<s>
The	O
elliptic	B-Algorithm
curve	I-Algorithm
only	I-Algorithm
hash	I-Algorithm
(	O
ECOH	B-Algorithm
)	O
algorithm	O
was	O
submitted	O
as	O
a	O
candidate	O
for	O
SHA-3	O
in	O
the	O
NIST	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
competition	I-Algorithm
.	O
</s>
<s>
The	O
ECOH	B-Algorithm
is	O
based	O
on	O
the	O
MuHASH	O
hash	B-Error_Name
algorithm	I-Error_Name
,	O
that	O
has	O
not	O
yet	O
been	O
successfully	O
attacked	O
.	O
</s>
<s>
The	O
main	O
difference	O
is	O
that	O
where	O
MuHASH	O
applies	O
a	O
random	B-Application
oracle	I-Application
,	O
ECOH	B-Algorithm
applies	O
a	O
padding	B-Algorithm
function	O
.	O
</s>
<s>
Assuming	O
random	B-Application
oracles	I-Application
,	O
finding	O
a	O
collision	O
in	O
MuHASH	O
implies	O
solving	O
the	O
discrete	O
logarithm	O
problem	O
.	O
</s>
<s>
MuHASH	O
is	O
thus	O
a	O
provably	O
secure	O
hash	B-Error_Name
,	O
i.e.	O
</s>
<s>
ECOH	B-Algorithm
does	O
not	O
use	O
random	B-Application
oracles	I-Application
and	O
its	O
security	O
is	O
not	O
strictly	O
directly	O
related	O
to	O
the	O
discrete	O
logarithm	O
problem	O
,	O
yet	O
it	O
is	O
still	O
based	O
on	O
mathematical	O
functions	O
.	O
</s>
<s>
ECOH	B-Algorithm
is	O
related	O
to	O
the	O
Semaev	O
's	O
problem	O
of	O
finding	O
low	O
degree	O
solutions	O
to	O
the	O
summation	O
polynomial	O
equations	O
over	O
binary	O
field	O
,	O
called	O
the	O
Summation	O
Polynomial	O
Problem	O
.	O
</s>
<s>
Under	O
certain	O
assumptions	O
,	O
finding	O
a	O
collision	O
in	O
ECOH	B-Algorithm
may	O
be	O
also	O
viewed	O
as	O
an	O
instance	O
of	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
ECOH	B-Algorithm
is	O
a	O
good	O
example	O
of	O
hash	B-Error_Name
function	I-Error_Name
that	O
is	O
based	O
on	O
mathematical	O
functions	O
(	O
with	O
the	O
provable	O
security	O
approach	O
)	O
rather	O
than	O
on	O
classical	O
ad	O
hoc	O
mixing	O
of	O
bits	O
to	O
obtain	O
the	O
hash	B-Error_Name
.	O
</s>
<s>
Given	O
,	O
ECOH	B-Algorithm
divides	O
the	O
message	O
into	O
blocks	O
.	O
</s>
<s>
One	O
additional	O
point	O
contains	O
the	O
padding	B-Algorithm
and	O
depends	O
only	O
on	O
the	O
message	O
length	O
.	O
</s>
<s>
The	O
result	O
is	O
truncated	B-Algorithm
to	O
get	O
the	O
hash	B-Error_Name
.	O
</s>
<s>
Finally	O
,	O
the	O
result	O
is	O
passed	O
through	O
an	O
output	O
transformation	O
function	O
f	O
to	O
get	O
the	O
hash	B-Error_Name
result	O
.	O
</s>
<s>
Four	O
ECOH	B-Algorithm
algorithms	O
were	O
proposed	O
,	O
ECOH-224	O
,	O
ECOH-256	O
,	O
ECOH-384	O
and	O
ECOH-512	O
.	O
</s>
<s>
ECOH-384	O
uses	O
the	O
curve	O
B-409	O
:	O
,	O
with	O
parameters	O
(	O
192	O
,	O
64	O
,	O
64	O
)	O
.	O
</s>
<s>
ECOH-512	O
uses	O
the	O
curve	O
B-571	O
:	O
,	O
with	O
parameters	O
(	O
256	O
,	O
128	O
,	O
128	O
)	O
.	O
</s>
<s>
It	O
can	O
hash	B-Error_Name
messages	O
of	O
bit	O
length	O
up	O
to	O
.	O
</s>
<s>
Incrementality	O
:	O
ECOH	B-Algorithm
of	O
a	O
message	O
can	O
be	O
updated	O
quickly	O
,	O
given	O
a	O
small	O
change	O
in	O
the	O
message	O
and	O
an	O
intermediate	O
value	O
in	O
ECOH	B-Algorithm
computation	O
.	O
</s>
<s>
Speed	O
:	O
The	O
ECOH	B-Algorithm
algorithm	O
is	O
about	O
thousand	O
times	O
slower	O
than	O
SHA-1	B-Algorithm
.	O
</s>
<s>
However	O
,	O
given	O
the	O
developments	O
in	O
desktop	O
hardware	O
towards	O
parallelization	B-Operating_System
and	O
carryless	B-Device
multiplication	I-Device
,	O
ECOH	B-Algorithm
may	O
in	O
a	O
few	O
years	O
be	O
as	O
fast	O
as	O
SHA-1	B-Algorithm
for	O
long	O
messages	O
.	O
</s>
<s>
For	O
short	O
messages	O
,	O
ECOH	B-Algorithm
is	O
relatively	O
slow	O
,	O
unless	O
extensive	O
tables	O
are	O
used	O
.	O
</s>
<s>
The	O
ECOH	B-Algorithm
hash	B-Error_Name
functions	I-Error_Name
are	O
based	O
on	O
concrete	O
mathematical	O
functions	O
.	O
</s>
<s>
They	O
were	O
designed	O
such	O
that	O
the	O
problem	O
of	O
finding	O
collisions	O
should	O
be	O
reducible	B-Algorithm
to	O
a	O
known	O
and	O
hard	O
mathematical	O
problem	O
(	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
)	O
.	O
</s>
<s>
Functions	O
with	O
these	O
properties	O
are	O
known	O
provably	O
secure	O
and	O
are	O
quite	O
unique	O
among	O
the	O
rest	O
of	O
hash	B-Error_Name
functions	I-Error_Name
.	O
</s>
<s>
The	O
problem	O
of	O
finding	O
collisions	O
in	O
ECOH	B-Algorithm
is	O
similar	O
to	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Solving	O
a	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
is	O
almost	O
as	O
hard	O
as	O
the	O
discrete	O
logarithm	O
problem	O
.	O
</s>
<s>
If	O
one	O
adopts	O
this	O
loose	O
heuristic	O
,	O
then	O
finding	O
an	O
internal	O
ECOH	B-Algorithm
collision	O
may	O
be	O
viewed	O
as	O
an	O
instance	O
of	O
the	O
subset	B-Algorithm
sum	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
It	O
requires	O
2143	O
time	O
for	O
ECOH-224	O
and	O
ECOH-256	O
,	O
2206	O
time	O
for	O
ECOH-384	O
,	O
and	O
2287	O
time	O
for	O
ECOH-512	O
.	O
</s>
<s>
For	O
this	O
attack	O
we	O
have	O
a	O
message	O
M	O
and	O
try	O
to	O
find	O
a	O
M'''	O
that	O
hashes	B-Error_Name
to	O
the	O
same	O
message	O
.	O
</s>
<s>
The	O
workload	O
we	O
have	O
to	O
do	O
here	O
is	O
two	O
times	O
K''	O
partial	O
hash	B-Error_Name
computations	O
.	O
</s>
<s>
ECOH-224	O
and	O
ECOH-256	O
use	O
the	O
elliptic	O
curve	O
B-283	O
with	O
approximately	O
points	O
on	O
the	O
curve	O
.	O
</s>
<s>
ECOH-384	O
uses	O
the	O
elliptic	O
curve	O
B-409	O
with	O
approximately	O
points	O
on	O
the	O
curve	O
.	O
</s>
<s>
ECOH-512	O
uses	O
the	O
elliptic	O
curve	O
B-571	O
with	O
approximately	O
points	O
on	O
the	O
curve	O
.	O
</s>
<s>
The	O
official	O
on	O
ECOH	B-Algorithm
included	O
a	O
proposal	O
called	O
ECOH2	O
that	O
doubles	O
the	O
elliptic	O
curve	O
size	O
in	O
an	O
effort	O
to	O
stop	O
the	O
Halcrow-Ferguson	O
second	O
preimage	O
attack	O
with	O
a	O
prediction	O
of	O
improved	O
or	O
similar	O
performance	O
.	O
</s>
