Asymptotically Optimal Hitting Sets Against Polynomials

Our main result is an efficient construction of a hitting setgenerator against the class of polynomials of degreediin the i-th variable. Theseed length of this generator is logD+Ō(log1/2D). Here, logD= Σilog(di+ 1) is a lower bound on the seed length of any hitting setgenerator against this class. Our construction is the first toachieve asymptotically optimal seed length for every choice of theparameters di. In fact, we present anearly linear time construction with this asymptotic guarantee.Furthermore, our results extend to classes of polynomialsparameterized by upper bounds on the number of nonzero terms ineach variable. Underlying our constructions is a general and novelframework that exploits the product structure common to the classesof polynomials we consider. This framework allows us to obtainefficient and asymptotically optimal hitting set generators fromprimitives that need not be optimal or efficient by themselves. As our main corollary, we obtain the first blackbox polynomialidentity tests with an asymptotically optimal randomnessconsumption.

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