We present a new method for secure two-party Random Access Memory (RAM) program computation that does not require taking a program and first turning it into a circuit. The method achieves logarithmic overhead compared to an insecure program execution. In the heart of our construction is a new Oblivious RAM construction where a client interacts with two non-communicating servers. Our two-server Oblivious RAM for n reads/writes requires O(n) memory for the servers, O(1) memory for the client, and

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... O(log n) amortized read/write overhead for data access. The constants in the big-O notation are tiny, and we show that the storage and data access overhead of our solution concretely compares favorably to the state-ofthe-art single-server schemes. Our protocol enjoys an important feature from a practical perspective as well. At the heart of almost all previous single-server Oblivious RAM solutions, a crucial but inefficient process known as oblivious sorting was required. In our two-server model, we describe a new technique to bypass oblivious sorting, and show how this can be carefully blended with existing techniques to attain a more practical Oblivious RAM protocol in comparison to all prior work. As alluded above, our two-server Oblivious RAM protocol leads to a novel application in the realm of secure two-party RAM program computation. We observe that in the secure two-party computation, Alice and Bob can play the roles of two non-colluding servers. We show that our Oblivious RAM construction can be composed with an extended version of the Ostrovsky-Shoup compiler to obtain a new method for secure two-party program computation with lower overhead than all existing constructions.