Obfuscation of Evasive Algebraic Set Membership

Abstract

We define the membership function of a set as the function that determines whether an input is an element of the set. Canetti, Rothblum, and Varia showed how to obfuscate evasive membership functions of hyperplanes over a finite field of order an exponentially large prime, assuming the hardness of a modified decisional Diffie-Hellman problem. Barak, Bitansky, Canetti, Kalai, Paneth, and Sahai extended their work from hyperplanes to hypersurfaces of bounded degree, assuming multilinear maps. Both works are limited to algebraic sets over large fields of prime orders, and are based on less standard assumptions, although they prove virtual black-box security. In this paper, we handle much more general algebraic sets based on more standard assumptions, and prove input-hiding security, which is not weaker nor stronger than virtual black-box security (i.e., they are incomparable). Our first obfuscator handles affine algebraic sets over finite fields of order an arbitrary prime power. It is based on the preimage-resistance property of cryptographic hash function families. Our second obfuscator applies to both affine and projective algebraic sets over finite fields of order a polynomial size prime power. It is based on the same hardness assumption(s) required by input-hiding small superset obfuscation. Our paper is the first to handle the obfuscation problem of projective algebraic sets over small finite fields