Middle-Product Learning with Errors

We introduce a new variant MP-LWE of the Learning With Errors problem (LWE) making use of the Middle Product between polynomials modulo an integer q. We exhibit a reduction from the PolynomialLWE problem (PLWE) parametrized by a polynomial f , to MP-LWE which is defined independently of any such f . The reduction only requires f to be monic with constant coefficient coprime with q. It incurs a noise growth proportional to the so-called expansion factor of f . We also describe a public-key encryption scheme with quasi-optimal asymptotic efficiency (the bit-sizes of the keys and the run-times of all involved algorithms are quasi-linear in the security parameter), which is secure against chosen plaintext attacks under the MP-LWE hardness assumption. The scheme is hence secure under the assumption that PLWE is hard for at least one polynomial f of degree n among a family of f ’s which is exponential in n.