On the complexity of parallel hardness amplification for one-way functions

Abstract

We prove complexity lower bounds for the tasks of hardness amplification of one-way functions and construction of pseudo-random generators from one-way functions, which are realized non-adaptively in black-box ways. First, we consider the task of converting a one-way function f : {0, 1}n → {0, 1}m into a harder one-way function f̄ : {0, 1} n̄ → {0, 1}m̄, with n̄,m̄ < poly(n), in a black-box way. The hardness is measured as the fraction of inputs any polynomial-size circuit must fail to invert. We show that to use a constant-depth circuit to amplify hardness beyond a polynomial factor, its size must exceed 2poly(n), and to amplify hardness beyond a 2 o(n) factor, its size must exceed 22o(n). Moreover, for a constant-depth circuit to amplify hardness beyond an n1+o(1) factor in a security preserving way (with n̄ = O(n)), it size must exceed 2 no(1) Next, we show that if a constant-depth polynomial-size circuit can amplify hardness beyond a polynomial factor in a weakly black-box way, then it must basically embed a hard function in itself. In fact, one can derive from such an amplification procedure a highly parallel one-way function, which is computable by an NC0 circuit (constant-depth polynomial-size circuit with bounded fan-in gates). Finally, we consider the task of constructing a pseudo-random generator G : {0, 1}n̄ → {0, 1} m̄ from a strongly one-way function f : {0, 1}n → {0, 1}m in a black-box way. We show that any such a construction realized by a constant-depth 2no(1) -size circuit can only have a sublinear stretch (with m̄ - n̄ = o(n̄)). © Springer-Verlag Berlin Heidelberg 2006.