Cryptography Reference
In-Depth Information
Suppose that we now select
t
= 754
at random and compute
βα
t
=4
22
754
3
2
·
≡
2
·
·
5
·
7 (mod 3361)
.
Thus, we have
log
22
(4) + 754
≡
log
22
(2) + 2 log
22
(3) + log
22
(5) + log
22
(7) (mod 3360)
.
Hence,
log
22
(4) = 2200
, and we check that indeed
22
2200
≡
4 (mod 3361)
.
D.5
Brands' Digital Cash Scheme
Now we turn to e-commerce and present the details of Brands' scheme dis-
cussed at the end of Section 5.8 on page 232.
Brands' Digital Cash Scheme
Setup Stage:
The bank performs the following steps:
(1) Choose a large prime
p
such that (
p
1)
/
2=
q
is also prime, and select
α
to be the square of a primitive root modulo
p
. Also, we assume that the
DLP in (
−
)
∗
is intractable.
Z
/p
Z
)
∗
, compute
g
1
≡
α
x
1
(mod
p
) and
(2)
Choose two random
x
1
,x
2
∈
(
Z
/q
Z
α
x
2
(mod
p
), then discard
x
1
,x
2
. (Note that by (1),
g
1
≡
g
2
≡
g
2
(mod
p
)
if and only if
x
1
≡
x
2
(mod
q
).) Make (
α,g
1
,g
2
) public.
)
∗
and compute
(3) Select a random secret
x
∈
(
Z
/q
Z
α
x
(mod
p
)
,
1
≡
g
1
(mod
p
),
g
2
(mod
p
)
.
h
≡
and
h
2
≡
Then (
h,h
1
,h
2
) is the bank's public key and
x
is the bank's private key.
(4) Choose two public cryptographic hash functions,
)
∗
)
5
)
∗
)
∗
)
4
)
∗
.
H
1
:((
Z
/p
Z
→
(
Z
/q
Z
and
H
2
:((
Z
/p
Z
→
(
Z
/q
Z
(5) The merchant registers identification number
M
with the bank.
Opening Alice's Account:
)
∗
at random and computes
(1) Alice generates
e
1
,e
2
∈
(
Z
/q
Z
g
e
1
g
e
2
A
≡
≡
1 (mod
p
)
,
2
which she sends to the bank.
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