Obfuscation for evasive functions

An evasive circuit family is a collection of circuits such that for every input x, a random circuit from outputs 0 on x with overwhelming probability. We provide a combination of definitional, constructive, and impossibility results regarding obfuscation for evasive functions: 1 The (average case variants of the) notions of virtual black box obfuscation (Barak et al, CRYPTO '01) and virtual gray box obfuscation (Bitansky and Canetti, CRYPTO '10) coincide for evasive function families. We also define the notion of input-hiding obfuscation for evasive function families, stipulating that for a random it is hard to find, given, a value outside the preimage of 0. Interestingly, this natural definition, also motivated by applications, is likely not implied by the seemingly stronger notion of average-case virtual black-box obfuscation. 2 If there exist average-case virtual gray box obfuscators for all evasive function families, then there exist (quantitatively weaker) average-case virtual gray obfuscators for all function families. 3 There does not exist a worst-case virtual black box obfuscator even for evasive circuits, nor is there an average-case virtual gray box obfuscator for evasive Turing machine families. 4 Let be an evasive circuit family consisting of functions that test if a low-degree polynomial (represented by an efficient arithmetic circuit) evaluates to zero modulo some large prime p. Then under a natural analog of the discrete logarithm assumption in a group supporting multilinear maps, there exists an input-hiding obfuscator for. Under a new perfectly-hiding multilinear encoding assumption, there is an average-case virtual black box obfuscator for the family.