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Journal Articles Journal of the ACM (JACM) Year : 2013

On Ideal Lattices and Learning with Errors over Rings


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). We 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. Specifically, we show that the ring-LWE distribution is pseudorandom, assuming that worst-case problems on ideal lattices are hard for polynomial- time quantum algorithms. Applications include the first truly practical lattice-based public-key cryptosys- tem with an efficient security reduction; moreover, many of the other applications of LWE can be made much more efficient through the use of ring-LWE
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hal-00921792 , version 1 (21-12-2013)



Vadim Lyubashevsky, Chris Peikert, Oded Regev. On Ideal Lattices and Learning with Errors over Rings. Journal of the ACM (JACM), 2013, 60 (6), ⟨10.1145/2535925⟩. ⟨hal-00921792⟩
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