Making RAMs Oblivious Requires Superlogarithmic Overhead

We prove a time-space tradeoff lower bound of T = Ω ( n log( n S ) log log( n S ) ) for randomized oblivious branching programs to compute 1GAP , also known as the pointer jumping problem, a problem for which there is a simple deterministic time n and space O(log n) RAM (random access machine) algorithm. Ajtai [3, 4] and Damgård, Meldgaard, and Nielsen [11] recently derived simulations of general RAMs by randomized oblivious RAMs with only a polylogarithmic factor increase in time and space. Our lower bound implies that a superlogarithmic factor increase is indeed necessary in any such simulation.