Learning with errors

Learning with errors (LWE) is a problem in machine learning that is conjectured to be hard to solve. Introduced by Oded Regev in 2005, it is a generalization of the parity learning problem. Regev showed, furthermore, that the LWE problem is as hard to solve as several worst-case lattice problems. The LWE problem has recently been used as a hardness assumption to create public-key cryptosystems. such as the ring learning with errors key exchange by Peikert.

An algorithm is said to solve the LWE problem if, when given access to samples where and , with the assurance, for some fixed linear function that with high probability and deviates from it according to some known noise model, the algorithm can recreate or some close approximation of it with high probability.

Definition

Denote by the additive group on reals modulo one. Denote by the distribution on obtained by choosing a vector uniformly at random, choosing according to a probability distribution on and outputting for some fixed vector . Here is the standard inner product , the division is done in the field of reals (or more formally, this "division by " is notation for the group homomorphism mapping to ), and the final addition is in .