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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