Kerr M. Number Theory and Cryptography

Lectures Notes. — Without publishing data. — 77 p.Standard introductory course on number theory and the basics of cryptography.Primes and divisibility
The Euclidean Algorithm
rimes and factorization
The distribution of primes
The prime number theorem
Congruences
Modular arithmetic
Consequences of Fermat’s theorem
The Chinese Remainder Theorem
Primality and compositeness testing
Groups, rings, and fields
Primitive roots
Prime power moduli and power residues
Introduction to cryptography
Symmetric ciphers
Public key cryptography
Discrete log problem
RSA Cryptosystem
Introduction to PARI
Breaking RSA
Diophantine equations
A first view of Diophantine equations
Quadratic Diophantine equations
Units in quadratic number rings
Pell’s equation and related problems
Unique factorization in number rings
Elliptic curves
Elliptic curves over F_p
Elliptic cryptosystems
Elliptic curve discrete log problem (ECDLP)
Elliptic curve cryptography
Lenstra’s factorization algorithm
Pairing-based cryptography
Divisors and the Weil pairing
Algebraic numbers
Algebraic number fields
Discriminants and algebraic integers
Ideals in number rings
The ideal class group
Fermat’s Last Theorem for regular exponents