Computationally Private Randomizing Polynomials and Their Applications

Abstract

Randomizing polynomials allow to represent a function f(x) by a low-degree randomized mapping f/spl circ/(x, r) whose output distribution on an input x is a randomized encoding of f(x). It is known that any function f in /spl oplus/L/poly (and in particular in NC/sup 1/) can be efficiently represented by degree-3 randomizing polynomials. Such a degree-3 representation gives rise to an NC/sub 4//sup 0/ representation, in which every bit of the output depends on only 4 bits of the input. In this paper, we study the relaxed notion of computationally private randomizing polynomials, where the output distribution of f/spl circ/(x, r) should only be computationally indistinguishable from a randomized encoding of f(x). We construct degree-3 randomizing polynomials of this type for every polynomial-time computable function, assuming the existence of a cryptographic pseudorandom generator (PRG) in /spl oplus/L/poly. (The latter assumption is implied by most standard intractability assumptions used in cryptography.) This result is obtained by combining a variant of Yao's garbled circuit technique with previous information-theoretic constructions of randomizing polynomials.