel protocol. 1. We prove that secret-key cryptography exists if $\widetilde{O}(\sqrt{n})$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. I.e., we extend to our setting (Micciancio and Regev's improved version of) Ajtai's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n)$-approximate SVP to SIS. 2. We prove that public-key cryptography exists if $\widetilde{O}(n)$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. This extends to our setting Regev's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n^{1.5})$-approximate SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical, generalizing Peikert's polynomial-time classical reduction from $\widetilde{O}(n^2)$-approximate SVP. 3. We show a $2^{\varepsilon n}$-time coAM protocol for $O(1)$-approximate CVP, generalizing the celebrated polynomial-time protocol for $O(\sqrt{n/\log n})$-CVP due to Goldreich and Goldwasser. These results show complexity-theoretic barriers to extending the recent line of fine-grained hardness results for CVP and SVP to larger approximation factors. (This result also extends to arbitrary norms.) 4. We show a $2^{\varepsilon n}$-time co-non-deterministic protocol for $O(\sqrt{\log n})$-approximate SVP, generalizing the (also celebrated!) polynomial-time protocol for $O(\sqrt{n})$-CVP due to Aharonov and Regev. 5. We give a novel coMA protocol for $O(1)$-approximate CVP with a $2^{\varepsilon n}$-time verifier. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs.