<s>
In	O
cryptography	O
,	O
SWIFFT	B-Algorithm
is	O
a	O
collection	O
of	O
provably	O
secure	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
It	O
is	O
based	O
on	O
the	O
concept	O
of	O
the	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
(	O
FFT	O
)	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
is	O
not	O
the	O
first	O
hash	B-Algorithm
function	I-Algorithm
based	O
on	O
FFT	O
,	O
but	O
it	O
sets	O
itself	O
apart	O
by	O
providing	O
a	O
mathematical	O
proof	O
of	O
its	O
security	O
.	O
</s>
<s>
It	O
also	O
uses	O
the	O
LLL	O
basis	O
reduction	B-Algorithm
algorithm	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
finding	O
collisions	O
in	O
SWIFFT	B-Algorithm
is	O
at	O
least	O
as	O
difficult	O
as	O
finding	O
short	O
vectors	O
in	O
cyclic/ideal	O
lattices	O
in	O
the	O
worst	O
case	O
.	O
</s>
<s>
By	O
giving	O
a	O
security	O
reduction	B-Algorithm
to	O
the	O
worst-case	B-General_Concept
scenario	O
of	O
a	O
difficult	O
mathematical	O
problem	O
,	O
SWIFFT	B-Algorithm
gives	O
a	O
much	O
stronger	O
security	O
guarantee	O
than	O
most	O
other	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
.	O
</s>
<s>
Unlike	O
many	O
other	O
provably	O
secure	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
,	O
the	O
algorithm	O
is	O
quite	O
fast	O
,	O
yielding	O
a	O
throughput	O
of	O
40Mbit/s	O
on	O
a	O
3.2GHz	O
Intel	O
Pentium	O
4	O
.	O
</s>
<s>
Although	O
SWIFFT	B-Algorithm
satisfies	O
many	O
desirable	O
cryptographic	O
and	O
statistical	O
properties	O
,	O
it	O
was	O
not	O
designed	O
to	O
be	O
an	O
"	O
all-purpose	O
"	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
it	O
is	O
not	O
a	O
pseudorandom	O
function	O
,	O
and	O
would	O
not	O
be	O
a	O
suitable	O
instantiation	O
of	O
a	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
The	O
algorithm	O
is	O
less	O
efficient	O
than	O
most	O
traditional	O
hash	B-Algorithm
functions	I-Algorithm
that	O
do	O
not	O
give	O
a	O
proof	O
of	O
their	O
collision-resistance	O
.	O
</s>
<s>
A	O
modification	O
of	O
SWIFFT	B-Algorithm
called	O
SWIFFTX	O
was	O
proposed	O
as	O
a	O
candidate	O
for	O
SHA-3	O
function	O
to	O
the	O
NIST	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
competition	I-Algorithm
and	O
was	O
rejected	O
in	O
the	O
first	O
round	O
.	O
</s>
<s>
Compute	O
the	O
Fourier	O
coefficients	O
of	O
each	O
using	O
SWIFFT	B-Algorithm
.	O
</s>
<s>
Define	O
the	O
Fourier	O
coefficients	O
of	O
,	O
so	O
that	O
they	O
are	O
fixed	O
and	O
depend	O
on	O
a	O
family	O
of	O
SWIFFT	B-Algorithm
.	O
</s>
<s>
The	O
SWIFFT	B-Algorithm
functions	O
can	O
be	O
described	O
as	O
a	O
simple	O
algebraic	O
expression	O
over	O
some	O
polynomial	O
ring	O
.	O
</s>
<s>
A	O
certain	O
function	O
in	O
the	O
SWIFFT	B-Algorithm
family	O
is	O
specified	O
by	O
fixed	O
elements	O
of	O
the	O
ring	O
,	O
that	O
are	O
called	O
multipliers	O
.	O
</s>
<s>
A	O
fast	O
way	O
to	O
compute	O
these	O
products	O
is	O
given	O
by	O
the	O
convolution	B-Language
theorem	O
.	O
</s>
<s>
Here	O
denotes	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
and	O
denotes	O
the	O
pointwise	O
product	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
of	O
the	O
convolution	B-Language
theorem	O
does	O
not	O
denote	O
multiplication	O
but	O
convolution	B-Language
.	O
</s>
<s>
It	O
can	O
however	O
be	O
shown	O
that	O
polynomial	O
multiplication	O
is	O
a	O
convolution	B-Language
.	O
</s>
<s>
For	O
finding	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
we	O
will	O
use	O
FFT	O
(	O
fast	O
Fourier	B-Algorithm
transform	I-Algorithm
)	O
which	O
finds	O
the	O
transform	O
in	O
time	O
.	O
</s>
<s>
Instead	O
of	O
the	O
normal	O
Fourier	B-Algorithm
transform	I-Algorithm
,	O
SWIFFT	B-Algorithm
uses	O
the	O
number-theoretic	O
transform	O
.	O
</s>
<s>
(	O
Universal	B-Algorithm
hashing	I-Algorithm
)	O
.	O
</s>
<s>
The	O
SWIFFT	B-Algorithm
family	O
of	O
functions	O
is	O
universal	B-Algorithm
.	O
</s>
<s>
SWIFFT	B-Algorithm
family	O
of	O
compression	O
functions	O
is	O
regular	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
is	O
a	O
randomness	O
extractor	O
.	O
</s>
<s>
For	O
hash	B-Algorithm
tables	O
and	O
related	O
applications	O
,	O
it	O
is	O
usually	O
desirable	O
for	O
the	O
outputs	O
of	O
the	O
hash	B-Algorithm
function	I-Algorithm
to	O
be	O
distributed	O
uniformly	O
(	O
or	O
as	O
close	O
to	O
uniformly	O
as	O
possible	O
)	O
,	O
even	O
when	O
the	O
inputs	O
are	O
not	O
uniform	O
.	O
</s>
<s>
Hash	B-Algorithm
functions	I-Algorithm
that	O
give	O
such	O
guarantees	O
are	O
known	O
as	O
randomness	O
extractors	O
,	O
because	O
they	O
distill	O
the	O
non-uniform	O
randomness	O
of	O
the	O
input	O
down	O
to	O
an	O
(	O
almost	O
)	O
uniformly	O
distributed	O
output	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
is	O
not	O
pseudorandom	O
,	O
due	O
to	O
linearity	O
.	O
</s>
<s>
It	O
is	O
not	O
claimed	O
by	O
the	O
authors	O
that	O
SWIFFT	B-Algorithm
functions	O
behave	O
like	O
a	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
A	O
function	O
is	O
said	O
to	O
behave	O
like	O
a	O
random	B-Application
oracle	I-Application
if	O
it	O
acts	O
like	O
a	O
truly	O
random	O
function	O
.	O
</s>
<s>
SWIFFT	B-Algorithm
family	O
is	O
provably	O
collision	O
resistant	O
(	O
in	O
an	O
asymptotic	O
sense	O
)	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>
SWIFFT	B-Algorithm
is	O
an	O
example	O
of	O
a	O
provably	O
secure	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
As	O
with	O
most	O
security	O
proofs	O
,	O
the	O
security	O
proof	O
of	O
SWIFFT	B-Algorithm
relies	O
on	O
a	O
reduction	B-Algorithm
to	O
a	O
certain	O
difficult	O
to	O
solve	O
mathematical	O
problem	O
.	O
</s>
<s>
Note	O
that	O
this	O
means	O
that	O
the	O
security	O
of	O
SWIFFT	B-Algorithm
relies	O
strongly	O
on	O
the	O
difficulty	O
of	O
this	O
mathematical	O
problem	O
.	O
</s>
<s>
The	O
reduction	B-Algorithm
in	O
the	O
case	O
of	O
SWIFFT	B-Algorithm
is	O
to	O
the	O
problem	O
of	O
finding	O
short	O
vectors	O
in	O
cyclic/ideal	O
lattices	O
.	O
</s>
<s>
Suppose	O
we	O
have	O
an	O
algorithm	O
that	O
for	O
a	O
random	O
version	O
of	O
SWIFFT	B-Algorithm
given	O
by	O
can	O
find	O
collisions	O
in	O
within	O
some	O
feasible	O
time	O
,	O
and	O
with	O
probability	O
.	O
</s>
<s>
It	O
is	O
allowed	O
that	O
the	O
algorithm	O
only	O
works	O
in	O
a	O
small	O
but	O
noticeable	O
fraction	O
of	O
the	O
family	O
SWIFFT	B-Algorithm
.	O
</s>
<s>
This	O
means	O
that	O
finding	O
collisions	O
in	O
SWIFFT	B-Algorithm
is	O
at	O
least	O
as	O
difficult	O
as	O
the	O
worst-case	B-General_Concept
scenario	O
of	O
finding	O
short	O
vectors	O
in	O
a	O
lattice	O
over	O
.	O
</s>
<s>
Note	O
that	O
this	O
ensures	O
that	O
there	O
is	O
no	O
significant	O
set	O
of	O
"	O
weak	O
instances	O
"	O
where	O
the	O
security	O
of	O
SWIFFT	B-Algorithm
is	O
weak	O
.	O
</s>
<s>
This	O
guarantee	O
is	O
not	O
given	O
by	O
most	O
other	O
provably	O
secure	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
.	O
</s>
