Introduction to Cryptography with Open-Source Software

Elliptic curves and cryptosystems 325

Now Trent can c ompute Alice and Bob’s secret keys S

A

and S

B

:

sage: QA = H1("[e-mail]"); QA

(29 : 22 : 1)

sage: QB = H1("[e-mail]"); QB

(17 : 5 : 1)

sage: SA = s*QA; SA

(23 : 24 : 1)

sage: SB = s*QB; SB

(1 : 28 : 1)

Let the message be m = 25. Here’s how Alice encrypts it (no te that XOR is

implemented in Sage with a double caret):

sage: u = e(QB,Q); u

X^2

sage: t = randint(1,p); t

19

sage: C1 = t*P; C1

(5 : 5 : 1)

sage: C2 = m^^H2(u^t); C2

13

So the ciphertext is the pair

(C

1

, C

2

) = ((5 : 5 : 1), 13).

Bob can decrypt this in one step, using

m = C

2

⊕ H

2

(e(S

B

, C

1

)).

In Sage:

sage: C2^^H2(e(SB,C1))

25

For s hort signatures, ...