Cryptography Reference
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From the analysis (a)-(d), it immediately has the following theorem.
Theorem 1.
If the parameter
p
satisfies the inequality (2), then the protocol
Π
for
2
-out-of-
2
rational secret sharing induces an
-Nash equilibrium with
=
μ
(
m
)(
a
b
)
,where
μ
(
m
)
is the negligible probability of successfully forging an
information-theoretic MAC.
Remark 1.
The 2-out-of-2 protocol in [10] used lists of length
l
−
−
1and
l
+
d
−
1 respectively, where
l
and
d
both were chosen according to a geometric
distribution with parameter
β
. Our protocol
Π
uses lists of length
l
1
and
l
2
respectively where
l
1
+
l
2
−
1 is chosen according to a geometric distribution
with parameter
p
.Sinceboth
β
and
p
are determined by the utility values under
the similar inequalities, we can simply regard
β
=
p
. Then the expected length
of lists in [10] are
1
1and
2
1
2
p
.
That is, we only need the list that is almost half as long as the shorter list in
[10], which means the expected size of shares in our protocol is smaller.
p
−
p
−
1, while our lists are both of length about
Remark 2.
Since in [10] the shorter list was just a prefix of the longer one and
every value alone could possibly be the secret, a player can certainly determine
the secret if he finds all his remain cells contain the same value. To fix this
problem, it masked each value by a random number for each cell. Thus the
cells in [10] contained both the masked value and share of the mask. But in our
protocol, the secret is jointly determined by all values contained in the two lists,
a player cannot determine the secret even if he sees all values in his list. Therefor
no mask is needed in our protocol and our lists consist of simpler cells.
4 Rational Secret Sharing: The
t
-Out-of-
n
Case
We now construct a
t
-out-of-
n
rational secret sharing protocol in the information
theoretic setting. Since it is in non-simultaneous channels and (
t
1)-resilience
is required, the protocol is not a simple extension of the protocol
Π
constructed
in Section 3. Denote the
t
-out-of-
n
protocol by
Π
. We still describe
Π
in terms
of Dealer's protocol and players' protocol separately.
−
Dealer's Protocol
1. Choose integers
l
∗
and
d
according to a geometric distribution with param-
eter
p
,where
p
is a constant to be determined later (in Theorem 2).
2. Randomly select
σ
such that
Prob
[
σ
=0]=
q
,where
q
is a constant
to be determined later (in Theorem 2).
3. Construct a list of length
l
=
l
∗
+
d
.For1
∈{
0
,
1
}
≤
j
≤
l
,the
j
-th cell contains:
-
Main
:(
s
j
,s
j
)
S
2
,where
S
is the secret-domain. In particular, it requires
s
l
∗
=
s
and the other values are randomly chosen.
-
Index
:(
I
j
,I
j
)
∈
2
where
∈{
0
,
1
}
=
1
,
if
j
=
1
,
if
j
=
l
∗
and
σ
=0
0
,
otherwise
1=
l
∗
and
σ
=1
0
,
otherwise
−
I
j
,I
j
.
For consistence, fix
I
1
=0.
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