Lecture 2 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 value 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  Solving ax+by=d=gcd(a,b)  Congruence  The Chinese Remainder Theorem  Fermat’s Little Theorem and Euler’s Theorem  Primitive Root  Inverting Matrices Mod n  Square Roots Mod n  Groups Rings Fields

1 Basic Notions 1.1 Divisibility

1.1 Divisibility (Continued)

1.2 Prime The primes less than 200:

1.2 Prime (Continued)

1.3 Greatest Common Divisor

1.3 Greatest Common Divisor (Continued)

2 Solving ax+by=d=gcd(a,b)

3 Congruences

3.1 Addition, Subtraction, Multiplication

3.1 Addition, Subtraction, Multiplication (Continued)

3.2 Division

3.2 Division (Continued)

3.3 Division (Continued)

3.2 Division (Continued)

4 The Chinese Remainder Theorem

4 The Chinese Remainder Theorem (Continued)

5 Fermat’s Little Theorem and Euler’s Theorem

5 Fermat’s Little Theorem and Euler’s Theorem (Continued)

6 Primitive Root

6 Primitive Root (Continued)

7 Inverting Matrices Mod n

7 Inverting Matrices Mod n (Continued)

8 Square Roots Mod n

8 Square Roots Mod n (Continued)

9 Groups, Rings, Fields 9.1 Groups

9.1 Groups (Continued)

9.2 Rings

9.2 Rings (Continued)

9.3 Fields

Thank you!