<s>
In	O
cryptography	O
,	O
the	O
fast	B-Algorithm
syndrome-based	I-Algorithm
hash	I-Algorithm
functions	O
(	O
FSB	O
)	O
are	O
a	O
family	O
of	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
introduced	O
in	O
2003	O
by	O
Daniel	O
Augot	O
,	O
Matthieu	O
Finiasz	O
,	O
and	O
Nicolas	O
Sendrier	O
.	O
</s>
<s>
Unlike	O
most	O
other	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
in	O
use	O
today	O
,	O
FSB	O
can	O
to	O
a	O
certain	O
extent	O
be	O
proven	O
to	O
be	O
secure	O
.	O
</s>
<s>
Several	O
versions	O
of	O
FSB	O
have	O
been	O
proposed	O
,	O
the	O
latest	O
of	O
which	O
was	O
submitted	O
to	O
the	O
SHA-3	B-Algorithm
cryptography	I-Algorithm
competition	I-Algorithm
but	O
was	O
rejected	O
in	O
the	O
first	O
round	O
.	O
</s>
<s>
FSB	O
is	O
slower	O
than	O
traditional	O
hash	B-Algorithm
functions	I-Algorithm
and	O
uses	O
quite	O
a	O
lot	O
of	O
memory	O
,	O
which	O
makes	O
it	O
impractical	O
on	O
memory	O
constrained	O
environments	O
.	O
</s>
<s>
This	O
last	O
problem	O
has	O
been	O
solved	O
in	O
recent	O
versions	O
by	O
simply	O
compressing	O
the	O
output	O
by	O
another	O
compression	O
function	O
called	O
Whirlpool	B-Algorithm
.	O
</s>
<s>
We	O
will	O
later	O
use	O
the	O
Merkle	B-Algorithm
–	I-Algorithm
Damgård	I-Algorithm
construction	I-Algorithm
to	O
extend	O
the	O
domain	O
to	O
inputs	O
of	O
arbitrary	O
lengths	O
.	O
</s>
<s>
For	O
security	O
purposes	O
as	O
well	O
as	O
to	O
get	O
a	O
faster	O
hash	B-Algorithm
speed	O
we	O
want	O
to	O
use	O
only	O
“	O
regular	O
words	O
of	O
weight	O
”	O
as	O
input	O
for	O
our	O
matrix	O
.	O
</s>
<s>
We	O
can	O
now	O
use	O
the	O
Merkle	B-Algorithm
–	I-Algorithm
Damgård	I-Algorithm
construction	I-Algorithm
to	O
generalize	O
the	O
compression	O
function	O
to	O
accept	O
inputs	O
of	O
arbitrary	O
lengths	O
.	O
</s>
<s>
The	O
Merkle	B-Algorithm
–	I-Algorithm
Damgård	I-Algorithm
construction	I-Algorithm
is	O
proven	O
to	O
base	O
its	O
security	O
only	O
on	O
the	O
security	O
of	O
the	O
used	O
compression	O
function	O
.	O
</s>
<s>
A	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
needs	O
to	O
be	O
secure	O
in	O
three	O
different	O
aspects	O
:	O
</s>
<s>
We	O
will	O
take	O
hard	O
to	O
mean	O
“	O
The	O
runtime	O
of	O
any	O
algorithm	O
that	O
finds	O
a	O
collision	O
or	O
pre-image	O
will	O
depend	O
exponentially	O
on	O
size	O
of	O
the	O
hash	B-Algorithm
value	O
”	O
.	O
</s>
<s>
This	O
means	O
that	O
by	O
relatively	O
small	O
additions	O
to	O
the	O
hash	B-Algorithm
size	O
,	O
we	O
can	O
quickly	O
reach	O
high	O
security	O
.	O
</s>
<s>
Syndrome	O
decoding	O
is	O
originally	O
a	O
problem	O
from	O
coding	B-Error_Name
theory	I-Error_Name
but	O
its	O
NP-completeness	O
makes	O
it	O
a	O
nice	O
application	O
for	O
cryptography	O
.	O
</s>
<s>
It	O
is	O
easy	O
to	O
see	O
that	O
finding	O
a	O
pre-image	O
of	O
a	O
given	O
hash	B-Algorithm
is	O
exactly	O
equivalent	O
to	O
this	O
problem	O
,	O
so	O
the	O
problem	O
of	O
finding	O
pre-images	O
in	O
FSB	O
must	O
also	O
be	O
NP-complete	O
.	O
</s>
<s>
Suppose	O
that	O
we	O
have	O
found	O
a	O
collision	O
,	O
so	O
we	O
have	O
Hash(m1 )	O
=	O
Hash(m2 )	O
with	O
.	O
</s>
<s>
We	O
then	O
have	O
;	O
is	O
a	O
sum	O
of	O
two	O
different	O
regular	O
words	O
and	O
so	O
must	O
be	O
a	O
2-regular	O
word	O
of	O
which	O
the	O
hash	B-Algorithm
is	O
zero	O
,	O
so	O
we	O
have	O
solved	O
an	O
instance	O
of	O
2-RNSD	O
.	O
</s>
<s>
The	O
latest	O
versions	O
of	O
FSB	O
use	O
the	O
compression	O
function	O
Whirlpool	B-Algorithm
to	O
further	O
compress	O
the	O
hash	B-Algorithm
output	O
.	O
</s>
<s>
Note	O
that	O
even	O
if	O
one	O
were	O
able	O
to	O
find	O
collisions	O
in	O
Whirlpool	B-Algorithm
,	O
one	O
would	O
still	O
need	O
to	O
find	O
the	O
collisions	O
pre-images	O
in	O
the	O
original	O
FSB	O
compression	O
function	O
to	O
find	O
a	O
collision	O
in	O
FSB	O
.	O
</s>
<s>
But	O
the	O
proof	O
is	O
a	O
reduction	O
to	O
a	O
problem	O
with	O
asymptotically	O
hard	O
worst-case	B-General_Concept
complexity	I-General_Concept
.	O
</s>
<s>
It	O
is	O
shown	O
that	O
the	O
hash	B-Algorithm
function	I-Algorithm
offers	O
minimal	O
pre-image	O
or	O
collision	O
resistance	O
when	O
the	O
message	O
space	O
is	O
chosen	O
in	O
a	O
specific	O
way	O
.	O
</s>
<s>
FSB	O
is	O
a	O
speed-up	O
version	O
of	O
syndrome-based	O
hash	B-Algorithm
function	I-Algorithm
(	O
SB	O
)	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
SB	O
the	O
compression	O
function	O
is	O
very	O
similar	O
to	O
the	O
encoding	O
function	O
of	O
Niederreiter	B-Algorithm
's	I-Algorithm
version	I-Algorithm
of	O
McEliece	O
cryptosystem	O
.	O
</s>
<s>
Both	O
the	O
block	O
size	O
of	O
the	O
hash	B-Algorithm
function	I-Algorithm
and	O
the	O
output	O
size	O
are	O
completely	O
scalable	O
.	O
</s>
<s>
This	O
problem	O
was	O
solved	O
in	O
the	O
related	O
hash	B-Algorithm
function	I-Algorithm
called	O
Improved	O
FSB	O
,	O
which	O
is	O
still	O
provably	O
secure	O
,	O
but	O
relies	O
on	O
slightly	O
stronger	O
assumptions	O
.	O
</s>
