This paper unifies and extends preliminary versions that appeared in STOC'87 and STOC'90 (authored by O.G. and R.O., respectively).
A machine is oblivious if the sequence in which it accesses memory locations is equivalent for any two inputs with the same running time. For example, an oblivious Turing Machine is one for which the movement of the heads on the tapes is identical for each computation. (Thus, the movement is independent of the actual input.) What is the slowdown in the running time of a machine, if it is required to be oblivious? In 1979 Pippenger and Fischer showed how a two-tape oblivious Turing Machine can simulate, on-line, a one-tape Turing Machine, with a logarithmic slowdown in the running time. We show an analogous result for the random-access machine (RAM) model of computation. In particular, we show how to do an on-line simulation of an arbitrary RAM by a probabilistic oblivious RAM with a poly-logarithmic slowdown in the running time. On the other hand, we show that a logarithmic slowdown is a lower bound.