We present a radically new approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), {\em without Gentry's bootstrapping procedure}.

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This paper aims to derandomize the following problems in the smoothed analysis of Spielman and Teng. Learn Disjunctive Normal Form (DNF), invert Fourier Transforms (FT), and verify small circuits' unsatisfiability. Learning algorithms must predict a future observation from the only $m$ i.i.d. samples of a fixed but unknown joint-distribution $P(G(x),y)$ ... more >>>

The Noisy Linear Equations problem involves finding the solution to a random linear system over a finite field given noisy evaluations. This is a generalisation of various learning problems widely used in cryptography, including Learning with Errors (LWE), Learning with Rounding (LWR), and Learning Parity with Noise (LPN).

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