is an immediate consequence of the Poisson Summation Formula (Lemma 2.5) that $\det(\mathcal{L})^{1/n} \le 2\eta_{1/2}(\mathcal{L})$. Notice that this inequality formalizes the intuitive notion that “a lattice cannot be smooth and sparse simultaneously.”
Dadush and Regev made the simple observation that the same statement is true when we consider projections of the lattice [DR16]. I.e., for any projection $\pi$ such that $\pi(\mathcal{L})$ is still a lattice, we have $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} \le 2\eta_{1/2}(\mathcal{L})$, where $\operatorname{rank}(\pi(\mathcal{L}))$ is the dimension of the span of $\pi(\mathcal{L})$. (Indeed, this fact is immediate from the above together with the identity $(\pi(\mathcal{L}))^* = \mathcal{L}^* \cap \operatorname{span}(\pi(\mathcal{L}))$.) Therefore, if we define
where the maximum is taken over all projections $\pi$ such that $\pi(\mathcal{L})$ is a lattice, then we have
Dadush and Regev conjectured that Equation (1.1) is tight up to a factor of polylog(n). I.e., up to polylog factors, a lattice is not smooth if and only if it has a sparse projection. Regev and Stephens-Davidowitz proved this conjecture [RS17], and the resulting theorem, presented below, will be our main technical tool.
Theorem 1.6 ([RS17]). For any lattice $\mathcal{L} \subset \mathbb{R}^n$,
I.e., if $\eta_{1/2}(\mathcal{L}) \ge 10(\log n + 2)$, then there exists a lattice projection $\pi$ such that $\det(\pi(\mathcal{L})) \ge 1$.
coNP proof system. Notice that Theorem 1.6 (together with Equation (1.1)) immediately implies that $O(\log n)$-GapSPP$\epsilon$ is in coNP for $\epsilon = 1/2$. Indeed, a projection $\pi$ such that $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} \ge \eta{1/2}(\mathcal{L})/O(\log n)$ can be used as a witness of “non-smoothness.” Theorem 1.6 shows that such a witness always exists, and Equation (1.1) shows that no such witness exists with $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} > 2\eta_{1/2}(\mathcal{L})$. In order to extend this result to all $\epsilon \in (0, 1)$, we use basic results about how $\eta_\epsilon(\mathcal{L})$ varies with $\epsilon$. (See Section 4.)
NISZK proof systems. We give two different NISZK proof systems for $O(\log(n)\sqrt{\log(1/\epsilon)})$-GapSPP$_\epsilon$, both of which rely on Theorem 1.6.
Our first proof system (shown in Figure 1, Section 3.1) uses many vectors $t_1, \dots, t_m$ sampled uniformly at random from a fundamental region of the lattice $\mathcal{L}$ as the common random string. The prover samples short vectors $e_i$ (for $i = 1, \dots, m$) from the discrete Gaussian distributions over the lattice cosets $e_i + \mathcal{L}$. The verifier accepts if and only if the matrix $E = \sum e_i e_i^T$ has small enough spectral norm. (I.e., the verifier accepts if the $e_i$ are “short in all directions.”) In fact, Peikert and Vaikuntanathan used the exact same proof system for the different lattice problem $O(\sqrt{n})$-coGapSVP, and their proofs of correctness and zero knowledge also apply to our setting. However, the proof of soundness is quite different: we show that, if the lattice has a sparse projection $\pi$, then $\operatorname{dist}(\pi(t_i), \pi(\mathcal{L}))$ will tend to be fairly large. It follows that $\sum ||\pi(e_i)||^2 = \operatorname{Tr}(\sum \pi(e_i)\pi(e_i)^T)$ will be fairly large with high probability, and therefore $\sum e_i e_i^T$ must have large spectral norm.
Our second proof system follows from a reduction to the Entropy Approximation problem, which asks to estimate the entropy of the output distribution of a circuit on random input. Goldreich, Sahai, and Vadhan [GSV99] showed that Entropy Approximation is NISZK-complete, so that a problem is in NISZK if and only if it can be (Karp-)reduced to approximating the entropy of a circuit. If $\eta_\epsilon(\mathcal{L})$ is small, then we