Number Theory and its Applications to Cryptography

Explore advanced number theory and its applications to cryptography through this comprehensive lecture series from the Jean-Morlet Chair semester program. Delve into the intricate connections between arithmetic structures and cryptographic systems, focusing on pseudorandomness, elliptic curve cryptography, and Frobenius distributions. Examine group structures of elliptic curves through Igor Shparlinski's foundational lectures, then advance to distributions of Frobenius with contributions from Chantal David and Nathan Jones. Investigate the Sato-Tate conjecture and its generalizations through detailed presentations by Francesc Fité, Andrew Sutherland, and Jean Pierre Serre. Study the Chebotarev density theorem and character sums with Peter Stevenhagen, while exploring unlikely intersections and polynomial orbits with Mike Zieve. Learn about Hilbert cubes in arithmetic sets, Galois types of Abelian surfaces, and limiting distributions of Frobenius traces. Master the mathematical foundations underlying pseudorandom number generators, integers of cryptographic interest, and the distribution of points on curves over finite fields. Gain insights into both theoretical aspects and practical applications, including quasi-Monte Carlo methods and elliptic curve cryptography, while exploring polynomial analogues of classical number theory results.