Sieve Method for Polynomial Linear Equivalence

Abstract

We consider the polynomial linear equivalence (PLE) problem arising from the multivariate public key cryptography, which is defined as to find an invertible linear transformation $ℒ$ satisfying $𝒫=𝒮\circ ℒ$ for given nonlinear polynomial maps $𝒫$ and $𝒮$ over a finite field ${𝔽}_q$. Some cryptographic and algebraic properties of PLE are discussed, and from the properties we derive three sieves called multiplicative, differential, and additive sieves. By combining the three sieves, we propose a sieve method for the PLE problem. As an application of our sieve method, we show that it is infeasible to construct public key encryption schemes from the PLE problem.