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S e c o n d E d i t i o n

I n t r o d u c tio n to C r y p to g r a p h y

with Coding Theory

W a d e T r a p p e

Wireless Inform ation N etw ork Laboratory

an d the E lectrical a nd C om puter Engineering Departm ent

Rutgers University

L a w r e n c e C . W a s h i n g t o n

Departm ent o f M athem atics

University o f M aryland

P e a r so n E d u c a tio n I n te r n a t io n a l

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Contents

P r ef ac e xi

1 O ve rv ie w o f C r y p to g ra p h y an d I ts A p p lic atio n s 1

1.1 Secure C om m unications.................................................................. 2

1.2 Cryptographic A pplications........................................................... 9

2 C lassica l C ry p to sy ste m s 12

2.1 Shift C ip h e rs ..................................................................................... 13

2.2 Affine C i p h e r s .................................................................................. 14

2.3 The Vigenere Cipher ..................................................................... 16

2.4 Substitution C ip hers........................................................................ 24

2.5 Sherlock H o lm es

.............................................................................. 27

2.6 The Playfair and ADFGX C ip h e rs.............................................. 30

2.7 Block C i p h e r s .................................................................................. 34

2.8 Binary Numbers and ASCII ........................................................ 38

2.9 One-Time Pads .............................................................................. 39

2.10 Pseudo-random Bit G en eration .................................................... 41

2.11 LFSR S eq uen ces............................................................................... 43

2.12 E n ig m a ............................................................................................... 50

2.13 E x e rc is es............................................................................................ 55

2.14 Com puter P ro b le m s........................................................................ 59

3 B asic N u m b er T h e o ry 63

3.1 Basic Notions .................................................................................. 63

3.2 Solving ax + by = d ........................................................................ 69

3.3 C o n grue n ce s..................................................................................... 70

3.4 T he Chinese Remainder T h e o r e m .............................................. 76

3.5 M odular Exponentiation .............................................................. 78

3.6 Ferm at and E u l e r ........................................................................... 79

3.7 Primitive R o o t s .............................................................................. 83

3.8 Inverting Matrices Mod n .............................................................. 85

3.9 Square Roots Mod n ..................................................................... 86

3.10 Legendre and Jacobi Symbols .................................................... 88

3.11 Finite F ie l d s ..................................................................................... 93

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