Cryptographic hardness of distribution-specific learning

We investigate cryptographic lower bounds on the learnability of Boolean formulas and constant depth circuits on the {niform distribution and other specifi; distributions. We first show that weakly learning Boolean formulas and constant depth threshold circuits with membership queries on the uniform distribution in polynomial time is as hard as factoring Blum integers (or inverting RSA, or deciding 1? quadratic residuosity . We formalize the notion of a trivially learnable distri ution and extend these hardness results to all non-trivial distributions. Moreover, we show that under appropriate assumptions on the hardness of factoring, the learnability of Boolean formulas and constant depth threshold circuits on any distribution is characterized by the distribution’s Renyi entropy. Furthermore, we show that a sub-exponential lower bound for factoring implies a Q(2’Og@ ‘‘) lower bound (for some constant ~) for learning Boolean circuits of depth d on the uniform distribution (with membership queries), which matches the upper bound of Linial, M ansour, and Nisan [19]. From this we conclude that, assuming such a lower bou-nd for factoring, there is no O(npOLy 10gn ) algorithm to learn all of ACO on the uniform distribution. We observe that, under cryptographic assumptions, all our bounds can be used to establish trade~trs between the running time and the number of samples necessary to learn.