On Approximating the Entropy of Polynomial Mappings

Abstract

We investigate the complexity of Polynomial Entropy Approximation (PEA): Given a low-degree polynomial mapping p : F^n-> F^m, where F is a finite field, approximate the output entropy H(p(U_n)), where U_n is the uniform distribution on F^n and H may be any of several entropy measures. We show: Approximating the Shannon entropy of degree 3 polynomials p : F_2^n->F_2^m over F_2 to within an additive constant (or even n^{.9}) is complete for SZKPL, the class of problems having statistical zero-knowledge proofs where the honest verifier and its simulator are computable in logarithmic space. (SZKPL contains most of the natural problems known to be in the full class SZKP.) For prime fields F\neq F_2 and homogeneous quadratic polynomials p : F^n->F^m, there is a probabilistic polynomial-time algorithm that distinguishes the case that p(U_n) has entropy smaller than k from the case that p(U_n) has min-entropy (or even Renyi entropy) greater than (2+o(1))k. For degree d polynomials p : F_2^n->F_2^m, there is a polynomial-time algorithm that distinguishes the case that p(U_n) has max-entropy smaller than k (where the max-entropy of a random variable is the logarithm of its support size) from the case that p(U_n) has max-entropy at least (1+o(1))k^d (for fixed d and large k).