Gaussians and the smoothing parameter. Gaussian measures have become an increasingly important tool in the study of lattices. For $s > 0$, the Gaussian measure of parameter (or width) $s$ on $\mathbb{R}^n$ is defined as $\rho_s(\mathbf{x}) = \exp(-\pi|\mathbf{x}|^2/s^2)$; for a lattice $\mathcal{L} \subset \mathbb{R}^n$, the Gaussian measure of the lattice is then
Gaussian measures on lattices have innumerable applications, including in worst-case to average-case reductions for lattice problems [MR04, Reg05], the construction of cryptographic primitives [GPV08], the design of algorithms for SVP and CVP [ADRS15, ADS15], and the study of the geometry of lattices [Ban93, Ban95, DR16, RS17].
In all of the above applications, a key quantity is the lattice smoothing parameter [MR04]. Informally, for a parameter $\epsilon > 0$ and a lattice $\mathcal{L}$, the smoothing parameter $\eta_\epsilon(\mathcal{L})$ is the minimal Gaussian parameter that “smooths out” the discrete structure of $\mathcal{L}$, up to error $\epsilon$. Formally, for $\epsilon > 0$ we define
where $\mathcal{L}^* := {\mathbf{w} \in \mathbb{R}^n : \forall \mathbf{y} \in \mathcal{L}, \langle \mathbf{w}, \mathbf{y} \rangle \in \mathbb{Z}}$ is the dual lattice of $\mathcal{L}$. All of the computational applications from the previous paragraph rely in some way on the “smoothness” of the Gaussian with parameter $s \ge \eta_\epsilon(\mathcal{L})$ where $2^{-n} \ll \epsilon < 1/2$.² For example, several of the proof systems from [PV08] start with deterministic reductions to an intermediate problem, which asks whether a lattice is “smooth” or well-separated.
The GapSPP problem. Given the prominence of the smoothing parameter in the theory of lattices, it is natural to ask about the complexity of computing it. Chung et al. [CDLP13] formally defined the problem $\gamma$-GapSPP$_\epsilon$ of approximating the smoothing parameter $\eta_\epsilon(\mathcal{L})$ to within a factor of $\gamma \ge 1$ and gave upper bounds on its complexity in the form of proof systems for remarkably low values of $\gamma$. For example, they showed that $\gamma$-GapSPP$_\epsilon \in$ SZK for $\gamma = O(1 + \sqrt{\log(1/\epsilon)/\log n})$. This in fact subsumes the prior result that $O(\sqrt{n/\log n})$-GapSVP $\in$ SZK of [GG98], via known relationships between the minimum distance and the smoothing parameter.
Chung et al. also showed a worst-case to average-case (quantum) reduction from $\tilde{O}(\sqrt{n}/\alpha)$-GapSPP to a very important average-case problem in lattice-based cryptography, Regev’s Learning With Errors (LWE), which asks us to decode from a random “q-ary” lattice under error proportional to $\alpha$ [Reg05]. Again, this subsumes the prior best reduction for GapSVP due to Regev. Most recently, Dadush and Regev [DR16] showed a similar worst-case to average-case reduction from GapSPP to the Short Integer Solution problem [Ajt96, MR04], another widely used average-case problem in lattice-based cryptography.
In hindsight, the proof systems and reductions of [GG98, Reg05, MR04] can most naturally be viewed as applying to GapSPP all along. This suggests that GapSPP may be a better problem than GapSVP on which to base the security of lattice-based cryptography. However, both [CDLP13] and [DR16] left open several questions and asked for a better understanding of the complexity of GapSPP. In particular, while interactive proof systems for this problem seem to be relatively well understood, nothing nontrivial was previously known about noninteractive proof systems (whether zero knowledge or not) for this problem.
²For $\epsilon = 2^{-\Omega(n)}$ the smoothing parameter is determined (up to a constant factor) by the dual minimum distance, so it is much less interesting to consider as a separate quantity. The upper bound of $1/2$ could be replaced by any constant less than one. For $\epsilon \ge 1$, $\eta_\epsilon(\mathcal{L})$ is still formally defined, but its interpretation in terms of the “smoothness” of the corresponding Gaussian measure over $\mathcal{L}$ is much less clear.