Lectures in course MATH-489: Number theory II.c - Cryptography

Explains the Fast Fourier Transform algorithm for polynomial multiplication efficiency.

Covers exponentiation, RSA encryption, big integer multiplication, and primitive roots.

Explores windowing exponentiation with base-B representation and precomputed small powers of x for efficient computation.

Explores common multiplicand speedup through logical operations like bitwise AND and XOR to optimize the multiplication process.

Covers Barrett reduction, Montgomery form of integers, and efficient Montgomery product computation.

Covers the basics of finite fields, including arithmetic operations and properties.

Explores the properties and applications of finite fields, including isomorphism and cyclic properties.

Introduces polynomial factorization over finite fields and efficient computation of greatest common divisors of polynomials.

Explores Berlekamp's algorithm for efficient polynomial factorization.

Delves into the complexity of factoring polynomials and the implications for security.

Explores methods and algorithms for integer factorization, including testing for B smoothness and computing small primes.

Explores smoothness in integer factorization and probabilities, including the analysis of Dixon's method.

Covers the Quadratic Sieve method for integer factorization, emphasizing the importance of choosing the right parameters for efficient factorization.

Explores integer factorization using the quadratic sieve method and the challenges of working with algebraic number fields.

Introduces Pollard's rho method for the discrete log problem and finding collisions efficiently in a cyclic group.

Explores the discrete log problem and index calculus method in cryptography.

Covers the basics of elliptic curves, their significance in cryptography, and their applications in public key cryptography.

Explores singular points, the group law, and the ambiguity in defining the sum of a point with itself on elliptic curves.

Covers the group law on elliptic curves and its role in key exchange protocols.

Explores Elliptic Curve Discrete Logarithm Problem, lattice-based cryptography, and cryptographic schemes like ECDSA.