TR18-109 Authors: Kasper Green Larsen, Jesper Buus Nielsen

Publication: 4th June 2018 19:57

Downloads: 1469

Keywords:

An Oblivious RAM (ORAM) introduced by Goldreich and Ostrovsky

[JACM'96] is a (possibly randomized) RAM, for which the memory access

pattern reveals no information about the operations

performed. The main performance metric of an ORAM is the bandwidth

overhead, i.e., the multiplicative factor extra memory blocks that must be

accessed to hide the operation sequence. In their seminal paper

introducing the ORAM, Goldreich and Ostrovsky proved an amortized

$\Omega(\lg n)$ bandwidth overhead lower bound for ORAMs with memory

size $n$. Their lower bound is

very strong in the sense that it applies to the ``offline'' setting in

which the ORAM knows the entire sequence of operations ahead of time.

However, as pointed out by Boyle and

Naor [ITCS'16] in the paper ``Is there an oblivious RAM lower bound?'', there are two caveats with the lower bound of

Goldreich and Ostrovsky: (1) it only applies to ``balls in bins'' algorithms,

i.e., algorithms where the ORAM may only shuffle blocks around and

not apply any sophisticated encoding of the data, and (2), it only

applies to statistically secure constructions. Boyle and Naor showed

that removing the ``balls in bins'' assumption would result in super

linear lower bounds for sorting circuits, a long standing open problem

in circuit complexity. As a way to circumventing this barrier, they also

proposed a notion of an ``online'' ORAM, which is an ORAM that remains

secure even if the operations arrive in an online manner. They argued that most

known ORAM constructions work in the online setting as well.

Our contribution is an

$\Omega(\lg n)$ lower bound on the bandwidth overhead of any online

ORAM, even if we require only computational security and allow arbitrary representations of data, thus greatly

strengthening the lower bound of Goldreich and Ostrovsky in the online setting.

Our lower bound applies to ORAMs with memory size $n$ and any word size

$r \geq 1$. The bound therefore asymptotically matches the

known upper bounds when $r = \Omega(\lg^2 n)$.