On Ideal Lattices and Learning with Errors Over Rings

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The "Learning With Errors" (LWE) problem is to distinguish random linear equations, which have been perturbed by a small amount of noise, from truly uniform ones. The problem has been shown to be as hard as worst-case lattice problems, and in recent years it has served as the foundation for a plethora of cryptographic applications. Unfortunately, these applications are rather inefficient due to an inherent quadratic overhead in the use of LWE. A main open question was whether LWE and its applications could be made truly efficient by exploiting extra algebraic structure, as was done for lattice-based hash functions (and related primitives). The authors resolve this question in the affirmative by introducing an algebraic variant of LWE called ring-LWE, and proving that it too enjoys very strong hardness guarantees.