Lecture 2-3 Basic Number Theory and Algebra
In modern cryptographic systems, the messages are represented by numerical values prior to being encrypted and transmitted. The encryption processes are mathematical operations that turn the input numerical values into output numerical values. Building, analyzing, and attacking these cryptosystem requires mathematical tools. The most important of these is number theory, especially the theory of congruences.
Outline Basic Notions Congruence Quadratic Residues Primitive Root Inverting Matrices Mod n Groups Rings Fields
1 Basic Notions 1.1 Divisibility
1.1 Divisibility (Continued)
1.1 Divisibility (Continued)
1.2 Prime The primes less than 200: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
1.2 Prime (Continued)
1.2 Prime (Continued)
1.2 Prime (Continued)
1.3 Greatest Common Divisor
1.3 Greatest Common Divisor (Continued)
1.3 Greatest Common Divisor (Continued)
1.3 Greatest Common Divisor (Continued)
1.3 Greatest Common Divisor (Continued)
1.4 The Fundamental Theorem of Arithmetic
1.4 The Fundamental Theorem of Arithmetic (Continued)
1.4 The Fundamental Theorem of Arithmetic (Continued)
1.4 The Fundamental Theorem of Arithmetic (Continued)
1.4 The Fundamental Theorem of Arithmetic (Continued)
1.5 Linear Diophantine Equations
2 Congruences 2.1 Introduction to Congruences
2.1 Introduction to Congruences (Continued)
2.1 Introduction to Congruences (Continued)
2.1 Introduction to Congruences (Continued)
2.1 Introduction to Congruences (Continued)
2.2 Linear Congruences
2.2 Linear Congruences (Continued)
2.2 Linear Congruences (Continued)
2.2 Linear Congruences (Continued)
2.3 The Chinese Remainder Theorem
2.3 The Chinese Remainder Theorem (Continued)
2.3 The Chinese Remainder Theorem (Continued)
2.4 Polynomial Modulo Prime
2.5 Fermat’s Little Theorem and Euler’s Theorem
2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)
2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)
2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)
2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)
2.5 Fermat’s Little Theorem and Euler’s Theorem (Continued)
3 Quadratic Residues 3.1 Quadratic Residues and Nonresidues
3.1 Quadratic Residues and Nonresidues (Continued)
3.1 Quadratic Residues and Nonresidues (Continued)
3.2 Modular Square Roots
3.2 Modular Square Roots (Continued)
4 Primitive Root 4.1 The Order of an Integer
4.1 The Order of an Integer (Continued)
4.1 The Order of an Integer (Continued)
4.2 Primitive Root
4.2 Primitive Root (Continued)
5 Inverting Matrices Mod n
5 Inverting Matrices Mod n (Continued)
5 Inverting Matrices Mod n (Continued)
6 Groups, Rings, Fields 6.1 Groups
6.1 Groups (Continued)
6.2 Rings
6.2 Rings (Continued)
6.3 Fields
Thank you!