Cryptography Reference
In-Depth Information
1. If the
H
1
(
ID
∗
)
=
w
0
,then
B
aborts.
randomly picks
r
1
,r
2
∈
Z
p
,astring
σ
∗
∈{
n
.
2. Otherwise,
B
0
,
1
}
B
sets the
,where
U
1
=
g
r
1
,
U
2
=
g
r
ciphertext
C
∗
=
U
1
,U
2
,V
∗
2
,
V
∗
is a random
n
.Notethat
g
r
1
1
=(
g
x−w
0
+
w
0
)
r
1
/x
,
g
r
2
2
=(
g
y−w
0
+
w
0
)
r
2
/y
,
{
0
,
1
}
string from
1
2
so the real random factors are
r
1
=
r
1
/x
,
r
2
=
r
2
/y
.
B
picks a random
λ
and returns
(
C
∗
,k
0
)to
A
.If
β
=1,
B
computes
k
1
=
H
2
(
σ
∗
) and responds to
A
with
(
C
∗
,k
1
).
We remark that
C
∗
is a valid ciphertext with probability at most 1
/p
. However,
this does not affect the final result of the security reduction.
bit
β
∈{
0
,
1
}
,if
β
=0,
B
picks a random string
k
0
∈{
0
,
1
}
Phase 2.
B
proceeds the same way as it did in Phase 1.
outputs its guess
β
for
β
.
Guess.
A
The remaining part of the proof is almost identical to the proof of the twin SK-
IBE scheme in Section 3. Due to lack of space, we include the complete proof in
the full version. We give the final result as follows. The advantage of
B
against
the twin strong BDHI problem is
1
Q
h
1
Adv-2BDHI
B
≥
2
·
(7)
Combined with Theorem 2.2, we have
1
1
Q
h
1
2
Q
h
1
p
Adv-BDHI
B
≥
2
·
−
B
is easy to be verified.
The running time of
5Con lu on
In this paper we propose a new computational problem, named the twin bilinear
Die-Hellman inversion assumption. We construct a new trapdoor test which
enables us to prove that the strong twin Die-Hellman inversion problem is at
least as hard as the original bilinear Die-Hellman inversion problem. Based
on this result, we show how to apply the twinning technique to SK-IBE and
SK-ID-KEM, respectively. It is worth to point out that the improvement on
the tightness of security reductions of SK-IBE [11] and SK-ID-KEM [12] comes
from two aspects, one which benefits from the twinning technique, the other one
which benefits from the using of self-decryption/self-decapsulation function.
Acknowledgements.
We would like to thank Eike Kiltz for helpful discussions,
and to the anonymous reviewers for their valuable suggestions. We also thank
Michael Scott for careful comments that improved the presentation of our results.
Yu Chen was supported by the China Scholarship Council.
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