Solving the learning parity with noiseʼs open question

Abstract

The classical cryptographic primitives are constructed on the assumptions that the private key is securely kept and uniformly distributed. Learning parity with noise is a famous problem used to construct several cryptographic primitives. This research studies the open question about the hardness of the learning parity with noise assumption when the secret vector is not uniform and has sufficient min-entropy. The proofs show that the standard learning parity with noise implies that it is secure even if the secret vector is sampled from an arbitrary distribution with sufficient entropy. Furthermore, this paper shows that the symmetric encryption scheme from learning parity with noise is secure even if the secret key has min-entropy at least k.