<s>
In	O
cryptography	O
,	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
or	O
zero-knowledge	B-Protocol
protocol	I-Protocol
is	O
a	O
method	O
by	O
which	O
one	O
party	O
(	O
the	O
prover	O
)	O
can	O
prove	O
to	O
another	O
party	O
(	O
the	O
verifier	O
)	O
that	O
a	O
given	O
statement	O
is	O
true	O
while	O
the	O
prover	O
avoids	O
conveying	O
any	O
additional	O
information	O
apart	O
from	O
the	O
fact	O
that	O
the	O
statement	O
is	O
indeed	O
true	O
.	O
</s>
<s>
The	O
essence	O
of	O
zero-knowledge	B-Protocol
proofs	I-Protocol
is	O
that	O
it	O
is	O
trivial	O
to	O
prove	O
that	O
one	O
possesses	O
knowledge	O
of	O
certain	O
information	O
by	O
simply	O
revealing	O
it	O
;	O
the	O
challenge	O
is	O
to	O
prove	O
such	O
possession	O
without	O
revealing	O
the	O
information	O
itself	O
or	O
any	O
additional	O
information	O
.	O
</s>
<s>
A	O
zero-knowledge	B-Protocol
proof	I-Protocol
of	O
knowledge	O
is	O
a	O
special	O
case	O
when	O
the	O
statement	O
consists	O
only	O
of	O
the	O
fact	O
that	O
the	O
prover	O
possesses	O
the	O
secret	O
information	O
.	O
</s>
<s>
Interactive	O
zero-knowledge	B-Protocol
proofs	I-Protocol
require	O
interaction	O
between	O
the	O
individual	O
(	O
or	O
computer	O
system	O
)	O
proving	O
their	O
knowledge	O
and	O
the	O
individual	O
validating	O
the	O
proof	O
.	O
</s>
<s>
A	O
protocol	O
implementing	O
zero-knowledge	B-Protocol
proofs	I-Protocol
of	O
knowledge	O
must	O
necessarily	O
require	O
interactive	O
input	O
from	O
the	O
verifier	O
.	O
</s>
<s>
Some	O
forms	O
of	O
non-interactive	O
zero-knowledge	B-Protocol
proofs	I-Protocol
exist	O
,	O
but	O
the	O
validity	O
of	O
the	O
proof	O
relies	O
on	O
computational	O
assumptions	O
(	O
typically	O
the	O
assumptions	O
of	O
an	O
ideal	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
)	O
.	O
</s>
<s>
There	O
is	O
a	O
well-known	O
story	O
presenting	O
the	O
fundamental	O
ideas	O
of	O
zero-knowledge	B-Protocol
proofs	I-Protocol
,	O
first	O
published	O
in	O
1990	O
by	O
Jean-Jacques	O
Quisquater	O
and	O
others	O
in	O
their	O
paper	O
"	O
How	O
to	O
Explain	O
Zero-Knowledge	B-Protocol
Protocols	I-Protocol
to	O
Your	O
Children	O
"	O
.	O
</s>
<s>
Using	O
the	O
common	O
Alice	O
and	O
Bob	O
anthropomorphic	O
thought	O
experiment	O
placeholders	O
,	O
the	O
two	O
parties	O
in	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
are	O
Peggy	O
as	O
the	O
prover	O
of	O
the	O
statement	O
,	O
and	O
Victor	O
,	O
the	O
verifier	O
of	O
the	O
statement	O
.	O
</s>
<s>
However	O
,	O
digital	O
cryptography	O
generally	O
"	O
flips	O
coins	O
"	O
by	O
relying	O
on	O
a	O
pseudo-random	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
,	O
which	O
is	O
akin	O
to	O
a	O
coin	O
with	O
a	O
fixed	O
pattern	O
of	O
heads	O
and	O
tails	O
known	O
only	O
to	O
the	O
coin	O
's	O
owner	O
.	O
</s>
<s>
If	O
Victor	O
's	O
coin	O
behaved	O
this	O
way	O
,	O
then	O
again	O
it	O
would	O
be	O
possible	O
for	O
Victor	O
and	O
Peggy	O
to	O
have	O
faked	O
the	O
"	O
experiment	O
"	O
,	O
so	O
using	O
a	O
pseudo-random	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
would	O
not	O
reveal	O
Peggy	O
's	O
knowledge	O
to	O
the	O
world	O
in	O
the	O
same	O
way	O
that	O
using	O
a	O
flipped	O
coin	O
would	O
.	O
</s>
<s>
A	O
zero-knowledge	B-Protocol
proof	I-Protocol
of	O
some	O
statement	O
must	O
satisfy	O
three	O
properties	O
:	O
</s>
<s>
Zero-knowledge	B-Protocol
proofs	I-Protocol
are	O
not	O
proofs	O
in	O
the	O
mathematical	O
sense	O
of	O
the	O
term	O
because	O
there	O
is	O
some	O
small	O
probability	O
,	O
the	O
soundness	O
error	O
,	O
that	O
a	O
cheating	O
prover	O
will	O
be	O
able	O
to	O
convince	O
the	O
verifier	O
of	O
a	O
false	O
statement	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
zero-knowledge	B-Protocol
proofs	I-Protocol
are	O
probabilistic	O
"	O
proofs	O
"	O
rather	O
than	O
deterministic	O
proofs	O
.	O
</s>
<s>
A	O
formal	O
definition	O
of	O
zero-knowledge	O
has	O
to	O
use	O
some	O
computational	O
model	O
,	O
the	O
most	O
common	O
one	O
being	O
that	O
of	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
Let	O
,	O
,	O
and	O
be	O
Turing	B-Architecture
machines	I-Architecture
.	O
</s>
<s>
The	O
prover	O
is	O
modeled	O
as	O
having	O
unlimited	O
computation	O
power	O
(	O
in	O
practice	O
,	O
usually	O
is	O
a	O
probabilistic	O
Turing	B-Architecture
machine	I-Architecture
)	O
.	O
</s>
<s>
If	O
is	O
truly	O
random	O
,	O
equally	O
distributed	O
between	O
zero	O
and	O
,	O
this	O
does	O
not	O
leak	O
any	O
information	O
about	O
(	O
see	O
one-time	B-Algorithm
pad	I-Algorithm
)	O
.	O
</s>
<s>
For	O
all	O
realistic	O
purposes	O
,	O
it	O
is	O
infeasibly	O
difficult	O
to	O
defeat	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
with	O
a	O
reasonable	O
number	O
of	O
rounds	O
in	O
this	O
way	O
.	O
</s>
<s>
Statistical	O
zero-knowledge	O
means	O
that	O
the	O
distributions	O
are	O
not	O
necessarily	O
exactly	O
the	O
same	O
,	O
but	O
they	O
are	O
statistically	B-General_Concept
close	I-General_Concept
,	O
meaning	O
that	O
their	O
statistical	O
difference	O
is	O
a	O
negligible	O
function	O
.	O
</s>
<s>
Witness	B-General_Concept
indistinguishable	I-General_Concept
proof	I-General_Concept
:	O
verifiers	O
cannot	O
know	O
which	O
witness	O
is	O
used	O
for	O
producing	O
the	O
proof	O
.	O
</s>
<s>
Multi-party	B-Protocol
computation	I-Protocol
:	O
while	O
each	O
party	O
can	O
keep	O
their	O
respective	O
secret	O
,	O
they	O
together	O
produce	O
a	O
result	O
.	O
</s>
<s>
Ring	B-Algorithm
signature	I-Algorithm
:	O
outsiders	O
have	O
no	O
idea	O
which	O
key	O
is	O
used	O
for	O
signing	O
.	O
</s>
<s>
Research	O
in	O
zero-knowledge	B-Protocol
proofs	I-Protocol
has	O
been	O
motivated	O
by	O
authentication	O
systems	O
where	O
one	O
party	O
wants	O
to	O
prove	O
its	O
identity	O
to	O
a	O
second	O
party	O
via	O
some	O
secret	O
information	O
(	O
such	O
as	O
a	O
password	O
)	O
but	O
does	O
n't	O
want	O
the	O
second	O
party	O
to	O
learn	O
anything	O
about	O
this	O
secret	O
.	O
</s>
<s>
This	O
is	O
called	O
a	O
"	O
zero-knowledge	B-Protocol
proof	I-Protocol
of	O
knowledge	O
"	O
.	O
</s>
<s>
However	O
,	O
a	O
password	O
is	O
typically	O
too	O
small	O
or	O
insufficiently	O
random	O
to	O
be	O
used	O
in	O
many	O
schemes	O
for	O
zero-knowledge	B-Protocol
proofs	I-Protocol
of	O
knowledge	O
.	O
</s>
<s>
A	O
zero-knowledge	B-General_Concept
password	I-General_Concept
proof	I-General_Concept
is	O
a	O
special	O
kind	O
of	O
zero-knowledge	B-Protocol
proof	I-Protocol
of	O
knowledge	O
that	O
addresses	O
the	O
limited	O
size	O
of	O
passwords	O
.	O
</s>
<s>
In	O
August	O
2021	O
,	O
Cloudflare	B-Application
,	O
an	O
American	O
web	O
infrastructure	O
and	O
security	O
company	O
decided	O
to	O
use	O
the	O
one-out-of-many	O
proofs	O
mechanism	O
for	O
private	O
web	O
verification	O
using	O
vendor	O
hardware	O
.	O
</s>
<s>
One	O
of	O
the	O
uses	O
of	O
zero-knowledge	B-Protocol
proofs	I-Protocol
within	O
cryptographic	B-Protocol
protocols	I-Protocol
is	O
to	O
enforce	O
honest	O
behavior	O
while	O
maintaining	O
privacy	O
.	O
</s>
<s>
Roughly	O
,	O
the	O
idea	O
is	O
to	O
force	O
a	O
user	O
to	O
prove	O
,	O
using	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
,	O
that	O
its	O
behavior	O
is	O
correct	O
according	O
to	O
the	O
protocol	O
.	O
</s>
<s>
Zero-knowledge	B-Protocol
proofs	I-Protocol
were	O
applied	O
in	O
the	O
Zerocoin	B-Protocol
and	O
Zerocash	B-Protocol
protocols	O
,	O
which	O
culminated	O
in	O
the	O
birth	O
of	O
Zcoin	B-Protocol
(	O
later	O
rebranded	O
as	O
Firo	B-Protocol
in	O
2020	O
)	O
and	O
Zcash	B-Protocol
cryptocurrencies	O
in	O
2016	O
.	O
</s>
<s>
Zerocoin	B-Protocol
has	O
a	O
built-in	O
mixing	O
model	O
that	O
does	O
not	O
trust	O
any	O
peers	O
or	O
centralised	O
mixing	O
providers	O
to	O
ensure	O
anonymity	O
.	O
</s>
<s>
Users	O
can	O
transact	O
in	O
a	O
base	O
currency	O
and	O
can	O
cycle	O
the	O
currency	O
into	O
and	O
out	O
of	O
Zerocoins	B-Protocol
.	O
</s>
<s>
The	O
Zerocash	B-Protocol
protocol	O
uses	O
a	O
similar	O
model	O
(	O
a	O
variant	O
known	O
as	O
a	O
non-interactive	O
zero-knowledge	B-Protocol
proof	I-Protocol
)	O
except	O
that	O
it	O
can	O
obscure	O
the	O
transaction	O
amount	O
,	O
while	O
Zerocoin	B-Protocol
cannot	O
.	O
</s>
<s>
Given	O
significant	O
restrictions	O
of	O
transaction	O
data	O
on	O
the	O
Zerocash	B-Protocol
network	O
,	O
Zerocash	B-Protocol
is	O
less	O
prone	O
to	O
privacy	O
timing	O
attacks	O
when	O
compared	O
to	O
Zerocoin	B-Protocol
.	O
</s>
<s>
However	O
,	O
this	O
additional	O
layer	O
of	O
privacy	O
can	O
cause	O
potentially	O
undetected	O
hyperinflation	O
of	O
Zerocash	B-Protocol
supply	O
because	O
fraudulent	O
coins	O
cannot	O
be	O
tracked	O
.	O
</s>
<s>
Bulletproofs	O
are	O
an	O
improvement	O
from	O
non-interactive	O
zero-knowledge	B-Protocol
proof	I-Protocol
where	O
trusted	O
setup	O
is	O
not	O
needed	O
.	O
</s>
<s>
It	O
was	O
later	O
implemented	O
into	O
the	O
Mimblewimble	O
protocol	O
(	O
which	O
the	O
Grin	O
and	O
Beam	O
cryptocurrencies	O
are	O
based	O
upon	O
)	O
and	O
Monero	B-Protocol
cryptocurrency	I-Protocol
.	O
</s>
<s>
In	O
2019	O
,	O
Firo	B-Protocol
implemented	O
the	O
Sigma	O
protocol	O
,	O
which	O
is	O
an	O
improvement	O
on	O
the	O
Zerocoin	B-Protocol
protocol	I-Protocol
without	O
trusted	O
setup	O
.	O
</s>
<s>
In	O
the	O
same	O
year	O
,	O
Firo	B-Protocol
introduced	O
the	O
Lelantus	O
protocol	O
,	O
an	O
improvement	O
on	O
the	O
Sigma	O
protocol	O
,	O
where	O
the	O
former	O
hides	O
the	O
origin	O
and	O
amount	O
of	O
a	O
transaction	O
.	O
</s>
<s>
Zero-knowledge	B-Protocol
proofs	I-Protocol
were	O
first	O
conceived	O
in	O
1985	O
by	O
Shafi	O
Goldwasser	O
,	O
Silvio	O
Micali	O
,	O
and	O
Charles	O
Rackoff	O
in	O
their	O
paper	O
"	O
The	O
Knowledge	O
Complexity	O
of	O
Interactive	O
Proof-Systems	O
"	O
.	O
</s>
<s>
They	O
also	O
gave	O
the	O
first	O
zero-knowledge	B-Protocol
proof	I-Protocol
for	O
a	O
concrete	O
problem	O
,	O
that	O
of	O
deciding	O
quadratic	O
nonresidues	O
mod	O
.	O
</s>
<s>
This	O
was	O
also	O
true	O
of	O
several	O
other	O
problems	O
for	O
which	O
zero-knowledge	B-Protocol
proofs	I-Protocol
were	O
subsequently	O
discovered	O
,	O
such	O
as	O
an	O
unpublished	O
proof	O
system	O
by	O
Oded	O
Goldreich	O
verifying	O
that	O
a	O
two-prime	O
modulus	O
is	O
not	O
a	O
Blum	O
integer	O
.	O
</s>
<s>
Oded	O
Goldreich	O
,	O
Silvio	O
Micali	O
,	O
and	O
Avi	O
Wigderson	O
took	O
this	O
one	O
step	O
further	O
,	O
showing	O
that	O
,	O
assuming	O
the	O
existence	O
of	O
unbreakable	O
encryption	O
,	O
one	O
can	O
create	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
system	O
for	O
the	O
NP-complete	O
graph	O
coloring	O
problem	O
with	O
three	O
colors	O
.	O
</s>
<s>
Since	O
every	O
problem	O
in	O
NP	O
can	O
be	O
efficiently	O
reduced	O
to	O
this	O
problem	O
,	O
this	O
means	O
that	O
,	O
under	O
this	O
assumption	O
,	O
all	O
problems	O
in	O
NP	O
have	O
zero-knowledge	B-Protocol
proofs	I-Protocol
.	O
</s>
<s>
On	O
top	O
of	O
this	O
,	O
they	O
also	O
showed	O
that	O
the	O
graph	O
nonisomorphism	O
problem	O
,	O
the	O
complement	O
of	O
the	O
graph	O
isomorphism	O
problem	O
,	O
has	O
a	O
zero-knowledge	B-Protocol
proof	I-Protocol
.	O
</s>
<s>
would	O
go	O
on	O
to	O
show	O
that	O
,	O
also	O
assuming	O
one-way	O
functions	O
or	O
unbreakable	O
encryption	O
,	O
that	O
there	O
are	O
zero-knowledge	B-Protocol
proofs	I-Protocol
for	O
all	O
problems	O
in	O
IP	O
=	O
PSPACE	O
,	O
or	O
in	O
other	O
words	O
,	O
anything	O
that	O
can	O
be	O
proved	O
by	O
an	O
interactive	O
proof	O
system	O
can	O
be	O
proved	O
with	O
zero	O
knowledge	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
,	O
without	O
any	O
intractability	O
assumptions	O
,	O
all	O
languages	O
in	O
NP	O
have	O
zero-knowledge	B-Protocol
proofs	I-Protocol
in	O
such	O
a	O
system	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
in	O
an	O
Internet-like	O
setting	O
,	O
where	O
multiple	O
protocols	O
may	O
be	O
executed	O
concurrently	O
,	O
building	O
zero-knowledge	B-Protocol
proofs	I-Protocol
is	O
more	O
challenging	O
.	O
</s>
<s>
The	O
line	O
of	O
research	O
investigating	O
concurrent	O
zero-knowledge	B-Protocol
proofs	I-Protocol
was	O
initiated	O
by	O
the	O
work	O
of	O
Dwork	O
,	O
Naor	O
,	O
and	O
Sahai	O
.	O
</s>
<s>
One	O
particular	O
development	O
along	O
these	O
lines	O
has	O
been	O
the	O
development	O
of	O
witness-indistinguishable	B-General_Concept
proof	I-General_Concept
protocols	O
.	O
</s>
<s>
Another	O
variant	O
of	O
zero-knowledge	B-Protocol
proofs	I-Protocol
are	O
non-interactive	O
zero-knowledge	B-Protocol
proofs	I-Protocol
.	O
</s>
<s>
The	O
most	O
popular	O
interactive	O
or	O
non-interactive	O
zero-knowledge	B-Protocol
proof	I-Protocol
(	O
e.g.	O
,	O
zk-SNARK	O
)	O
protocols	O
can	O
be	O
broadly	O
categorized	O
in	O
the	O
following	O
four	O
categories	O
:	O
Succinct	O
Non-Interactive	O
ARguments	O
of	O
Knowledge	O
(	O
SNARK	O
)	O
,	O
Scalable	O
Transparent	O
ARgument	O
of	O
Knowledge	O
(	O
STARK	O
)	O
,	O
Verifiable	O
Polynomial	O
Delegation	O
(	O
VPD	O
)	O
,	O
and	O
Succinct	O
Non-interactive	O
ARGuments	O
(	O
SNARG	O
)	O
.	O
</s>
<s>
A	O
list	O
of	O
zero-knowledge	B-Protocol
proof	I-Protocol
protocols	O
and	O
libraries	O
is	O
provided	O
below	O
along	O
with	O
comparisons	O
based	O
on	O
transparency	O
,	O
universality	O
,	O
plausible	O
post-quantum	O
security	O
,	O
and	O
programming	O
paradigm	O
.	O
</s>
