On Approximating the Entropy of Polynomial Mappings
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چکیده
We investigate the complexity of the following computational problem: POLYNOMIAL ENTROPY APPROXIMATION (PEA): Given a low-degree polynomial mapping p : F → F, where F is a finite field, approximate the output entropy H(p(Un)), where Un is the uniform distribution on F and H may be any of several entropy measures. We show: • Approximating the Shannon entropy of degree 3 polynomials p : F2 → F2 over F2 to within an additive constant (or even n) 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 6= F2 and homogeneous quadratic polynomials p : F → F, there is a probabilistic polynomial-time algorithm that distinguishes the case that p(Un) has entropy smaller than k from the case that p(Un) has min-entropy (or even Renyi entropy) greater than (2 + o(1))k. • For degree d polynomials p : F2 → F2 , there is a polynomial-time algorithm that distinguishes the case that p(Un) 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(Un) has max-entropy at least (1 + o(1)) · k (for fixed d and large k).
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تاریخ انتشار 2010