Middle-Product Learning with Errors

We introduce a new variant \(\mathsf {MP}\text {-}\mathsf {LWE}\) of the Learning With Errors problem (\(\mathsf {LWE}\)) making use of the Middle Product between polynomials modulo an integer q. We exhibit a reduction from the Polynomial-\(\mathsf {LWE}\) problem (\(\mathsf {PLWE}\)) parametrized by a polynomial f, to \(\mathsf {MP}\text {-}\mathsf {LWE}\) 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 \(\mathsf {MP}\text {-}\mathsf {LWE}\) hardness assumption. The scheme is hence secure under the assumption that \(\mathsf {PLWE}\) is hard for at least one polynomial f of degree n among a family of f’s which is exponential in n.

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