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Courses
Group Theory, Florian Luca
(University of the Witwatersrand)
Abstract:
- Basic definitions, properties, subgroups, quotient groups and the basic isomorphism theorem.
- Specific examples include finite groups (order, exponent),
symmetric group (permutations, cycles, order of a permutation), cyclic
groups (generators), finitely generated abelian groups (torsion and
rank, structure theorem), multiplicative groups mod n (Euler function, Carmichael function), GL2(Z/nZ).
- Mention solvable and nilpotent groups.
- Present the basic notions of representation theory such as
irreducible representations, their characters and their main properties
(the number of them and the orthogonality relations).
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Galois Theory, Alain Togbe
(Purdue University Northwest)
Abstract: In this course, we will introduce the
Galois theory. So we will start from the basic definitions of field
automorphisms to gently introduce Galois theory. We will state the
fundamental theorem of Galois theory. We will also discuss the Galois
groups of polynomials, solvable and radical extensions, and
transcendental extensions. |
Algebraic Number Theory, Francesco Pappalardi
(Università degli Studi Roma Tre)
Abstract:
- Basics of algebraic number theory (Number fields, rings of integers, splitting of primes)
- Splitting of primes in cyclotomic fields, quadratic fields and general number fields. Primes in arithmetic progressions
- Unity and the Dirichlet units theorem. Finiteness of the class number
- Dedekind zeta functions and the class number formula.
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Introduction à la cryptographie, Nitaj Abderrahmane
(Universite de Caen)
Abstract: Ce cours aborde les principaux thèmes
mathématiques pour s'initier á la cryptographie. Le cours est composé
des chapitres suivants:
- L'arithmétique modulaire, les tests de primalité, la factorisation des nombres entiers, les corps finis, les générateurs.
- Le protocole de Diffie-Hellman, le système RSA, le système d'El Gamal.
- Cryptanalyses élémentaires de RSA et du logarithme discret.
Chaque chapitre sera mis en pratique à l'aide du système de calcul Python.
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Introduction to Number Theory based Cryptography, Djiby Sow
(Université Cheikh Anta Diop)
Abstract:
- For non post quantum cryptography: overview on discrete logarithm
problem and its applications on elliptic curves based cryptography (key
exchange, signature, encoding and hashing).
- For post quantum cryptography: introduction to isogeny (on elliptic curved) based cryptography.
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Introduction to SAGEMath, Cécile Armana
(Université de Franche-Comté)
Abstract:Explicit computations and numerical
experiments play an increasingly important role in number theory.
SageMath is a computer algebra system which covers many aspects of
mathematics, including algebra, combinatorics, graph theory, numerical
analysis, number theory, calculus and statistics. The software is free
and open-source. This course will consist in an introduction to SageMath
and a series of
tutorial sessions in number theory, in connection with the topics of
this school.
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Introduction to Cryptographic Proof Techniques, Augustin Sarr
(Université Gaston Berger de Saint-Louis)
Abstract:We propose an introduction to the
computational cryptographic proof techniques, with a focus on the
central paradigms (computational hardness, computational
indistinguishability, security game, etc.) used in modeling and
proofs. For many cryptographic tasks (encryption, entity authentication,
signature, etc.), we will present the games that define the attributes,
and analyze some constructions that achieve the security goals.
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