<s>
Verifiable	B-Operating_System
computing	I-Operating_System
(	O
or	O
verified	O
computation	O
or	O
verified	B-Operating_System
computing	I-Operating_System
)	O
enables	O
a	O
computer	O
to	O
offload	B-Operating_System
the	O
computation	O
of	O
some	O
function	O
,	O
to	O
other	O
perhaps	O
untrusted	O
clients	B-Operating_System
,	O
while	O
maintaining	O
verifiable	O
results	O
.	O
</s>
<s>
The	O
other	O
clients	B-Operating_System
evaluate	O
the	O
function	O
and	O
return	O
the	O
result	O
with	O
a	O
proof	O
that	O
the	O
computation	O
of	O
the	O
function	O
was	O
carried	O
out	O
correctly	O
.	O
</s>
<s>
The	O
introduction	O
of	O
this	O
notion	O
came	O
as	O
a	O
result	O
of	O
the	O
increasingly	O
common	O
phenomenon	O
of	O
"	O
outsourcing	B-Architecture
"	O
computation	O
to	O
untrusted	O
users	O
in	O
projects	O
such	O
as	O
SETI	B-Application
@home	I-Application
and	O
also	O
to	O
the	O
growing	O
desire	O
of	O
weak	O
clients	B-Operating_System
to	O
outsource	B-Architecture
computational	O
tasks	O
to	O
a	O
more	O
powerful	O
computation	O
service	O
like	O
in	O
cloud	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
)	O
,	O
"	O
delegating	O
computations	O
"	O
,	O
"	O
certified	O
computation	O
"	O
,	O
and	O
verifiable	B-Operating_System
computing	I-Operating_System
.	O
</s>
<s>
The	O
term	O
verifiable	B-Operating_System
computing	I-Operating_System
itself	O
was	O
formalized	O
by	O
Rosario	O
Gennaro	O
,	O
Craig	O
Gentry	O
,	O
and	O
Bryan	O
Parno	O
,	O
and	O
echoes	O
Micali	O
's	O
"	O
certified	O
computation	O
"	O
.	O
</s>
<s>
The	O
growing	O
desire	O
to	O
outsource	B-Architecture
computational	O
tasks	O
from	O
a	O
relatively	O
weak	O
computational	O
device	O
(	O
client	O
)	O
to	O
a	O
more	O
powerful	O
computation	O
services	O
(	O
worker	O
)	O
,	O
and	O
the	O
problem	O
of	O
dishonest	O
workers	O
who	O
modify	O
their	O
client	O
's	O
software	O
to	O
return	O
plausible	O
results	O
without	O
performing	O
the	O
actual	O
work	O
motivated	O
the	O
formalization	O
of	O
the	O
notion	O
of	O
Verifiable	O
Computation	O
.	O
</s>
<s>
Verifiable	B-Operating_System
computing	I-Operating_System
is	O
not	O
only	O
concerned	O
with	O
getting	O
the	O
result	O
of	O
the	O
outsourced	O
function	O
on	O
the	O
client	O
's	O
input	O
and	O
the	O
proof	O
of	O
its	O
correctness	O
,	O
but	O
also	O
with	O
the	O
client	O
being	O
able	O
to	O
verify	O
the	O
proof	O
with	O
significantly	O
less	O
computational	O
effort	O
than	O
computing	O
the	O
function	O
from	O
scratch	O
.	O
</s>
<s>
These	O
verifications	O
are	O
either	O
interactive	O
which	O
require	O
the	O
client	O
to	O
interact	O
with	O
the	O
worker	O
to	O
verify	O
the	O
correctness	O
proof	O
,	O
or	O
are	O
non-interactive	O
protocols	O
which	O
can	O
be	O
proven	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
.	O
</s>
<s>
The	O
largest	O
verified	O
computation	O
(	O
SETI	B-Application
@home	I-Application
)	O
uses	O
verification	O
by	O
replication	O
.	O
</s>
<s>
The	O
SETI	B-Application
@home	I-Application
verification	O
process	O
involves	O
one	O
client	O
machine	O
and	O
many	O
worker	O
machines	O
.	O
</s>
<s>
The	O
public	O
information	O
is	O
sent	O
to	O
the	O
worker	O
to	O
compute	O
F	O
on	O
the	O
input	B-General_Concept
data	I-General_Concept
.	O
</s>
<s>
Output	B-General_Concept
computation	O
and	O
verification	O
.	O
</s>
<s>
In	O
this	O
stage	O
,	O
the	O
worker	O
uses	O
the	O
public	O
information	O
associated	O
with	O
the	O
function	O
F	O
and	O
the	O
input	O
,	O
which	O
are	O
calculated	O
in	O
the	O
previous	O
two	O
phases	O
,	O
to	O
compute	O
an	O
encoded	B-Algorithm
output	B-General_Concept
of	O
the	O
function	O
F	O
on	O
the	O
provided	O
input	O
.	O
</s>
<s>
This	O
result	O
is	O
then	O
returned	O
to	O
the	O
client	O
to	O
verify	O
its	O
correctness	O
by	O
computing	O
the	O
actual	O
value	O
of	O
the	O
output	B-General_Concept
by	O
decoding	O
the	O
result	O
returned	O
by	O
the	O
worker	O
using	O
the	O
private	O
information	O
calculated	O
in	O
the	O
previous	O
phases	O
.	O
</s>
<s>
defined	O
a	O
verifiable	O
computation	O
scheme	O
for	O
any	O
function	O
F	O
using	O
Yao	B-Protocol
's	I-Protocol
garbled	I-Protocol
circuit	I-Protocol
combined	O
with	O
a	O
fully	O
homomorphic	O
encryption	O
system	O
.	O
</s>
<s>
The	O
public	B-Application
key	I-Application
encodes	O
the	O
target	O
function	O
F	O
and	O
is	O
sent	O
to	O
the	O
worker	O
to	O
compute	O
F	O
.	O
On	O
the	O
other	O
hand	O
,	O
the	O
secret	O
key	O
is	O
kept	O
private	O
by	O
the	O
client	O
.	O
</s>
<s>
Compute(PK, σx )	O
→	O
σy	O
:	O
The	O
worker	O
computes	O
an	O
encoded	B-Algorithm
value	O
σy	O
of	O
the	O
function	O
's	O
output	B-General_Concept
y	O
=	O
F(x )	O
using	O
the	O
client	O
's	O
public	B-Application
key	I-Application
PK	O
and	O
the	O
encoded	B-Algorithm
input	O
σx	O
.	O
</s>
<s>
VerifySK	O
( τx	O
,	O
σy	O
)	O
→	O
y	O
∪	O
⊥	O
:	O
The	O
verification	O
algorithm	O
converts	O
the	O
worker	O
's	O
encoded	B-Algorithm
output	B-General_Concept
σy	O
into	O
the	O
actual	O
output	B-General_Concept
of	O
the	O
function	O
F	O
using	O
both	O
the	O
secret	O
key	O
SK	O
and	O
the	O
secret	O
“	O
decoding	O
”	O
τx	O
.	O
</s>
<s>
It	O
outputs	O
y	O
=	O
F(x )	O
if	O
the	O
σy	O
represents	O
a	O
valid	O
output	B-General_Concept
of	O
F	O
on	O
x	O
,	O
or	O
outputs	O
⊥	O
otherwise	O
.	O
</s>
<s>
The	O
public	B-Application
key	I-Application
(	O
PK	O
)	O
is	O
composed	O
of	O
all	O
the	O
ciphertexts	O
that	O
represent	O
the	O
garbled	B-Protocol
circuit	I-Protocol
,	O
and	O
the	O
secret	O
key	O
(	O
SK	O
)	O
is	O
composed	O
of	O
all	O
the	O
random	O
wire	O
labels	O
.	O
</s>
<s>
This	O
algorithm	O
first	O
generates	O
a	O
new	O
pair	O
of	O
public	O
and	O
secret	O
keys	O
for	O
the	O
homomorphic	O
encryption	O
scheme	O
,	O
and	O
then	O
uses	O
these	O
keys	O
with	O
the	O
homomorphic	O
scheme	O
to	O
encrypt	O
the	O
correct	O
input	O
wires	O
,	O
represented	O
as	O
the	O
secret	O
key	O
of	O
the	O
garbled	B-Protocol
circuit	I-Protocol
.	O
</s>
<s>
The	O
produced	O
ciphertexts	O
represent	O
the	O
public	O
encoding	B-Algorithm
of	O
the	O
input	O
( σx	O
)	O
that	O
is	O
given	O
to	O
the	O
worker	O
,	O
while	O
the	O
secret	O
key	O
( τx	O
)	O
is	O
kept	O
private	O
by	O
the	O
client	O
.	O
</s>
<s>
After	O
that	O
,	O
the	O
worker	O
applies	O
the	O
computation	O
steps	O
of	O
the	O
Yao	B-Protocol
's	I-Protocol
protocol	I-Protocol
over	O
the	O
ciphertexts	O
generated	O
by	O
the	O
problem	O
generation	O
algorithm	O
.	O
</s>
<s>
This	O
is	O
done	O
by	O
recursively	O
decrypting	O
the	O
gate	O
ciphertexts	O
until	O
arriving	O
to	O
the	O
final	O
output	B-General_Concept
wire	O
values	O
( σy	O
)	O
.	O
</s>
<s>
The	O
homomorphic	O
properties	O
of	O
the	O
encryption	O
scheme	O
enable	O
the	O
worker	O
to	O
obtain	O
an	O
encryption	O
of	O
the	O
correct	O
output	B-General_Concept
wire	O
.	O
</s>
<s>
Finally	O
,	O
the	O
worker	O
returns	O
the	O
ciphertexts	O
of	O
the	O
output	B-General_Concept
to	O
the	O
client	O
who	O
decrypts	O
them	O
to	O
compute	O
the	O
actual	O
output	B-General_Concept
y	O
=	O
F(x )	O
or	O
⊥	O
.	O
</s>
<s>
Scheme	O
Correctness	O
is	O
achieved	O
if	O
the	O
problem	O
generation	O
algorithm	O
produces	O
values	O
that	O
enable	O
an	O
honest	O
worker	O
to	O
compute	O
encoded	B-Algorithm
output	B-General_Concept
values	O
that	O
will	O
verify	O
successfully	O
and	O
correspond	O
to	O
the	O
evaluation	O
of	O
F	O
on	O
those	O
inputs	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
a	O
verifiable	O
computation	O
scheme	O
is	O
secure	O
if	O
a	O
malicious	O
worker	O
cannot	O
convince	O
the	O
verification	O
algorithm	O
to	O
accept	O
an	O
incorrect	O
output	B-General_Concept
for	O
a	O
given	O
function	O
F	O
and	O
input	O
x	O
.	O
</s>
<s>
Although	O
it	O
was	O
shown	O
that	O
verifiable	B-Operating_System
computing	I-Operating_System
is	O
possible	O
in	O
theory	O
(	O
using	O
fully	O
homomorphic	O
encryption	O
or	O
via	O
probabilistically	O
checkable	O
proofs	O
)	O
,	O
most	O
of	O
the	O
known	O
constructions	O
are	O
very	O
expensive	O
in	O
practice	O
.	O
</s>
<s>
In	O
the	O
2010s	O
,	O
verifiable	B-Operating_System
computing	I-Operating_System
techniques	O
have	O
seen	O
an	O
increase	O
of	O
practical	O
applications	O
in	O
blockchain	O
technology	O
.	O
</s>
