Introduction to Cryptography with Open-Source Software

240 Cryptography with Open-Source Software

Show that these opera tions form a group, with juxtapo sition as the

operation, so that, for example R

2

◦F R consists of R

2

followed by F R,

the r e sult being F R

3

.

12. Produce a Cayley table for D

4

, and show that it is non-abelian.

13. Show that the g roup defined in question 9 is the dihedral group of or-

der 3: the group formed by r otations and reflections of an equilateral

triangle.

14. Let p a nd q be two distinct primes. Show that Z

p

⊕ Z

q

is c yclic.

15. Show that if f : G → H is an isomorphism, then the identity element of

G is mapped by f onto the identity element of H.

16. Using the field Z

7

and the tables given on page 220, eva luate the follow-

ing:

(a) (2 + 3)(4 + 5)

−1

, (b) (3

4

)/(4

5

), (c) 1/2 + 3/4 + 4