<s>
Secure	B-Protocol
multi-party	I-Protocol
computation	I-Protocol
(	O
also	O
known	O
as	O
secure	B-Protocol
computation	I-Protocol
,	O
multi-party	B-Protocol
computation	I-Protocol
(	O
MPC	O
)	O
or	O
privacy-preserving	B-Protocol
computation	I-Protocol
)	O
is	O
a	O
subfield	O
of	O
cryptography	O
with	O
the	O
goal	O
of	O
creating	O
methods	O
for	O
parties	O
to	O
jointly	O
compute	O
a	O
function	O
over	O
their	O
inputs	O
while	O
keeping	O
those	O
inputs	O
private	O
.	O
</s>
<s>
Unlike	O
traditional	O
cryptographic	O
tasks	O
,	O
where	O
cryptography	O
assures	O
security	O
and	O
integrity	O
of	O
communication	O
or	O
storage	O
and	O
the	O
adversary	B-General_Concept
is	O
outside	O
the	O
system	O
of	O
participants	O
(	O
an	O
eavesdropper	O
on	O
the	O
sender	O
and	O
receiver	O
)	O
,	O
the	O
cryptography	O
in	O
this	O
model	O
protects	O
participants	O
 '	O
privacy	O
from	O
each	O
other	O
.	O
</s>
<s>
The	O
foundation	O
for	O
secure	B-Protocol
multi-party	I-Protocol
computation	I-Protocol
started	O
in	O
the	O
late	O
1970s	O
with	O
the	O
work	O
on	O
mental	B-Algorithm
poker	I-Algorithm
,	O
cryptographic	O
work	O
that	O
simulates	O
game	O
playing/computational	O
tasks	O
over	O
distances	O
without	O
requiring	O
a	O
trusted	O
third	O
party	O
.	O
</s>
<s>
Later	O
,	O
secure	B-Protocol
computation	I-Protocol
was	O
formally	O
introduced	O
as	O
secure	B-General_Concept
two-party	I-General_Concept
computation	I-General_Concept
(	O
2PC	O
)	O
in	O
1982	O
(	O
for	O
the	O
so-called	O
Millionaires	O
 '	O
Problem	O
,	O
a	O
specific	O
problem	O
which	O
is	O
a	O
Boolean	O
predicate	O
)	O
,	O
and	O
in	O
generality	O
(	O
for	O
any	O
feasible	O
computation	O
)	O
in	O
1986	O
by	O
Andrew	O
Yao	O
.	O
</s>
<s>
The	O
area	O
is	O
also	O
referred	O
to	O
as	O
Secure	B-General_Concept
Function	I-General_Concept
Evaluation	I-General_Concept
(	O
SFE	O
)	O
.	O
</s>
<s>
The	O
computation	O
is	O
based	O
on	O
secret	O
sharing	O
of	O
all	O
the	O
inputs	O
and	O
zero-knowledge	O
proofs	O
for	O
a	O
potentially	O
malicious	O
case	O
,	O
where	O
the	O
majority	O
of	O
honest	O
players	O
in	O
the	O
malicious	O
adversary	B-General_Concept
case	O
assure	O
that	O
bad	O
behavior	O
is	O
detected	O
and	O
the	O
computation	O
continues	O
with	O
the	O
dishonest	O
person	O
eliminated	O
or	O
his	O
input	O
revealed	O
.	O
</s>
<s>
This	O
work	O
introduced	O
an	O
approach	O
,	O
known	O
as	O
GMW	O
paradigm	O
,	O
for	O
compiling	O
a	O
multi-party	B-Protocol
computation	I-Protocol
protocol	O
which	O
is	O
secure	O
against	O
semi-honest	O
adversaries	B-General_Concept
to	O
a	O
protocol	O
that	O
is	O
secure	O
against	O
malicious	O
adversaries	B-General_Concept
.	O
</s>
<s>
The	O
above	O
results	O
are	O
in	O
a	O
model	O
where	O
the	O
adversary	B-General_Concept
is	O
limited	O
to	O
polynomial	O
time	O
computations	O
,	O
and	O
it	O
observes	O
all	O
communications	O
,	O
and	O
therefore	O
the	O
model	O
is	O
called	O
the	O
`computational	O
model	O
 '	O
.	O
</s>
<s>
The	O
above	O
results	O
established	O
that	O
it	O
is	O
possible	O
under	O
the	O
above	O
variations	O
to	O
achieve	O
secure	B-Protocol
computation	I-Protocol
when	O
the	O
majority	O
of	O
users	O
are	O
honest	O
.	O
</s>
<s>
The	O
next	O
question	O
to	O
solve	O
was	O
the	O
case	O
of	O
secure	O
communication	O
channels	O
where	O
the	O
point-to-point	O
communication	O
is	O
not	O
available	O
to	O
the	O
adversary	B-General_Concept
;	O
in	O
this	O
case	O
it	O
was	O
shown	O
that	O
solutions	O
can	O
be	O
achieved	O
with	O
up	O
to	O
1/3	O
of	O
the	O
parties	O
being	O
misbehaving	O
and	O
malicious	O
,	O
and	O
the	O
solutions	O
apply	O
no	O
cryptographic	O
tools	O
(	O
since	O
secure	O
communication	O
is	O
available	O
)	O
.	O
</s>
<s>
Over	O
the	O
years	O
,	O
the	O
notion	O
of	O
general	O
purpose	O
multi-party	O
protocols	O
became	O
a	O
fertile	O
area	O
to	O
investigate	O
basic	O
and	O
general	O
protocol	O
issues	O
properties	O
on	O
,	O
such	O
as	O
universal	O
composability	O
or	O
mobile	B-General_Concept
adversary	I-General_Concept
as	O
in	O
proactive	O
secret	O
sharing	O
.	O
</s>
<s>
The	O
first	O
large-scale	O
and	O
practical	O
application	O
of	O
multi-party	B-Protocol
computation	I-Protocol
was	O
the	O
execution	O
of	O
an	O
electronic	O
double	O
auction	O
in	O
the	O
Danish	O
Sugar	O
Beet	O
Auction	O
,	O
which	O
took	O
place	O
in	O
January	O
2008	O
.	O
</s>
<s>
Informally	O
speaking	O
,	O
the	O
most	O
basic	O
properties	O
that	O
a	O
multi-party	B-Protocol
computation	I-Protocol
protocol	O
aims	O
to	O
ensure	O
are	O
:	O
</s>
<s>
A	O
multi-party	B-Protocol
computation	I-Protocol
protocol	O
must	O
be	O
secure	O
to	O
be	O
effective	O
.	O
</s>
<s>
Nevertheless	O
,	O
it	O
is	O
not	O
always	O
possible	O
to	O
formalize	O
the	O
cryptographic	B-Protocol
protocol	I-Protocol
security	O
verification	O
based	O
on	O
the	O
party	O
knowledge	O
and	O
the	O
protocol	O
correctness	O
.	O
</s>
<s>
Nonetheless	O
,	O
in	O
1987	O
it	O
was	O
demonstrated	O
that	O
any	O
function	O
can	O
be	O
securely	O
computed	O
,	O
with	O
security	O
for	O
malicious	O
adversaries	B-General_Concept
and	O
the	O
other	O
initial	O
works	O
mentioned	O
before	O
.	O
</s>
<s>
Unconditionally	O
or	O
information-theoretically	O
secure	O
MPC	O
is	O
closely	O
related	O
and	O
builds	O
on	O
to	O
the	O
problem	O
of	O
secret	O
sharing	O
,	O
and	O
more	O
specifically	O
verifiable	O
secret	O
sharing	O
(	O
VSS	O
)	O
,	O
which	O
many	O
secure	O
MPC	O
protocols	O
use	O
against	O
active	O
adversaries	B-General_Concept
.	O
</s>
<s>
Unlike	O
traditional	O
cryptographic	O
applications	O
,	O
such	O
as	O
encryption	O
or	O
signature	O
,	O
one	O
must	O
assume	O
that	O
the	O
adversary	B-General_Concept
in	O
an	O
MPC	O
protocol	O
is	O
one	O
of	O
the	O
players	O
engaged	O
in	O
the	O
system	O
(	O
or	O
controlling	O
internal	O
parties	O
)	O
.	O
</s>
<s>
Adversaries	B-General_Concept
faced	O
by	O
the	O
different	O
protocols	O
can	O
be	O
categorized	O
according	O
to	O
how	O
willing	O
they	O
are	O
to	O
deviate	O
from	O
the	O
protocol	O
.	O
</s>
<s>
There	O
are	O
essentially	O
two	O
types	O
of	O
adversaries	B-General_Concept
,	O
each	O
giving	O
rise	O
to	O
different	O
forms	O
of	O
security	O
(	O
and	O
each	O
fits	O
into	O
different	O
real	O
world	O
scenario	O
)	O
:	O
</s>
<s>
This	O
is	O
a	O
naive	O
adversary	B-General_Concept
model	O
,	O
yielding	O
weak	O
security	O
in	O
real	O
situations	O
.	O
</s>
<s>
Malicious	O
(	O
Active	O
)	O
Security	O
:	O
In	O
this	O
case	O
,	O
the	O
adversary	B-General_Concept
may	O
arbitrarily	O
deviate	O
from	O
the	O
protocol	O
execution	O
in	O
its	O
attempt	O
to	O
cheat	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
majority	O
of	O
misbehaving	O
parties	O
:	O
The	O
only	O
thing	O
that	O
an	O
adversary	B-General_Concept
can	O
do	O
in	O
the	O
case	O
of	O
dishonest	O
majority	O
is	O
to	O
cause	O
the	O
honest	O
parties	O
to	O
"	O
abort	O
"	O
having	O
detected	O
cheating	O
.	O
</s>
<s>
Security	O
against	O
active	O
adversaries	B-General_Concept
typically	O
leads	O
to	O
a	O
reduction	O
in	O
efficiency	O
that	O
leads	O
to	O
covert	O
security	O
,	O
a	O
relaxed	O
form	O
of	O
active	O
security	O
.	O
</s>
<s>
Covert	O
security	O
captures	O
more	O
realistic	O
situations	O
,	O
where	O
active	O
adversaries	B-General_Concept
are	O
willing	O
to	O
cheat	O
but	O
only	O
if	O
they	O
are	O
not	O
caught	O
.	O
</s>
<s>
In	O
a	O
way	O
,	O
covert	O
adversaries	B-General_Concept
are	O
active	O
ones	O
forced	O
to	O
act	O
passively	O
due	O
to	O
external	O
non-cryptographic	O
(	O
e.g.	O
</s>
<s>
Like	O
many	O
cryptographic	B-Protocol
protocols	I-Protocol
,	O
the	O
security	O
of	O
an	O
MPC	O
protocol	O
can	O
rely	O
on	O
different	O
assumptions	O
:	O
</s>
<s>
The	O
model	O
might	O
assume	O
that	O
participants	O
use	O
a	O
synchronized	O
network	O
,	O
where	O
a	O
message	O
sent	O
at	O
a	O
"	O
tick	O
"	O
always	O
arrives	O
at	O
the	O
next	O
"	O
tick	O
"	O
,	O
or	O
that	O
a	O
secure	O
and	O
reliable	O
broadcast	O
channel	O
exists	O
,	O
or	O
that	O
a	O
secure	O
communication	O
channel	O
exists	O
between	O
every	O
pair	O
of	O
participants	O
where	O
an	O
adversary	B-General_Concept
cannot	O
read	O
,	O
modify	O
or	O
generate	O
messages	O
in	O
the	O
channel	O
,	O
etc	O
.	O
</s>
<s>
The	O
set	O
of	O
honest	O
parties	O
that	O
can	O
execute	O
a	O
computational	O
task	O
is	O
related	O
to	O
the	O
concept	O
of	O
access	B-General_Concept
structure	I-General_Concept
.	O
</s>
<s>
Adversary	B-General_Concept
structures	O
can	O
be	O
static	O
,	O
where	O
the	O
adversary	B-General_Concept
chooses	O
its	O
victims	O
before	O
the	O
start	O
of	O
the	O
multi-party	B-Protocol
computation	I-Protocol
,	O
or	O
dynamic	O
,	O
where	O
it	O
chooses	O
its	O
victims	O
during	O
the	O
course	O
of	O
execution	O
of	O
the	O
multi-party	B-Protocol
computation	I-Protocol
making	O
the	O
defense	O
harder	O
.	O
</s>
<s>
An	O
adversary	B-General_Concept
structure	O
can	O
be	O
defined	O
as	O
a	O
threshold	O
structure	O
or	O
as	O
a	O
more	O
complex	O
structure	O
.	O
</s>
<s>
In	O
a	O
threshold	O
structure	O
the	O
adversary	B-General_Concept
can	O
corrupt	O
or	O
read	O
the	O
memory	O
of	O
a	O
number	O
of	O
participants	O
up	O
to	O
some	O
threshold	O
.	O
</s>
<s>
There	O
are	O
major	O
differences	O
between	O
the	O
protocols	O
proposed	O
for	O
two	O
party	O
computation	O
(	O
2PC	O
)	O
and	O
multi-party	B-Protocol
computation	I-Protocol
(	O
MPC	O
)	O
.	O
</s>
<s>
Indeed	O
,	O
secure	B-Protocol
multi-party	I-Protocol
computation	I-Protocol
(	O
in	O
fact	O
the	O
restricted	O
case	O
of	O
secure	B-General_Concept
function	I-General_Concept
evaluation	I-General_Concept
,	O
where	O
only	O
a	O
single	O
function	O
is	O
evaluated	O
)	O
was	O
first	O
presented	O
in	O
the	O
two-party	O
setting	O
.	O
</s>
<s>
The	O
original	O
work	O
is	O
often	O
cited	O
as	O
being	O
from	O
one	O
of	O
the	O
two	O
papers	O
of	O
Yao	O
;	O
although	O
the	O
papers	O
do	O
not	O
actually	O
contain	O
what	O
is	O
now	O
known	O
as	O
Yao	B-Protocol
's	I-Protocol
garbled	I-Protocol
circuit	I-Protocol
protocol	I-Protocol
.	O
</s>
<s>
Yao	O
's	O
basic	O
protocol	O
is	O
secure	O
against	O
semi-honest	O
adversaries	B-General_Concept
and	O
is	O
extremely	O
efficient	O
in	O
terms	O
of	O
number	O
of	O
rounds	O
,	O
which	O
is	O
constant	O
,	O
and	O
independent	O
of	O
the	O
target	O
function	O
being	O
evaluated	O
.	O
</s>
<s>
At	O
a	O
high	O
level	O
,	O
the	O
sender	O
prepares	O
the	O
garbled	B-Protocol
circuit	I-Protocol
and	O
sends	O
it	O
to	O
the	O
receiver	O
,	O
who	O
obliviously	O
evaluates	O
the	O
circuit	O
,	O
learning	O
the	O
encodings	O
corresponding	O
to	O
both	O
his	O
and	O
the	O
sender	O
's	O
output	O
.	O
</s>
<s>
In	O
more	O
detail	O
,	O
the	O
garbled	B-Protocol
circuit	I-Protocol
is	O
computed	O
as	O
follows	O
.	O
</s>
<s>
If	O
one	O
is	O
considering	O
malicious	O
adversaries	B-General_Concept
,	O
further	O
mechanisms	O
to	O
ensure	O
correct	O
behavior	O
of	O
both	O
parties	O
need	O
to	O
be	O
provided	O
.	O
</s>
<s>
By	O
construction	O
it	O
is	O
easy	O
to	O
show	O
security	O
for	O
the	O
sender	O
if	O
the	O
OT	O
protocol	O
is	O
already	O
secure	O
against	O
malicious	O
adversary	B-General_Concept
,	O
as	O
all	O
the	O
receiver	O
can	O
do	O
is	O
to	O
evaluate	O
a	O
garbled	B-Protocol
circuit	I-Protocol
that	O
would	O
fail	O
to	O
reach	O
the	O
circuit-output	O
wires	O
if	O
he	O
deviated	O
from	O
the	O
instructions	O
.	O
</s>
<s>
For	O
example	O
,	O
he	O
may	O
send	O
an	O
incorrect	O
garbled	B-Protocol
circuit	I-Protocol
that	O
computes	O
a	O
function	O
revealing	O
the	O
receiver	O
's	O
input	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
possible	O
to	O
efficiently	O
apply	O
Zero-Knowledge	O
proofs	O
to	O
make	O
this	O
protocol	O
secure	O
against	O
malicious	O
adversaries	B-General_Concept
with	O
a	O
small	O
overhead	O
comparing	O
to	O
the	O
semi-honest	O
protocol	O
.	O
</s>
<s>
Two	O
types	O
of	O
secret	O
sharing	O
schemes	O
are	O
commonly	O
used	O
;	O
Shamir	B-Architecture
secret	I-Architecture
sharing	I-Architecture
and	O
additive	O
secret	O
sharing	O
.	O
</s>
<s>
Secret	O
sharing	O
schemes	O
can	O
tolerate	O
an	O
adversary	B-General_Concept
controlling	O
up	O
to	O
t	O
parties	O
out	O
of	O
n	O
total	O
parties	O
,	O
where	O
t	O
varies	O
based	O
on	O
the	O
scheme	O
,	O
the	O
adversary	B-General_Concept
can	O
be	O
passive	O
or	O
active	O
,	O
and	O
different	O
assumptions	O
are	O
made	O
on	O
the	O
power	O
of	O
the	O
adversary	B-General_Concept
.	O
</s>
<s>
The	O
Shamir	B-Architecture
secret	I-Architecture
sharing	I-Architecture
scheme	O
is	O
secure	O
against	O
a	O
passive	O
adversary	B-General_Concept
when	O
and	O
an	O
active	O
adversary	B-General_Concept
when	O
while	O
achieving	O
information-theoretic	O
security	O
,	O
meaning	O
that	O
even	O
if	O
the	O
adversary	B-General_Concept
has	O
unbounded	O
computational	O
power	O
,	O
they	O
cannot	O
learn	O
any	O
information	O
about	O
the	O
secret	O
underlying	O
a	O
share	O
.	O
</s>
<s>
Additive	O
secret	O
sharing	O
schemes	O
can	O
tolerate	O
the	O
adversary	B-General_Concept
controlling	O
all	O
but	O
one	O
party	O
,	O
that	O
is	O
,	O
while	O
maintaining	O
security	O
against	O
a	O
passive	O
and	O
active	O
adversary	B-General_Concept
with	O
unbounded	O
computational	O
power	O
.	O
</s>
<s>
Some	O
protocols	O
require	O
a	O
setup	O
phase	O
,	O
which	O
may	O
only	O
be	O
secure	O
against	O
a	O
computationally	O
bounded	O
adversary	B-General_Concept
.	O
</s>
<s>
The	O
most	O
popular	O
is	O
SPDZ	O
,	O
which	O
implements	O
MPC	O
with	O
additive	O
secret	O
shares	O
and	O
is	O
secure	O
against	O
active	O
adversaries	B-General_Concept
.	O
</s>
<s>
In	O
2014	O
a	O
"	O
model	O
of	O
fairness	O
in	O
secure	B-Protocol
computation	I-Protocol
in	O
which	O
an	O
adversarial	O
party	O
that	O
aborts	O
on	O
receiving	O
output	O
is	O
forced	O
to	O
pay	O
a	O
mutually	O
predefined	O
monetary	O
penalty	O
"	O
has	O
been	O
described	O
for	O
the	O
Bitcoin	B-Protocol
network	O
or	O
for	O
fair	O
lottery	O
.	O
</s>
<s>
The	O
second	O
component	O
can	O
then	O
garble	O
the	O
circuit	O
and	O
execute	O
a	O
protocol	O
to	O
securely	O
evaluate	O
the	O
garbled	B-Protocol
circuit	I-Protocol
.	O
</s>
<s>
As	O
well	O
as	O
two-party	O
computation	O
based	O
on	O
Yao	B-Protocol
's	I-Protocol
protocol	I-Protocol
,	O
Fairplay	O
can	O
also	O
carry	O
out	O
multi-party	O
protocols	O
.	O
</s>
<s>
More	O
recently	O
,	O
there	O
has	O
been	O
a	O
focus	O
on	O
highly	O
parallel	O
implementations	O
based	O
on	O
garbled	B-Protocol
circuits	I-Protocol
,	O
designed	O
to	O
be	O
run	O
on	O
CPUs	B-General_Concept
with	O
many	O
cores	O
.	O
</s>
<s>
To	O
accomplish	O
this	O
they	O
developed	O
a	O
custom	O
,	O
better	O
optimized	O
circuit	O
compiler	O
than	O
Fairplay	O
and	O
several	O
new	O
optimizations	O
such	O
as	O
pipelining	O
,	O
whereby	O
transmission	O
of	O
the	O
garbled	B-Protocol
circuit	I-Protocol
across	O
the	O
network	O
begins	O
while	O
the	O
rest	O
of	O
the	O
circuit	O
is	O
still	O
being	O
generated	O
.	O
</s>
<s>
Meanwhile	O
,	O
another	O
group	O
of	O
researchers	O
has	O
investigated	O
using	O
consumer-grade	O
GPUs	B-Architecture
to	O
achieve	O
similar	O
levels	O
of	O
parallelism	O
.	O
</s>
<s>
They	O
utilize	O
OT	O
extensions	O
and	O
some	O
other	O
novel	O
techniques	O
to	O
design	O
their	O
GPU-specific	O
protocol	O
.	O
</s>
<s>
The	O
authors	O
obtain	O
a	O
timing	O
of	O
2.7	O
seconds	O
per	O
AES	O
block	O
on	O
a	O
standard	O
desktop	O
,	O
with	O
a	O
standard	O
GPU	B-Architecture
.	O
</s>
<s>
One	O
of	O
the	O
primary	O
applications	O
of	O
secure	B-Protocol
multi-party	I-Protocol
computation	I-Protocol
is	O
allowing	O
analysis	O
of	O
data	O
that	O
is	O
held	O
by	O
multiple	O
parties	O
,	O
or	O
blind	O
analysis	O
of	O
data	O
by	O
third	O
parties	O
without	O
allowing	O
the	O
data	O
custodian	O
to	O
understand	O
the	O
kind	O
of	O
data	O
analysis	O
being	O
performed	O
.	O
</s>
