The link to this page is:      https://www.mat.uniroma2.it/~geatti/KAP2026/AlgebraIV.html

Program: Review of the ring Z/nZ and its properties. Primality of Large Numbers. Large number factorization. Cryptography. The RSA Cryptosystem. Digital Signatures. Safety of RSA. Discrete Logarithms. Diffie-Hellman protocol.

Prerequisites: The Algebra course, parts I, II, III. A good working knowledge of modular arithmetic is especially desirable (see Exercises 1 below).

References:
• P. Stevenhagen, Algebra 1, Chapter 7.
• PARI/GP computer algebra system for doing fast computations in number theory.
• VIDEOS of the lectures will be recorded. The links to the videos will be posted on this page.


Contacts: Our email adderesses are      geatti@mat.uniroma2.it      lea.terracini@unito.it


• Video1: Introduction to the course link
• Video2: The division algorithm and its applications: the Euclidean algorithm for computing the gcd of two integers, Bezout identity, expressing an integer in different bases. link
• Video3: Review of the ring Z/n Z link
• Video4: The Euler φ function link
• Video5: Inverses modulo n link
• Video6: Congruences modulo n link
• Video7: Chinese Remainder Theorem and systems of congruences link
• Video8: Euler's Theorem, Fermat Little Theorem and prime numbers link
• Video9: Primality testing and the Prime Number Theorem (part 1) link      PNT: examples pdf
• Video10: Introduction to Cryptography (part 1) link
• Video11: Introduction to Cryptography (part 2) link
• Video12: Primality testing and the Prime Number Theorem (part 2) link
• Video13: RSA examples and digital signatures link      RSA examples pdf
• Video14: The discrete logarithm (Part 1) link
• Video15: The discrete logarithm (Part 2) link
• Video16: The discrete logarithm (Part 3) link
• Video17: Review exercises link



Exercises
• Exercises 1 (review of modular arithmetic) pdf
• Exercises 1: solutions pdf
• Exercises 2 (the ring Z/n Z, RSA, the discrete logarithm, Diffie-Hellman key exchange) pdf