Introduction to Cryptography with Open-Source Software

376 Cryptography with Open-Source Software

Withdrawing a coin

If the User wishes to withdraw a coin, the following steps must be followed.

Step 1. The Bank first choo ses a random w ∈ Z

q

and computes

a = g

w

(mod p)

b = (Ig

2

)

w

(mod p)

which are sent to the User. A new w must be chosen for each tra nsaction.

Step 2. The User then cho oses at random s, x

1

, x

2

, u, v ∈ Z

q

(these s hould

be freshly chosen for each new coin) and computes

A = (Ig

2

)

s

B = g

x

1

1

g

x

2

2

z

′

= z

s

a

′

= a

u

g

v

b

′

= b

su

A

v

where all arithmetic is done modulo p. The value A = 1 is not permitted:

this would assume either a value s = 0(modq), o r that Ig

2

= 1(modp)

which means that the User has managed to find U to solve the discrete

logarithm problem g

U

1

= g

−1

2

( mod p). The User next computes the hash

value

H