Elementary Abstract Algebra- Examples and Applications, 2019a

CONTENTS 11
11.8 Hints for “Permutations” exercises . . . . . . . . . . . . . . . 494
11.8.1 Hints for additional exercises (Section 11.7) . . . . . . 494
12 Introduction to Groups 495
12.1 Formal definition of a group . . . . . . . . . . . . . . . . . . . 496
12.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
12.2.1 The group of units of Z n . . . . . . . . . . . . . . . . 505
12.2.2 Groups of matrices . . . . . . . . . . . . . . . . . . . . 507
12.3 Basic properties of groups . . . . . . . . . . . . . . . . . . . . 510
12.4 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 520
12.5 Cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . 526
12.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . 526
12.5.2 Orbits (cyclic subgroups) . . . . . . . . . . . . . . . . 530
12.5.3 Subgroups of cyclic groups . . . . . . . . . . . . . . . 536
12.6 Additional group and subgroup exercises . . . . . . . . . . . . 538
12.7 Hints for “Abstract Groups: Definitions and Basic Properties”
exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
13 Further Topics in Cryptography 542
13.1 Diffie-Hellman key exchange . . . . . . . . . . . . . . . . . . . 542
13.1.1 Man in the middle attack . . . . . . . . . . . . . . . . 550
13.2 Elliptic curve cryptography . . . . . . . . . . . . . . . . . . . 551
13.2.1 Definition of elliptic curves . . . . . . . . . . . . . . . 553
13.2.2 Elliptic curve arithmetic . . . . . . . . . . . . . . . . . 554
13.2.3 Elliptic curve groups . . . . . . . . . . . . . . . . . . . 559
13.2.4 Elliptic curves over Z p . . . . . . . . . . . . . . . . . . 561
13.2.5 An encryption system using elliptic curves . . . . . . . 563
13.2.6 Next steps . . . . . . . . . . . . . . . . . . . . . . . . . 567
13.3 References and suggested reading . . . . . . . . . . . . . . . . 568
13.4 Hints for “Further Topics in Cryptography” exercises . . . . . 570