<s>
Shamir	B-Architecture
's	I-Architecture
secret	I-Architecture
sharing	I-Architecture
(	O
SSS	O
)	O
is	O
an	O
efficient	B-General_Concept
secret	O
sharing	O
algorithm	O
for	O
distributing	O
private	O
information	O
(	O
the	O
"	O
secret	O
"	O
)	O
among	O
a	O
group	O
so	O
that	O
the	O
secret	O
cannot	O
be	O
revealed	O
unless	O
a	O
quorum	B-General_Concept
of	O
the	O
group	O
acts	O
together	O
to	O
pool	O
their	O
knowledge	O
.	O
</s>
<s>
SSS	O
has	O
the	O
property	O
of	O
information-theoretic	B-Algorithm
security	I-Algorithm
,	O
meaning	O
that	O
even	O
if	O
an	O
attacker	O
steals	O
some	O
shares	O
,	O
it	O
is	O
impossible	O
for	O
the	O
attacker	O
to	O
reconstruct	O
the	O
secret	O
unless	O
they	O
have	O
stolen	O
the	O
quorum	B-General_Concept
number	O
of	O
shares	O
.	O
</s>
<s>
Shamir	B-Architecture
's	I-Architecture
secret	I-Architecture
sharing	I-Architecture
is	O
used	O
in	O
some	O
applications	O
to	O
share	O
the	O
access	O
keys	O
to	O
a	O
master	O
secret	O
.	O
</s>
<s>
No	O
information	O
about	O
the	O
secret	O
can	O
be	O
gained	O
from	O
any	O
number	O
of	O
shares	O
below	O
than	O
the	O
threshold	O
(	O
a	O
property	O
called	O
perfect	B-Algorithm
secrecy	I-Algorithm
)	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
SSS	O
is	O
a	O
generalisation	O
of	O
the	O
one-time	B-Algorithm
pad	I-Algorithm
(	O
which	O
can	O
be	O
viewed	O
as	O
SSS	O
with	O
a	O
two-share	O
threshold	O
and	O
two	O
shares	O
in	O
total	O
)	O
.	O
</s>
<s>
sharing	O
of	O
the	O
passphrase	O
used	O
to	O
recreate	O
a	O
master	O
secret	O
,	O
which	O
is	O
in	O
turn	O
used	O
to	O
access	O
a	O
cryptocurrency	B-Protocol
wallet	I-Protocol
.	O
</s>
<s>
Secure	O
:	O
The	O
scheme	O
has	O
information-theoretic	B-Algorithm
security	I-Algorithm
.	O
</s>
<s>
Single	B-Architecture
point	I-Architecture
of	I-Architecture
failure	I-Architecture
:	O
The	O
secret	O
must	O
exist	O
in	O
one	O
place	O
when	O
it	O
is	O
split	O
into	O
shares	O
,	O
and	O
again	O
in	O
one	O
place	O
when	O
it	O
is	O
reassembled	O
.	O
</s>
<s>
These	O
are	O
attack	O
points	O
,	O
and	O
other	O
schemes	O
including	O
multisignature	O
eliminate	O
at	O
least	O
one	O
of	O
these	O
single	B-Architecture
points	I-Architecture
of	I-Architecture
failure	I-Architecture
.	O
</s>
<s>
The	O
scheme	O
exploits	O
the	O
Lagrange	O
interpolation	B-Algorithm
theorem	O
,	O
specifically	O
that	O
points	O
on	O
the	O
polynomial	O
uniquely	O
determines	O
a	O
polynomial	O
of	O
degree	O
less	O
than	O
or	O
equal	O
to	O
.	O
</s>
<s>
Shamir	B-Architecture
's	I-Architecture
secret	I-Architecture
sharing	I-Architecture
is	O
an	O
ideal	O
and	O
perfect	O
-threshold	O
scheme	O
based	O
on	O
polynomial	O
interpolation	B-Algorithm
over	O
finite	O
fields	O
.	O
</s>
<s>
Given	O
any	O
subset	O
of	O
of	O
these	O
pairs	O
,	O
can	O
be	O
obtained	O
using	O
interpolation	B-Algorithm
,	O
with	O
one	O
possible	O
formula	O
for	O
doing	O
so	O
being	O
,	O
where	O
the	O
list	O
of	O
points	O
on	O
the	O
polynomial	O
is	O
given	O
as	O
pairs	O
of	O
the	O
form	O
.	O
</s>
<s>
Therefore	O
the	O
example	O
below	O
does	O
not	O
provide	O
perfect	B-Algorithm
secrecy	I-Algorithm
and	O
is	O
not	O
a	O
proper	O
example	O
of	O
Shamir	O
's	O
scheme	O
.	O
</s>
<s>
Using	O
the	O
formula	O
for	O
polynomial	O
interpolation	B-Algorithm
,	O
is	O
:	O
</s>
<s>
Using	O
polynomial	O
interpolation	B-Algorithm
to	O
find	O
a	O
coefficient	O
in	O
a	O
source	O
polynomial	O
using	O
Lagrange	O
polynomials	O
is	O
not	O
efficient	B-General_Concept
,	O
since	O
unused	O
constants	O
are	O
calculated	O
.	O
</s>
