Oblivious RAM with logarithmic overhead

1. A method comprising:

executing, by data processing hardware, an instruction to execute a query (q) for a data block (B), the data block (B) associated with a corresponding memory level (l i ) of a logarithmic number of memory levels (l i ) of memory, each memory level (l i ) comprising physical memory (RAM i ) residing on memory hardware of a distributed system in communication with the data processing hardware;

retrieving, by the data processing hardware, a value (v) associated with the data block (B) from an oblivious hash table using a corresponding key (k);

extracting, by the data processing hardware, un-queried key value pairs (k, v) from the oblivious hash table associated with un-queried data blocks after executing a threshold number of queries (q) for data blocks; and

executing, by the data processing hardware, a multi-array shuffle routine on the extracted key value pairs from the oblivious hash table to generate an output array containing the un-queried key value pairs (k, v).

2. The method of claim 1 , wherein executing the multi-array shuffle routine comprises:

merging one or more input arrays of data blocks (B) each having a capacity less than a threshold capacity into a new array; and

shuffling the new array obliviously.

3. The method of claim 1 , wherein the logarithmic number of memory levels (l i ) comprise a logarithmic number of memory levels (l i ) of increasing size where each memory level (l i ) has a storage capacity equal to the joint capacity of all preceding smaller memory levels (l i ).

4. The method of claim 1 , wherein the instruction to execute the query (q) for the data block (B) comprises an operation including a read or write operation on the data block and an address indicating a memory location for the data block (B).

5. The method of claim 4 , wherein the instruction to execute the query (q) for the data block (B) further comprises data when the operation includes the write operation.

6. The method of claim 1 , further comprising, when the corresponding memory level (l i ) associated with the data block (B) is not a lowest memory level (l l ), moving the data block (B) to the lowest memory level (l l ) after executing the instruction to execute the query (q).

7. The method of claim 6 , further comprising, after moving the data block (B) to the lowest memory level (l l ), updating, by the data processing hardware, a memory-level map in communication with the data processing hardware to indicate that the corresponding memory level (l i ) associated with the data block (B) now includes the lowest memory level (l l ).

8. The method of claim 1 , wherein the data processing hardware resides on a client device.

9. The method of claim 8 , wherein the client device stores a memory-level map in memory hardware of the client device, the memory-level map mapping each data block stored on the memory hardware of the distributed system to a corresponding query memory level (l q ).

10. The method of claim 1 , further comprising:

executing, by the data processing hardware, an instruction to execute a new query (q) for another data block (B);

determining, by the data processing hardware, whether the other data block (B) is stored locally on memory hardware of a client device; and

when the other data block (B) is stored locally on the memory hardware of the client device, retrieving, by the data processing hardware, the other data block (B) from the memory hardware of the client device.

11. The method of claim 10 , further comprising, when the other data block (B) is stored locally on the memory hardware of the client device, issuing, by the data processing hardware, one or more fake queries to the distributed system for retrieving a corresponding dummy block (D) to conceal the retrieval of the other data block (B) from the memory hardware of the client device.

12. A system comprising:

data processing hardware; and

memory hardware in communication with the data processing hardware and storing instructions, that when executed by the data processing hardware, cause the data processing hardware to perform operations comprising:

executing an instruction to execute a query (q) for a data block (B), the data block (B) associated with a corresponding memory level (l i ) of a logarithmic number of memory levels (l i ) of memory, each memory level (l i ) comprising physical memory (RAM i ) residing on a storage abstraction of a distributed system in communication with the data processing hardware;

retrieving a value (v) associated with the data block (B) from an oblivious hash table using a corresponding key (k);

extracting un-queried key value pairs (k, v) from the oblivious hash table associated with un-queried data blocks after executing a threshold number of queries (q) for data blocks; and

executing a multi-array shuffle routine on the extracted key value pairs from the oblivious hash table to generate an output array containing the un-queried key value pairs (k, v).

13. The system of claim 12 , wherein executing the multi-array shuffle routine comprises:

merging one or more input arrays of data blocks (B) each having a capacity less than a threshold capacity into a new array; and

shuffling the new array obliviously.

14. The system of claim 12 , wherein the logarithmic number of memory levels (l i ) comprise a logarithmic number of memory levels (l i ) of increasing size where each memory level (l i ) has a storage capacity equal to the joint capacity of all preceding smaller memory levels (l i ).

15. The system of claim 12 , wherein the instruction to execute the query (q) for the data block (B) comprises an operation including a read or write operation on the data block and an address indicating a memory location for the data block (B).

16. The system of claim 15 , wherein the instruction to execute the query (q) for the data block (B) further comprises data when the operation includes the write operation.

17. The system of claim 12 , wherein the operations further comprise, when the corresponding memory level (l i ) associated with the data block (B) is not a lowest memory level (l l ), moving the data block (B) to the lowest memory level (l i ) after executing the instruction to execute the query (q).

18. The system of claim 17 , wherein the operations further comprise, after moving the data block (B) to the lowest memory level (l l ), updating a memory-level map stored on the memory hardware to indicate that the corresponding memory level (l i ) associated with the data block (B) now includes the lowest memory level (l l ).

19. The system of claim 12 , wherein the data processing hardware and the memory hardware reside on a client device.

20. The system of claim 12 , wherein the memory hardware stores a memory-level map, the memory-level map mapping each data block stored on the storage abstraction of the distributed system to a corresponding query memory level (l q ).

21. The system of claim 12 , wherein the operations further comprise:

executing an instruction to execute a new query (q) for another data block (B);

determining whether the other data block (B) is stored locally on the memory hardware; and

when the other data block (B) is stored locally on the memory hardware, retrieving the other data block (B) from the memory hardware.

22. The system of claim 21 , wherein the operations further comprise, when the other data block (B) is stored locally on the memory hardware, issuing one or more fake queries to the storage abstraction for retrieving a corresponding dummy block (D) to conceal the retrieval of the other data block (B) from the memory hardware.