<s>
An	O
oblivious	B-Application
RAM	I-Application
(	O
ORAM	B-Application
)	O
simulator	O
is	O
a	O
compiler	B-Language
that	O
transforms	O
algorithms	O
in	O
such	O
a	O
way	O
that	O
the	O
resulting	O
algorithms	O
preserve	O
the	O
input-output	B-General_Concept
behavior	O
of	O
the	O
original	O
algorithm	O
but	O
the	O
distribution	O
of	O
memory	O
access	O
pattern	O
of	O
the	O
transformed	O
algorithm	O
is	O
independent	O
of	O
the	O
memory	O
access	O
pattern	O
of	O
the	O
original	O
algorithm	O
.	O
</s>
<s>
The	O
use	O
of	O
ORAMs	B-Application
is	O
motivated	O
by	O
the	O
fact	O
that	O
an	O
adversary	O
can	O
obtain	O
nontrivial	O
information	O
about	O
the	O
execution	O
of	O
a	O
program	O
and	O
the	O
nature	O
of	O
the	O
data	O
that	O
it	O
is	O
dealing	O
with	O
,	O
just	O
by	O
observing	O
the	O
pattern	O
in	O
which	O
various	O
locations	O
of	O
memory	O
are	O
accessed	O
during	O
its	O
execution	O
.	O
</s>
<s>
The	O
definition	O
suits	O
equally	O
well	O
to	O
the	O
settings	O
of	O
protected	O
programs	O
running	O
on	O
unprotected	O
shared	B-Operating_System
memory	I-Operating_System
as	O
well	O
as	O
a	O
client	O
running	O
a	O
program	O
on	O
its	O
system	O
by	O
accessing	O
previously	O
stored	O
data	O
on	O
a	O
remote	O
server	O
.	O
</s>
<s>
A	O
Turing	B-Architecture
machine	I-Architecture
(	O
TM	O
)	O
,	O
the	O
mathematical	O
abstraction	O
of	O
a	O
real	O
computer	O
(	O
program	O
)	O
,	O
is	O
said	O
to	O
be	O
oblivious	O
if	O
for	O
any	O
two	O
inputs	O
of	O
the	O
same	O
length	O
,	O
the	O
motions	O
of	O
the	O
tape	O
heads	O
remain	O
the	O
same	O
.	O
</s>
<s>
A	O
more	O
realistic	O
model	O
of	O
computation	O
is	O
the	O
RAM	B-Application
model	I-Application
.	O
</s>
<s>
In	O
the	O
RAM	B-Application
model	I-Application
of	O
computation	O
,	O
there	O
is	O
a	O
CPU	B-Device
that	O
can	O
execute	O
the	O
basic	O
mathematical	O
,	O
logical	O
and	O
control	O
instructions	O
.	O
</s>
<s>
The	O
CPU	B-Device
is	O
also	O
associated	O
with	O
a	O
few	O
registers	B-General_Concept
and	O
a	O
physical	O
random	O
access	O
memory	O
,	O
where	O
it	O
stores	O
the	O
operands	O
of	O
its	O
instructions	O
.	O
</s>
<s>
The	O
CPU	B-Device
in	O
addition	O
has	O
instructions	O
to	O
read	O
the	O
contents	O
of	O
a	O
memory	O
cell	O
and	O
write	O
a	O
specific	O
value	O
to	O
a	O
memory	O
cell	O
.	O
</s>
<s>
The	O
definition	O
of	O
ORAMs	B-Application
capture	O
a	O
similar	O
notion	O
of	O
obliviousness	O
memory	O
accesses	O
in	O
this	O
model	O
.	O
</s>
<s>
Informally	O
,	O
an	O
ORAM	B-Application
is	O
an	O
algorithm	O
at	O
the	O
interface	O
of	O
a	O
protected	O
CPU	B-Device
and	O
the	O
physical	O
RAM	O
such	O
that	O
it	O
acts	O
like	O
a	O
RAM	O
to	O
the	O
CPU	B-Device
by	O
querying	O
the	O
physical	O
RAM	O
for	O
the	O
CPU	B-Device
while	O
hiding	O
information	O
about	O
the	O
actual	O
memory	O
access	O
pattern	O
of	O
the	O
CPU	B-Device
from	O
the	O
physical	O
RAM	O
.	O
</s>
<s>
This	O
description	O
will	O
still	O
make	O
sense	O
if	O
the	O
CPU	B-Device
is	O
replaced	O
by	O
a	O
client	O
with	O
a	O
small	O
storage	O
and	O
the	O
physical	O
RAM	O
is	O
replaced	O
with	O
a	O
remote	O
server	O
with	O
a	O
large	O
storage	O
capacity	O
,	O
where	O
the	O
data	O
of	O
the	O
client	O
resides	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
formal	O
definition	O
of	O
ORAMs	B-Application
.	O
</s>
<s>
Let	O
denote	O
a	O
program	O
requiring	O
a	O
memory	O
of	O
size	O
when	O
executing	O
on	O
an	O
input	B-General_Concept
.	O
</s>
<s>
A	O
polynomial-time	O
algorithm	O
,	O
is	O
an	O
Oblivious	B-Application
RAM	I-Application
(	O
ORAM	B-Application
)	O
compiler	B-Language
with	O
computational	O
overhead	O
and	O
memory	O
overhead	O
,	O
if	O
given	O
and	O
a	O
deterministic	B-General_Concept
RAM	I-General_Concept
program	I-General_Concept
with	O
memory-size	O
outputs	O
a	O
program	O
with	O
memory-size	O
such	O
that	O
for	O
any	O
input	B-General_Concept
,	O
the	O
running-time	O
of	O
is	O
bounded	O
by	O
where	O
is	O
the	O
running-time	O
of	O
,	O
and	O
there	O
exists	O
a	O
negligible	O
function	O
such	O
that	O
the	O
following	O
properties	O
hold	O
:	O
</s>
<s>
Note	O
that	O
the	O
above	O
definition	O
uses	O
the	O
notion	O
of	O
statistical	B-Algorithm
security	I-Algorithm
.	O
</s>
<s>
The	O
main	O
result	O
in	O
this	O
work	O
is	O
that	O
there	O
exists	O
an	O
ORAM	B-Application
compiler	B-Language
that	O
uses	O
server	O
space	O
and	O
incurs	O
a	O
running	O
time	O
overhead	O
of	O
when	O
making	O
a	O
program	O
that	O
uses	O
memory	O
cells	O
oblivious	O
.	O
</s>
<s>
This	O
work	O
initiated	O
a	O
series	O
of	O
works	O
in	O
the	O
construction	O
of	O
oblivious	B-Application
RAMs	I-Application
that	O
is	O
going	O
on	O
till	O
date	O
.	O
</s>
<s>
There	O
are	O
several	O
attributes	O
that	O
need	O
to	O
be	O
considered	O
when	O
we	O
compare	O
various	O
ORAM	B-Application
constructions	O
.	O
</s>
<s>
The	O
most	O
important	O
parameters	O
of	O
an	O
ORAM	B-Application
construction	O
are	O
the	O
amounts	O
of	O
client	O
storage	O
,	O
the	O
amount	O
of	O
server	O
storage	O
and	O
the	O
time	O
overhead	O
in	O
making	O
one	O
memory	O
access	O
.	O
</s>
<s>
is	O
the	O
best	O
known	O
ORAM	B-Application
construction	O
.	O
</s>
<s>
Another	O
important	O
attribute	O
of	O
an	O
ORAM	B-Application
construction	O
is	O
whether	O
the	O
access	O
overhead	O
is	O
amortized	B-General_Concept
or	O
worst-case	B-General_Concept
.	O
</s>
<s>
Several	O
of	O
the	O
earlier	O
ORAM	B-Application
constructions	O
have	O
good	O
amortized	B-General_Concept
access	O
overhead	O
guarantees	O
,	O
but	O
have	O
worst-case	B-General_Concept
access	O
overheads	O
.	O
</s>
<s>
Some	O
of	O
the	O
ORAM	B-Application
constructions	O
with	O
polylogarithmic	O
worst-case	B-General_Concept
computational	O
overheads	O
are	O
.	O
</s>
<s>
The	O
authors	O
of	O
also	O
achieve	O
an	O
access	O
overhead	O
of	O
,	O
which	O
is	O
just	O
a	O
log-factor	O
away	O
from	O
the	O
best	O
known	O
ORAM	B-Application
access	O
overhead	O
.	O
</s>
<s>
One	O
of	O
the	O
only	O
known	O
lower	O
bounds	O
on	O
the	O
access	O
overhead	O
of	O
ORAMs	B-Application
is	O
due	O
to	O
Goldreich	O
et	O
al	O
.	O
</s>
<s>
They	O
show	O
a	O
lower	O
bound	O
for	O
ORAM	B-Application
access	O
overhead	O
,	O
where	O
is	O
the	O
data	O
size	O
.	O
</s>
<s>
There	O
is	O
also	O
a	O
conditional	O
lower	O
bound	O
on	O
the	O
access	O
overhead	O
of	O
ORAMs	B-Application
due	O
to	O
Boyle	O
et	O
al	O
.	O
</s>
<s>
A	O
trivial	O
ORAM	B-Application
simulator	O
construction	O
,	O
for	O
each	O
read	O
or	O
write	O
operation	O
,	O
reads	O
from	O
and	O
writes	O
to	O
every	O
single	O
element	O
in	O
the	O
array	O
,	O
only	O
performing	O
a	O
meaningful	O
action	O
for	O
the	O
address	O
specified	O
in	O
that	O
single	O
operation	O
.	O
</s>
<s>
A	O
simple	O
version	O
of	O
a	O
statistically	O
secure	O
ORAM	B-Application
compiler	B-Language
constructed	O
by	O
Chung	O
and	O
Pass	O
is	O
described	O
in	O
the	O
following	O
along	O
with	O
an	O
overview	O
of	O
the	O
proof	O
of	O
its	O
correctness	O
.	O
</s>
<s>
The	O
compiler	B-Language
on	O
input	B-General_Concept
and	O
a	O
program	O
with	O
its	O
memory	O
requirement	O
,	O
outputs	O
an	O
equivalent	O
oblivious	O
program	O
.	O
</s>
<s>
If	O
the	O
input	B-General_Concept
program	O
uses	O
registers	B-General_Concept
,	O
the	O
output	B-General_Concept
program	O
will	O
need	O
registers	B-General_Concept
,	O
where	O
is	O
a	O
parameter	O
of	O
the	O
construction	O
.	O
</s>
<s>
uses	O
memory	O
and	O
its	O
(	O
worst-case	B-General_Concept
)	O
access	O
overhead	O
is	O
.	O
</s>
<s>
The	O
ORAM	B-Application
compiler	B-Language
is	O
very	O
simple	O
to	O
describe	O
.	O
</s>
<s>
The	O
ORAM	B-Application
compiler	B-Language
,	O
when	O
constructing	O
,	O
simply	O
replaces	O
each	O
and	O
instructions	O
with	O
subroutines	O
and	O
and	O
keeps	O
the	O
rest	O
of	O
the	O
program	O
the	O
same	O
.	O
</s>
<s>
It	O
may	O
be	O
noted	O
that	O
this	O
construction	O
can	O
be	O
made	O
to	O
work	O
even	O
for	O
memory	O
requests	O
coming	O
in	O
an	O
online	B-Algorithm
fashion	O
.	O
</s>
<s>
To	O
keep	O
track	O
of	O
this	O
association	O
,	O
also	O
stores	O
a	O
data	O
structure	O
called	O
position	O
map	O
,	O
denoted	O
by	O
,	O
using	O
registers	B-General_Concept
.	O
</s>
<s>
The	O
program	O
starts	O
by	O
initializing	O
its	O
memory	O
as	O
well	O
as	O
registers	B-General_Concept
to	O
.	O
</s>
<s>
For	O
the	O
sub-routine	O
,	O
the	O
input	B-General_Concept
is	O
just	O
a	O
memory	O
location	O
and	O
it	O
is	O
almost	O
the	O
same	O
as	O
.	O
</s>
<s>
Let	O
stand	O
for	O
the	O
ORAM	B-Application
compiler	B-Language
that	O
was	O
described	O
above	O
.	O
</s>
<s>
Let	O
denote	O
the	O
execution	O
of	O
the	O
program	O
on	O
an	O
input	B-General_Concept
using	O
memory	O
cells	O
.	O
</s>
<s>
Let	O
denote	O
a	O
function	O
such	O
that	O
for	O
any	O
,	O
for	O
any	O
program	O
and	O
for	O
any	O
input	B-General_Concept
,	O
the	O
probability	O
that	O
outputs	O
an	O
overflow	O
is	O
at	O
most	O
.	O
</s>
<s>
Given	O
a	O
program	O
,	O
with	O
probability	O
at	O
least	O
,	O
the	O
output	B-General_Concept
of	O
is	O
identical	O
to	O
the	O
output	B-General_Concept
of	O
.	O
</s>
<s>
The	O
following	O
lemma	O
completes	O
the	O
proof	O
of	O
correctness	O
of	O
the	O
ORAM	B-Application
scheme	O
.	O
</s>
<s>
Overflow	O
LemmaThere	O
exists	O
a	O
negligible	O
function	O
such	O
that	O
for	O
every	O
program	O
,	O
every	O
and	O
input	B-General_Concept
,	O
the	O
program	O
outputs	O
overflow	O
with	O
probability	O
at	O
most	O
.	O
</s>
