<s>
In	O
cryptography	O
,	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
can	O
be	O
divided	O
into	O
two	O
main	O
categories	O
.	O
</s>
<s>
In	O
the	O
first	O
category	O
are	O
those	O
functions	O
whose	O
designs	O
are	O
based	O
on	O
mathematical	O
problems	O
,	O
and	O
whose	O
security	O
thus	O
follows	O
from	O
rigorous	O
mathematical	O
proofs	O
,	O
complexity	O
theory	O
and	O
formal	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
These	O
functions	O
are	O
called	O
Provably	O
Secure	O
Cryptographic	B-Algorithm
Hash	I-Algorithm
Functions	I-Algorithm
.	O
</s>
<s>
In	O
the	O
second	O
category	O
are	O
functions	O
which	O
are	O
not	O
based	O
on	O
mathematical	O
problems	O
,	O
but	O
on	O
an	O
ad-hoc	O
constructions	O
,	O
in	O
which	O
the	O
bits	O
of	O
the	O
message	O
are	O
mixed	O
to	O
produce	O
the	O
hash	B-Algorithm
.	O
</s>
<s>
Almost	O
all	O
hash	B-Algorithm
functions	I-Algorithm
in	O
widespread	O
use	O
reside	O
in	O
this	O
category	O
.	O
</s>
<s>
See	O
Hash	B-Algorithm
function	I-Algorithm
security	I-Algorithm
summary	I-Algorithm
.	O
</s>
<s>
Generally	O
,	O
the	O
basic	O
security	B-Algorithm
of	I-Algorithm
cryptographic	I-Algorithm
hash	I-Algorithm
functions	I-Algorithm
can	O
be	O
seen	O
from	O
different	O
angles	O
:	O
pre-image	O
resistance	O
,	O
second	O
pre-image	O
resistance	O
,	O
collision	O
resistance	O
,	O
and	O
pseudo-randomness	O
.	O
</s>
<s>
Pre-image	O
resistance	O
:	O
given	O
a	O
hash	B-Algorithm
it	O
should	O
be	O
hard	O
to	O
find	O
any	O
message	O
such	O
that	O
.	O
</s>
<s>
Such	O
a	O
pair	O
is	O
called	O
a	O
(	O
cryptographic	O
)	O
hash	B-Algorithm
collision	O
.	O
</s>
<s>
It	O
requires	O
a	O
hash	B-Algorithm
value	O
at	O
least	O
twice	O
as	O
long	O
as	O
what	O
is	O
required	O
for	O
pre-image	O
resistance	O
,	O
otherwise	O
collisions	O
may	O
be	O
found	O
by	O
a	O
birthday	O
attack	O
.	O
</s>
<s>
Pseudo-randomness	O
:	O
it	O
should	O
be	O
hard	O
to	O
distinguish	O
the	O
pseudo-random	O
number	O
generator	O
based	O
on	O
the	O
hash	B-Algorithm
function	I-Algorithm
from	O
a	O
random	O
number	O
generator	O
,	O
e.g.	O
,	O
it	O
passes	O
usual	O
randomness	O
tests	O
.	O
</s>
<s>
If	O
problem	O
A	O
is	O
hard	O
,	O
there	O
exists	O
a	O
formal	O
security	B-Algorithm
reduction	I-Algorithm
from	O
a	O
problem	O
which	O
is	O
widely	O
considered	O
unsolvable	O
in	O
polynomial	O
time	O
,	O
such	O
as	O
integer	O
factorization	O
problem	O
or	O
discrete	O
logarithm	O
problem	O
.	O
</s>
<s>
For	O
example	O
,	O
RSA	B-Architecture
public	I-Architecture
key	I-Architecture
cryptography	I-Architecture
relies	O
on	O
the	O
difficulty	O
of	O
integer	O
factorization	O
.	O
</s>
<s>
Also	O
,	O
if	O
the	O
set	O
of	O
inputs	O
to	O
the	O
hash	B-Algorithm
is	O
relatively	O
small	O
or	O
is	O
ordered	O
by	O
likelihood	O
in	O
some	O
way	O
,	O
then	O
a	O
brute	O
force	O
search	O
may	O
be	O
practical	O
,	O
regardless	O
of	O
theoretical	O
security	O
.	O
</s>
<s>
Likelihood	O
of	O
recovering	O
the	O
preimage	O
depends	O
on	O
the	O
input	O
set	O
size	O
and	O
the	O
speed	O
or	O
cost	O
of	O
computing	O
the	O
hash	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
A	O
common	O
example	O
is	O
the	O
use	O
of	O
hashes	B-Algorithm
to	O
store	O
password	O
validation	O
data	O
.	O
</s>
<s>
Rather	O
than	O
store	O
the	O
plaintext	O
of	O
user	O
passwords	O
,	O
an	O
access	O
control	O
system	O
typically	O
stores	O
a	O
hash	B-Algorithm
of	O
the	O
password	O
.	O
</s>
<s>
If	O
the	O
stored	O
validation	O
data	O
is	O
stolen	O
,	O
the	O
thief	O
will	O
only	O
have	O
the	O
hash	B-Algorithm
values	O
,	O
not	O
the	O
passwords	O
.	O
</s>
<s>
However	O
most	O
users	O
choose	O
passwords	O
in	O
predictable	O
ways	O
and	O
often	O
passwords	O
are	O
short	O
enough	O
so	O
that	O
all	O
possible	O
combinations	O
can	O
be	O
tested	O
if	O
fast	O
hashes	B-Algorithm
are	O
used	O
.	O
</s>
<s>
Special	O
hashes	B-Algorithm
called	O
key	B-Algorithm
derivation	I-Algorithm
functions	I-Algorithm
have	O
been	O
created	O
to	O
slow	O
searches	O
.	O
</s>
<s>
Most	O
hash	B-Algorithm
functions	I-Algorithm
are	O
built	O
on	O
an	O
ad	O
hoc	O
basis	O
,	O
where	O
the	O
bits	O
of	O
the	O
message	O
are	O
nicely	O
mixed	O
to	O
produce	O
the	O
hash	B-Algorithm
.	O
</s>
<s>
rotations	O
)	O
,	O
modular	O
additions	O
and	O
compression	B-Algorithm
functions	I-Algorithm
are	O
used	O
in	O
iterative	O
mode	O
to	O
ensure	O
high	O
complexity	O
and	O
pseudo-randomness	O
of	O
the	O
output	O
.	O
</s>
<s>
Only	O
a	O
few	O
years	O
ago	O
,	O
one	O
of	O
the	O
most	O
popular	O
hash	B-Algorithm
functions	I-Algorithm
,	O
SHA-1	B-Algorithm
,	O
was	O
shown	O
to	O
be	O
less	O
secure	O
than	O
its	O
length	O
suggested	O
:	O
collisions	O
could	O
be	O
found	O
in	O
only	O
251	O
tests	O
,	O
rather	O
than	O
the	O
brute-force	O
number	O
of	O
280	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
most	O
of	O
the	O
hash	B-Algorithm
functions	I-Algorithm
in	O
use	O
nowadays	O
are	O
not	O
provably	O
collision-resistant	O
.	O
</s>
<s>
These	O
hashes	B-Algorithm
are	O
not	O
based	O
on	O
purely	O
mathematical	O
functions	O
.	O
</s>
<s>
The	O
famous	O
case	O
is	O
MD5	B-Algorithm
.	O
</s>
<s>
In	O
this	O
approach	O
,	O
we	O
base	O
the	O
security	O
of	O
hash	B-Algorithm
function	I-Algorithm
on	O
some	O
hard	O
mathematical	O
problem	O
and	O
we	O
prove	O
that	O
finding	O
collisions	O
of	O
the	O
hash	B-Algorithm
functions	I-Algorithm
is	O
as	O
hard	O
as	O
breaking	O
the	O
underlying	O
problem	O
.	O
</s>
<s>
A	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
has	O
provable	O
security	O
against	O
collision	O
attacks	O
if	O
finding	O
collisions	O
is	O
provably	O
polynomial-time	B-Algorithm
reducible	I-Algorithm
from	O
problem	O
P	O
which	O
is	O
supposed	O
to	O
be	O
unsolvable	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Hash	B-Algorithm
functions	I-Algorithm
with	O
the	O
proof	O
of	O
their	O
security	O
are	O
based	O
on	O
mathematical	O
functions	O
.	O
</s>
<s>
Current	O
collision-resistant	O
hash	B-Algorithm
algorithms	O
that	O
have	O
provable	O
security	O
reductions	B-Algorithm
are	O
too	O
inefficient	O
to	O
be	O
used	O
in	O
practice	O
.	O
</s>
<s>
In	O
comparison	O
to	O
classical	O
hash	B-Algorithm
functions	I-Algorithm
,	O
they	O
tend	O
to	O
be	O
relatively	O
slow	O
and	O
do	O
not	O
always	O
meet	O
all	O
of	O
criteria	O
traditionally	O
expected	O
of	O
cryptographic	B-Algorithm
hashes	I-Algorithm
.	O
</s>
<s>
Very	B-Algorithm
smooth	I-Algorithm
hash	I-Algorithm
is	O
an	O
example	O
.	O
</s>
<s>
Constructing	O
a	O
hash	B-Algorithm
function	I-Algorithm
with	O
provable	O
security	O
is	O
much	O
more	O
difficult	O
than	O
using	O
a	O
classical	O
approach	O
where	O
we	O
just	O
hope	O
that	O
the	O
complex	O
mixing	O
of	O
bits	O
in	O
the	O
hashing	O
algorithm	O
is	O
strong	O
enough	O
to	O
prevent	O
adversary	O
from	O
finding	O
collisions	O
.	O
</s>
<s>
The	O
proof	O
is	O
often	O
a	O
reduction	O
to	O
a	O
problem	O
with	O
asymptotically	O
hard	O
worst-case	B-General_Concept
or	O
average-case	O
complexity	O
.	O
</s>
<s>
Worst-case	B-General_Concept
measures	O
the	O
difficulty	O
of	O
solving	O
pathological	O
cases	O
rather	O
than	O
typical	O
cases	O
of	O
the	O
underlying	O
problem	O
.	O
</s>
<s>
For	O
example	O
,	O
early	O
versions	O
of	O
Fast	B-Algorithm
Syndrome	I-Algorithm
Based	I-Algorithm
Hash	I-Algorithm
turned	O
out	O
to	O
be	O
insecure	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
is	O
an	O
example	O
of	O
a	O
hash	B-Algorithm
function	I-Algorithm
that	O
circumvents	O
these	O
security	O
problems	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
for	O
any	O
algorithm	O
that	O
can	O
break	O
SWIFFT	B-Algorithm
with	O
probability	O
P	O
within	O
an	O
estimated	O
time	O
T	O
,	O
we	O
can	O
find	O
an	O
algorithm	O
that	O
solves	O
the	O
worst-case	B-General_Concept
scenario	O
of	O
a	O
certain	O
difficult	O
mathematical	O
problem	O
within	O
time	O
T	O
 '	O
depending	O
on	O
T	O
and	O
P	O
.	O
</s>
<s>
Hash(m )	O
=	O
xm	O
mod	O
n	O
where	O
n	O
is	O
hard	O
to	O
factor	O
composite	O
number	O
,	O
and	O
x	O
is	O
some	O
prespecified	O
base	O
value	O
.	O
</s>
<s>
VSH	B-Algorithm
-	I-Algorithm
Very	I-Algorithm
Smooth	I-Algorithm
Hash	I-Algorithm
function	I-Algorithm
-	O
a	O
provably	O
secure	O
collision-resistant	O
hash	B-Algorithm
function	I-Algorithm
assuming	O
the	O
hardness	O
of	O
finding	O
nontrivial	O
modular	O
square	O
roots	O
modulo	O
composite	O
number	O
(	O
this	O
is	O
proven	O
to	O
be	O
as	O
hard	O
as	O
factoring	O
)	O
.	O
</s>
<s>
ECOH	B-Algorithm
-	I-Algorithm
Elliptic	I-Algorithm
Curve	I-Algorithm
Only	I-Algorithm
hash	I-Algorithm
function	I-Algorithm
-	O
based	O
on	O
the	O
concept	O
of	O
Elliptic	O
curves	O
,	O
Subset	B-Algorithm
Sum	I-Algorithm
Problem	I-Algorithm
and	O
summation	O
of	O
polynomials	O
.	O
</s>
<s>
FSB	B-Algorithm
-	I-Algorithm
Fast	I-Algorithm
Syndrome-Based	I-Algorithm
hash	I-Algorithm
function	I-Algorithm
-	O
it	O
can	O
be	O
proven	O
that	O
breaking	O
FSB	O
is	O
at	O
least	O
as	O
difficult	O
as	O
solving	O
a	O
certain	O
NP-complete	O
problem	O
known	O
as	O
Regular	O
Syndrome	O
Decoding	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
-	O
SWIFFT	B-Algorithm
is	O
based	O
on	O
the	O
Fast	O
Fourier	O
transform	O
and	O
is	O
provably	O
collision	O
resistant	O
,	O
under	O
a	O
relatively	O
mild	O
assumption	O
about	O
the	O
worst-case	B-General_Concept
difficulty	O
of	O
finding	O
short	O
vectors	O
in	O
cyclic/ideal	O
lattices	O
.	O
</s>
<s>
Chaum	O
,	O
van	O
Heijst	O
,	O
Pfitzmann	O
hash	B-Algorithm
function	I-Algorithm
-	O
A	O
compression	O
function	O
where	O
finding	O
collisions	O
is	O
as	O
hard	O
as	O
solving	O
the	O
discrete	O
logarithm	O
problem	O
in	O
a	O
finite	O
group	O
.	O
</s>
<s>
Knapsack-based	O
hash	B-Algorithm
functions	I-Algorithm
-	O
A	O
family	O
of	O
hash	B-Algorithm
functions	I-Algorithm
based	O
on	O
the	O
Knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
Zémor-Tillich	O
hash	B-Algorithm
function	I-Algorithm
-	O
A	O
family	O
of	O
hash	B-Algorithm
functions	I-Algorithm
that	O
rely	O
on	O
the	O
arithmetic	O
of	O
the	O
group	O
of	O
matrices	O
SL2	O
.	O
</s>
<s>
For	O
this	O
hash	B-Algorithm
,	O
an	O
attack	O
was	O
eventually	O
discovered	O
with	O
a	O
time	O
complexity	O
close	O
to	O
.	O
</s>
<s>
This	O
beat	O
by	O
far	O
the	O
birthday	O
bound	O
and	O
ideal	O
pre-image	O
complexities	O
which	O
are	O
and	O
for	O
the	O
Zémor-Tillich	O
hash	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Hash	B-Algorithm
functions	I-Algorithm
from	O
Sigma	O
Protocols	O
-	O
there	O
exists	O
a	O
general	O
way	O
of	O
constructing	O
a	O
provably	O
secure	O
hash	B-Algorithm
,	O
specifically	O
from	O
any	O
(	O
suitable	O
)	O
sigma	O
protocol	O
.	O
</s>
<s>
A	O
faster	O
version	O
of	O
VSH	B-Algorithm
(	O
called	O
VSH*	O
)	O
could	O
be	O
obtained	O
in	O
this	O
way	O
.	O
</s>
