The traditional notion of program obfuscation requires that an obfuscation ˜ P of a program P computes the exact same function as P, but beyond that, the code of ˜ P should not leak any information about P. This strong notion of virtual black-box security was shown by Barak et al. (CRYPTO 2001) to be impossible to achieve, for certain unobfuscatable function families. The same work raised the question of approximate obfuscation, where the obfuscated ˜ P is only required to approximate P; that is, ˜ P only agrees with P with high enough probability on some input distribution. We show that, assuming trapdoor permutations, there exist families of robust unobfuscatable functions for which even approximate obfuscation is impossible. Specifically, obfuscation is impossible even if the obfuscated ˜ P is only required to agree with P with probability slightly more than 1 2, on a uniformly sampled input (below 1 2-agreement, the function obfuscated by ˜ P is not uniquely defined). Additionally, assuming only one-way functions, we rule out approximate obfuscation where ˜P may output ⊥ with probability close to 1, but otherwise must agree with P
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