Cryptography




aviso

New:

* The grades of extra exercise are in Moodle.
* The third handout has been published.


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temario

Course details

This a course mainly on modular forms with some references to cryptography by administrative requirements.

The lectures are scheduled on Monday and Wednesday 2:30pm - 4:00pm in room 320 (Building 17, Science Faculty, Mathematics). They will be in Spanish unless some students object. Most likely all the information related to the course (e.g. lecture notes) will be released in English.
The course is composed by five blocks corresponding to those indicated in the teaching guide:
  • [1] Algebra and geometry of the modular group.
  • [2] Modular forms.
  • [3] Theta functions.
  • [4] Hecke operators.
  • [5] Introduction to the Eichler-Shimura theory.
In the lectures, blocks [2] and [3] have been put together under the name "The basic theory of modular forms".
For general reference, please check the master program website in the Department of Mathematics.


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apuntes

Notes


I shall post periodically notes on the topics of the course.

Algebra and geometry of the modular group

 Version
  • The modular group and some relatives 4/Feb/2025
  • *Mixing with matrices 10/Feb/2025
  • Depicting groups 24/Feb/2025
  • Elliptic galore 03/Mar/2025
  • *Cubics to hide messages 12/Mar/2025

The basic theory of modular forms

 Version
  • What they are and how many they are 26/Mar/2025
  • * Curves for codes 02/Apr/2025
  • Hecke explaining Ramanujan 27/Apr/2025
  • L-functions elegance 30/Apr/2025

Theta functions

 Version
  • New groups and new weights 17/May/2025
Focusing on theta **/***/2025
Adding squares **/***/2025

The topics preceded by a star corresponds to sections related to cryptography. Typically they are shorter and less profound.

Errata:
  • Lecture 2, Lemma 2.2. Missing (-1)^n in front of P(\lambda).
  • Lecture 4, Proof of Theorem 4.12, l.3, z_0 is z_1.
  • Lecture 6, Proof of Theorem 1.17, p.6, l.-5, z^{-2} missing in front of f(-1/z).
  • Lecture 7, p.5, first displayed formula, c_{k-1} should read m_{k-1}.
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ejercicios

Exercises

The following sheets are home assignments that contribute to the grading:

Handouts
Sheet 1 Sheet 2 Sheet 3

Extra exercise. This is part of the extra activities. This exercise is much harder than those in the handouts.
Extra exercise 2. This is part of the extra activities. This exercise is much harder than those in the handouts.
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examenes

Grading


The grading follows the formula max(0.3E+0.7C,F) where C means three collections of exercises, F a non mandatory final exam and E extra activities (to be announced).
In few words, the course can be passed handing in a number of exercises, but getting a grade over 7 requires additional effort.





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