<s>
In	O
cryptography	O
,	O
Very	B-Algorithm
Smooth	I-Algorithm
Hash	I-Algorithm
(	O
VSH	O
)	O
is	O
a	O
secure	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
invented	O
in	O
2005	O
by	O
Scott	O
Contini	O
,	O
Arjen	O
Lenstra	O
and	O
Ron	O
Steinfeld	O
.	O
</s>
<s>
Unlike	O
other	O
secure	O
collision-resistant	O
hashes	B-Algorithm
,	O
VSH	O
is	O
efficient	O
and	O
usable	O
in	O
practice	O
.	O
</s>
<s>
Asymptotically	O
,	O
it	O
only	O
requires	O
a	O
single	O
multiplication	O
per	O
log(n )	O
message-bits	O
and	O
uses	O
RSA-type	O
arithmetic	O
.	O
</s>
<s>
Both	O
versions	O
have	O
similar	O
efficiency	B-General_Concept
.	O
</s>
<s>
VSH	O
is	O
not	O
suitable	O
as	O
a	O
substitute	O
for	O
a	O
random	B-Application
oracle	I-Application
,	O
but	O
can	O
be	O
used	O
to	O
build	O
a	O
secure	O
randomized	O
trapdoor	O
hash	B-Algorithm
function	I-Algorithm
.	O
</s>
<s>
This	O
function	O
can	O
replace	O
the	O
trapdoor	O
function	O
used	O
in	O
the	O
Cramer	B-Algorithm
–	I-Algorithm
Shoup	I-Algorithm
signature	I-Algorithm
scheme	I-Algorithm
,	O
maintaining	O
its	O
provable	O
security	O
while	O
speeding	O
up	O
verification	O
time	O
by	O
about	O
50%	O
.	O
</s>
<s>
All	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
that	O
are	O
now	O
widely	O
used	O
are	O
not	O
based	O
on	O
hard	O
mathematical	O
problems	O
.	O
</s>
<s>
For	O
the	O
basic	O
version	O
of	O
Very	B-Algorithm
Smooth	I-Algorithm
Hash	I-Algorithm
function	O
,	O
this	O
hard	O
problem	O
is	O
to	O
find	O
modular	O
square	O
roots	O
(	O
VSSR	O
)	O
of	O
certain	O
special	O
numbers	O
(	O
VSN	O
)	O
.	O
</s>
<s>
Let	O
be	O
a	O
large	O
RSA	B-Architecture
composite	O
and	O
let	O
the	O
sequence	O
of	O
primes	O
.	O
</s>
<s>
To	O
compute	O
the	O
hash	B-Algorithm
of	O
:	O
</s>
<s>
Nonetheless	O
,	O
the	O
hash	B-Algorithm
function	I-Algorithm
VSH	O
is	O
collision-resistant	O
based	O
on	O
the	O
VSSR	O
assumption	O
.	O
</s>
<s>
Since	O
the	O
output	O
length	O
of	O
VSH	O
is	O
the	O
length	O
of	O
a	O
secure	O
RSA	B-Architecture
modulus	O
,	O
VSH	O
seems	O
quite	O
suitable	O
in	O
practice	O
for	O
constructing	O
"	O
hash-then-sign	O
"	O
RSA	B-Architecture
signatures	I-Architecture
for	O
arbitrarily	O
long	O
messages	O
.	O
</s>
<s>
Efficiency	B-General_Concept
:	O
The	O
cost	O
of	O
each	O
iteration	O
is	O
less	O
than	O
the	O
cost	O
of	O
3	O
modular	O
multiplications	O
.	O
</s>
<s>
important	O
hash	B-Algorithm
function	I-Algorithm
properties	O
,	O
and	O
the	O
authors	O
state	O
that	O
"	O
VSH	O
should	O
not	O
be	O
used	O
to	O
model	O
random	B-Application
oracles	I-Application
,	O
"	O
and	O
cannot	O
be	O
substituted	O
into	O
constructions	O
that	O
depend	O
upon	O
them	O
(	O
RSA	B-Architecture
signatures	I-Architecture
,	O
some	O
MACs	B-Algorithm
)	O
.	O
</s>
<s>
VSH	O
should	O
not	O
be	O
considered	O
a	O
general-purpose	O
hash	B-Algorithm
function	I-Algorithm
as	O
usually	O
understood	O
in	O
security	O
engineering	O
.	O
</s>
<s>
trade-off	O
attack	O
that	O
applies	O
to	O
multiplicative	O
and	O
additive	O
hashes	B-Algorithm
.	O
</s>
<s>
VSH	O
produces	O
a	O
very	O
long	O
hash	B-Algorithm
(	O
typically	O
1024	O
bits	O
)	O
.	O
</s>
<s>
a	O
truncated	O
VSH	O
hash	B-Algorithm
offers	O
security	O
that	O
is	O
commensurate	O
with	O
the	O
hash	B-Algorithm
length	O
.	O
</s>
<s>
Total	O
cost	O
:	O
roughly	O
,	O
rather	O
than	O
as	O
expected	O
from	O
a	O
hash	B-Algorithm
function	I-Algorithm
with	O
good	O
pseudorandomness	O
properties	O
.	O
</s>
<s>
This	O
probably	O
rules	O
out	O
the	O
applicability	O
of	O
VSH	O
in	O
digital	O
signature	O
schemes	O
which	O
produce	O
signatures	O
shorter	O
than	O
the	O
VSH	O
hash	B-Algorithm
result	O
,	O
such	O
as	O
elliptic-curve	O
signature	O
schemes	O
.	O
</s>
