Zero-Knowledge for Multivariate Polynomials

In [12] a Zero-Knowledge scheme ZK(2) was designed from a solution of a set of multivariate quadratic equations over a finite field. In this paper we will give two methods to generalize this construction for polynomials of any degree d, i.e. we will design two Zero-Knowledge schemes ZK(d) and ̃ ZK(d) from a set of polynomial equations of degree d. We will show that ̃ ZK(d) is optimal in term of the number of computations to be performed and that ZK(d) is optimal in term of the number of bits to be send. Moreover this property is still true for all kinds of polynomials: for example if the polynomials are sparse or dense. Finally, we will present two examples of applications: with Brent equations, or with morphisms of polynomials.