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A family of trapdoor functions is one-way under correlated inputs if no efficient adversary can invert it even when given the value of the function on multiple correlated inputs. This powerful primitive was introduced at TCC 2009 by Rosen and Segev, who use it in an elegant black box construction of a chosen ciphertext secure public key encryption. In this work we continue the study of security under correlated inputs, and prove that there is no black box construction of correlation secure injective trapdoor functions from classic trapdoor permutations, even if the latter is assumed to be one-way for inputs from high entropy, rather than uniform distributions. Our negative result holds for all input distributions where each xi is determined by the remaining n−1 coordinates. The techniques we employ for proving lower bounds about trapdoor permutations are new and quite general, and we believe that they will find other applications in the area of black-box separations.

Two Is a Crowd? A Black-Box Separation of One-Wayness and Security under Correlated Inputs