We show direct and conceptually simple reductions between the classical learning with errors (LWE) problem and its continuous analog, CLWE (Bruna, Regev, Song and Tang, STOC 2021). This allows us to bring to bear the powerful machinery of LWE-based cryptography to the applications of CLWE. For example, we obtain the hardness of CLWE under the classical worst-case hardness of the gap shortest vector problem. Previously, this was known only under quantum worst-case hardness of lattice problems. More broadly, with our reductions between the two problems, any future developments to LWE will also apply to CLWE and its downstream applications.As a concrete application, we show an improved hardness result for density estimation for mixtures of Gaussians. In this computational problem, given sample access to a mixture of Gaussians, the goal is to output a function that estimates the density function of the mixture. Under the (plausible and widely believed) exponential hardness of the classical LWE problem, we show that Gaussian mixture density estimation in R n with roughly log n Gaussian components given poly(n) samples requires time quasi-polynomial in n. Under the (conservative) polynomial hardness of LWE, we show hardness of density estimation for n ǫ Gaussians for any constant ǫ > 0, which improves on Bruna, Regev, Song and Tang (STOC 2021), who show hardness for at least √ n Gaussians under polynomial (quantum) hardness assumptions. Our key technical tool is a reduction from classical LWE to LWE with k-sparse secrets where the multiplicative increase in the noise is only O( √ k), independent of the ambient dimension n.

In recent literature, moonshine has been explored for some groups beyond the Monster, for example the sporadic O'Nan and Thompson groups. This collection of examples may suggest that moonshine is a rare phenomenon, but a fundamental and largely unexplored question is how general the correspondence is between modular forms and finite groups. For every finite group G, we give constructions of infinitely many graded infinite-dimensional C[G]-modules where the McKay-Thompson series for a conjugacy class [g] is a weakly holomorphic modular function properly on Γ 0 (ord(g)). As there are only finitely many normalized Hauptmoduln, groups whose McKay-Thompson series are normalized Hauptmoduln are rare, but not as rare as one might naively expect. We give bounds on the powers of primes dividing the order of groups which have normalized Hauptmoduln of level ord(g) as the graded trace functions for any conjugacy class [g], and completely classify the finite abelian groups with this property. In particular, these include (Z/5Z) 5 and (Z/7Z) 4 , which are not subgroups of the Monster.

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions [4], and is provably not hard under "local" reductions computable in TIME(n 0.49 ) [26]. The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of

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