Obfuscation of Evasive Algebraic Set Membership

Abstract

Canetti, Rothblum, and Varia showed how to obfuscate membership testing in a hyperplane over a finite field of exponentially large prime order, assuming the membership predicate is evasive and the under a modified DDH assumption. Barak, Bitansky, Canetti, Kalai, Paneth, and Sahai extended their work from hyperplanes to hypersurfaces (of bounded degree), assuming multi-linear maps. In this paper we give much more general obfuscation tools that allow to obfuscate evasive membership testing in arbitrary algebraic sets (including projective sets) over finite fields of arbitrary (prime power) order. We give two schemes and prove input-hiding security based on relatively standard assumptions. The first scheme is based on the preimage resistance property of cryptographic hash functions; and the second scheme is based on the hardness assumptions required for small superset obfuscation. We also introduce a new security notion called span-hiding, and prove that the second scheme achieves span-hiding assuming small superset obfuscation. One special case of algebraic sets over finite fields is boolean polynomials, which means our methods can be applied to obfuscate any evasive function defined by a polynomial-size collection of boolean polynomials. As a corollary, we obtain an input-hiding obfuscator for evasive functions defined by circuits in NC^0