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Course

MATH-489: Number theory II.d - Cryptography

Lectures in this course (26)

Explains the Fast Fourier Transform algorithm for polynomial multiplication efficiency.

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

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.

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

Explores Hermite's Theorem in lattices, including Minkowski's first theorem and properties of centrally symmetric lattices.

Covers the LLL algorithm for lattice reduction and discusses Hermite's constant and Minkowski's theorem.

Covers Coppersmith's method for attacking RSA encryption by efficiently finding small roots of polynomials modulo N.

Covers Coppersmith's method for finding small roots of polynomials modulo N efficiently.

Covers Fully Homomorphic Encryption (FHE) based on Gentry's 2009 scheme, explaining encryption, decryption, and multiplication processes.

Covers homomorphic encryption, focusing on Fully Homomorphic Encryption (FHE) and its applications in reducing noise in encrypted data.

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