 
Summary: Computationally Private Randomizing Polynomials
and Their Applications
Benny Applebaum Yuval Ishai Eyal Kushilevitz
Computer Science Department, Technion
{abenny,yuvali,eyalk}@cs.technion.ac.il
March 5, 2006
Abstract
Randomizing polynomials allow to represent a function f(x) by a lowdegree randomized mapping
^f(x, r) whose output distribution on an input x is a randomized encoding of f(x). It is known that any
function f in uniformL/poly (and in particular in NC1
) can be efficiently represented by degree3
randomizing polynomials. Such a degree3 representation gives rise to an NC0
4 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(x, r) should only be computationally indistinguishable from a ran
domized encoding of f(x). We construct degree3 randomizing polynomials of this type for every
polynomialtime computable function, assuming the existence of a cryptographic pseudorandom gen
erator (PRG) in uniformL/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
