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
=Î
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.
"¹
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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