Lattice Problems beyond Polynomial Time

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time. Specifically, we revisit four foundational results in this context—two protocols and two worst-case to average-case reductions. We show how to improve the approximation factor in each result by a factor of roughly √n/logn when running the protocol or reduction in 2є n time instead of polynomial time, and we show a novel protocol with no polynomial-time analog. Our results are as follows. (1) We show a worst-case to average-case reduction proving that secret-key cryptography (specifically, collision-resistant hash functions) exists if the (decision version of the) Shortest Vector Problem (SVP) cannot be approximated to within a factor of Õ(√n) in 2є n time. This extends to our setting Ajtai’s celebrated polynomial-time reduction for the Short Integer Solutions (SIS) problem (1996),which showed (after improvements by Micciancio and Regev (2004, 2007)) that secret-key cryptography exists if SVP cannot be approximated to within a factor of Õ(n) in polynomial time. (2) We show another worst-case to average-case reduction proving that public-key cryptography exists if SVP cannot be approximated to within a factor of Õ(n) in 2є n time. This extends Regev’s celebrated polynomial-time reduction for the Learning with Errors (LWE) problem (2005, 2009), which achieved an approximation factor of Õ(n1.5). In fact, Regev’s reduction is quantum, but we prove our result under a classical reduction, generalizing Peikert’s polynomial-time classical reduction (2009), which achieved an approximation factor of Õ(n2). (3) We show that the (decision version of the) Closest Vector Problem (CVP) with a constant approximation factor has a coAM protocol with a 2є n-time verifier. We prove this via a (very simple) generalization of the celebrated polynomial-time protocol due to Goldreich and Goldwasser (1998, 2000). It follows that the recent series of 2є n-time and even 2(1−є)n-time hardness results for CVP cannot be extended to large constant approximation factors γ unless AMETH is false. We also rule out 2(1−є)n-time lower bounds for any constant approximation factor γ > √2, under plausible complexity-theoretic assumptions. (These results also extend to arbitrary norms, with different constants.) (4) We show that O(√logn)-approximate SVP has a coNTIME protocol with a 2є n-time verifier. Here, the analogous (also celebrated! ) polynomial-time result is due to Aharonov and Regev (2005), who showed a polynomial-time protocol achieving an approximation factor of √n (for both SVP and CVP, while we only achieve this result for CVP). This result implies similar barriers to hardness, with a larger approximation factor under a weaker complexity-theoretic conjectures (as does the next result). (5) Finally, we give a novel coMA protocol for constant-factor-approximate CVP with a 2є n-time verifier. Unlike our other results, this protocol has no known analog in the polynomial-time regime. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs. In particular, the tradeoffs for the first four results smoothly interpolate from the polynomial-time results in prior work to our new results in the exponential-time world.