Cryptography Reference
In-Depth Information
For a given message Î q
mZ , choose a random Î q
rZ and set
, mr
Cham - Hash (
v mr
, )
=
g
y
(4.30)
Note that the above chameleon hash function is collision-resistant for any user UV .
User V can always find any other message ¹
ˆ mm and compute the appropriate r to
find the same hash value, because he knows x and can easily solve +=+
m r m r .
ˆ
ˆ
*
q
x
vg G is the public key of user u .
Key-Gen :
x Î is the secret key and
1
1
Noninteractive Ring Signing : Given public keys v 1 ,…, v n G 1 , a message m Z q ,
and a private key x corresponding to one of the public keys v s for some s , do the
following:
1. Choose randomly Î q
ˆ
rZ and compute
hH
=
(Cham-Hash (
mr G
, ))
Î .
v
2
*
q
2. Choose randomly Î"¹
aZ i
s .
3. Set
1/
x
æ
ö ÷
ç
ˆ
h
÷
ç
÷
ç
σ
=
÷
(4.31)
ç
÷
s
ç
÷
a
i
ψ
ç
(
v
÷ ÷
i
ç è
ø
is
¹
σ
= 2 a
4.
i
s , set
g
.
i
n
5. The ring signature is
σσσ
=
(
,...,
)
Î .
G
1
n
2
Verification : Given public keys v 1 ,…, v n G 1 , a message m M , a ring signature
and r , compute
ˆ
hH
=
(Cham - Hash (
mr
, ))
(4.32)
V
and verify
n
= 1
egh
(,)
ev
( , )
σ
(4.33)
i
i
=
i
4.3.4.5 Zhang-Safavi-Susilo (ZSS) Short Signature Scheme
The ZSS scheme is as follows (Zhang et al. 2004).
Key-Gen : Let H 1 be a hash function. The secret key is Î q
xZ and the public key
is P 0 = xP for a signer.
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