<s>
Whereas	O
insecure	O
secret	O
sharing	O
allows	O
an	O
attacker	B-General_Concept
to	O
gain	O
more	O
information	O
with	O
each	O
share	O
,	O
secure	O
secret	O
sharing	O
is	O
'	O
all	O
or	O
nothing	O
 '	O
(	O
where	O
'	O
all	O
 '	O
means	O
the	O
necessary	O
number	O
of	O
shares	O
)	O
.	O
</s>
<s>
Secret	O
sharing	O
schemes	O
are	O
important	O
in	O
cloud	B-Architecture
computing	I-Architecture
environments	O
.	O
</s>
<s>
Secret	O
sharing	O
has	O
also	O
been	O
suggested	O
for	O
sensor	B-Architecture
networks	I-Architecture
where	O
the	O
links	O
are	O
liable	O
to	O
be	O
tapped	O
by	O
sending	O
the	O
data	O
in	O
shares	O
which	O
makes	O
the	O
task	O
of	O
the	O
eavesdropper	O
harder	O
.	O
</s>
<s>
In	O
contrast	O
,	O
consider	O
the	O
secret	O
sharing	O
scheme	O
where	O
X	O
is	O
the	O
secret	O
to	O
be	O
shared	O
,	O
Pi	O
are	O
public	O
asymmetric	B-Application
encryption	I-Application
keys	O
and	O
Qi	O
their	O
corresponding	O
private	B-Application
keys	I-Application
.	O
</s>
<s>
Each	O
player	O
J	O
is	O
provided	O
with	O
In	O
this	O
scheme	O
,	O
any	O
player	O
with	O
private	B-Application
key	I-Application
1	O
can	O
remove	O
the	O
outer	O
layer	O
of	O
encryption	O
,	O
a	O
player	O
with	O
keys	O
1	O
and	O
2	O
can	O
remove	O
the	O
first	O
and	O
second	O
layer	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
A	O
player	O
with	O
fewer	O
than	O
N	O
keys	O
can	O
never	O
fully	O
reach	O
the	O
secret	O
X	O
without	O
first	O
needing	O
to	O
decrypt	O
a	O
public-key-encrypted	O
blob	O
for	O
which	O
he	O
does	O
not	O
have	O
the	O
corresponding	O
private	B-Application
key	I-Application
–	O
a	O
problem	O
that	O
is	O
currently	O
believed	O
to	O
be	O
computationally	O
infeasible	O
.	O
</s>
<s>
Additionally	O
we	O
can	O
see	O
that	O
any	O
user	O
with	O
all	O
N	O
private	B-Application
keys	I-Application
is	O
able	O
to	O
decrypt	O
all	O
of	O
the	O
outer	O
layers	O
to	O
obtain	O
X	O
,	O
the	O
secret	O
,	O
and	O
consequently	O
this	O
system	O
is	O
a	O
secure	O
secret	O
distribution	O
system	O
.	O
</s>
<s>
Several	O
secret-sharing	O
schemes	O
are	O
said	O
to	O
be	O
information-theoretically	B-Algorithm
secure	I-Algorithm
and	O
can	O
be	O
proven	O
to	O
be	O
so	O
,	O
while	O
others	O
give	O
up	O
this	O
unconditional	B-Algorithm
security	I-Algorithm
for	O
improved	O
efficiency	O
while	O
maintaining	O
enough	O
security	O
to	O
be	O
considered	O
as	O
secure	O
as	O
other	O
common	O
cryptographic	O
primitives	O
.	O
</s>
<s>
For	O
example	O
,	O
they	O
might	O
allow	O
secrets	O
to	O
be	O
protected	O
by	O
shares	O
with	O
entropy	O
of	O
128	O
bits	O
each	O
,	O
since	O
each	O
share	O
would	O
be	O
considered	O
enough	O
to	O
stymie	O
any	O
conceivable	O
present-day	O
adversary	B-General_Concept
,	O
requiring	O
a	O
brute	O
force	O
attack	O
of	O
average	O
size	O
2127	O
.	O
</s>
<s>
There	O
is	O
sometimes	O
a	O
workaround	O
for	O
this	O
limitation	O
by	O
first	O
compressing	B-General_Concept
the	O
secret	O
before	O
sharing	O
it	O
,	O
but	O
this	O
is	O
often	O
not	O
possible	O
because	O
many	O
secrets	O
(	O
keys	O
for	O
example	O
)	O
look	O
like	O
high-quality	O
random	O
data	O
and	O
thus	O
are	O
hard	O
to	O
compress	O
.	O
</s>
<s>
If	O
an	O
insider	O
can	O
gain	O
any	O
more	O
knowledge	O
about	O
the	O
secret	O
than	O
an	O
outsider	O
can	O
,	O
then	O
the	O
system	O
no	O
longer	O
has	O
information	B-Algorithm
theoretic	I-Algorithm
security	I-Algorithm
.	O
</s>
<s>
If	O
the	O
players	O
store	O
their	O
shares	O
on	O
insecure	O
computer	O
servers	O
,	O
an	O
attacker	B-General_Concept
could	O
crack	O
in	O
and	O
steal	O
the	O
shares	O
.	O
</s>
<s>
All	O
of	O
the	O
non-updated	O
shares	O
the	O
attacker	B-General_Concept
accumulated	O
become	O
useless	O
.	O
</s>
<s>
An	O
attacker	B-General_Concept
can	O
only	O
recover	O
the	O
secret	O
if	O
he	O
can	O
find	O
enough	O
other	O
non-updated	O
shares	O
to	O
reach	O
the	O
threshold	O
.	O
</s>
<s>
Additionally	O
,	O
an	O
attacker	B-General_Concept
cannot	O
recover	O
any	O
information	O
about	O
the	O
original	O
secret	O
from	O
the	O
update	O
files	O
because	O
they	O
contain	O
only	O
random	O
information	O
.	O
</s>
<s>
Tal	O
Rabin	O
and	O
Michael	O
Ben-Or	O
devised	O
a	O
multiparty	B-Protocol
computing	I-Protocol
(	O
MPC	O
)	O
system	O
that	O
allows	O
players	O
to	O
detect	O
dishonesty	O
on	O
the	O
part	O
of	O
the	O
dealer	O
or	O
on	O
part	O
of	O
up	O
to	O
one	O
third	O
of	O
the	O
threshold	O
number	O
of	O
players	O
,	O
even	O
if	O
those	O
players	O
are	O
coordinated	O
by	O
an	O
"	O
adaptive	O
"	O
attacker	B-General_Concept
who	O
can	O
change	O
strategies	O
in	O
realtime	O
depending	O
on	O
what	O
information	O
has	O
been	O
revealed	O
.	O
</s>
<s>
Alternate	O
techniques	O
have	O
been	O
proposed	O
for	O
greatly	O
increasing	O
the	O
efficiency	O
of	O
secret	O
sharing	O
schemes	O
,	O
by	O
giving	O
up	O
the	O
requirement	O
of	O
unconditional	B-Algorithm
security	I-Algorithm
.	O
</s>
<s>
One	O
of	O
these	O
techniques	O
,	O
known	O
as	O
secret	O
sharing	O
made	O
short	O
,	O
combines	O
Rabin	O
's	O
information	O
dispersal	O
algorithm	O
(	O
IDA	O
)	O
with	O
Shamir	B-Architecture
's	I-Architecture
secret	I-Architecture
sharing	I-Architecture
.	O
</s>
<s>
An	O
information-theoretically	B-Algorithm
secure	I-Algorithm
k-of-n	O
secret-sharing	O
scheme	O
generates	O
n	O
shares	O
,	O
each	O
of	O
size	O
at	O
least	O
that	O
of	O
the	O
secret	O
itself	O
,	O
leading	O
to	O
the	O
total	O
required	O
storage	O
being	O
at	O
least	O
n-fold	O
larger	O
than	O
the	O
secret	O
.	O
</s>
<s>
In	O
multi-secret	O
sharing	O
designed	O
by	O
Matthew	O
K	O
.	O
Franklin	O
and	O
Moti	O
Yung	O
,	O
multiple	O
points	O
of	O
the	O
polynomial	O
host	O
secrets	O
;	O
the	O
method	O
was	O
found	O
useful	O
in	O
numerous	O
applications	O
from	O
coding	O
to	O
multi-party	B-Protocol
computations	I-Protocol
.	O
</s>
<s>
sensor	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
An	O
attacker	B-General_Concept
would	O
have	O
to	O
intercept	O
all	O
t	O
shares	O
to	O
recover	O
the	O
secret	O
,	O
a	O
task	O
which	O
is	O
more	O
difficult	O
than	O
intercepting	O
a	O
single	O
file	O
,	O
especially	O
if	O
the	O
shares	O
are	O
sent	O
using	O
different	O
media	O
(	O
e.g.	O
</s>
<s>
Secret	O
sharing	O
is	O
an	O
important	O
primitive	O
in	O
several	O
protocols	O
for	O
secure	B-Protocol
multiparty	I-Protocol
computation	I-Protocol
.	O
</s>
