Computational tools for Number Theory and Algebra
Computational tools for Number Theory and Algebra
Date
Tue-Thu, 15h-17h, from Sept. 23, 2014
Registration
–
Location
Aula IMUB, Facultat de Matemàtiques, UB
REMARK: The course will include a number of laboratory sessions to practice the main topics with SAGE and Magma.
Date
Tue-Thu, 15h-17h, from Sept. 23, 2014
Registration
–
Location
Aula IMUB, Facultat de Matemàtiques, UB
REMARK: The course will include a number of laboratory sessions to practice the main topics with SAGE and Magma.
Lecturers
Jordi Guàrdia (UPC)
Enric Nart (UAB)
Contents
- Integer arithmetic. Fast arithmetic. Euclid’s algorithm. Modular computations. Basic primality tests and factorization algorithms.
- Polynomial arithmetic. Basic arithmetic. Euclid’s algorithm. Resultants and discriminants. Basic factorization algorithms.
- Finite fields. Computational representations. Factorization of polynomials over finite fields.
- Local techniques. The ring of p-adic numbers. Power series rings. Hensel lemma and Hensel lift. Factorization of polynomials over local fields. Local computation of discriminants and resultants.
- Modules and lattices. Applications. Modules over Z. Hermite normal form. Lattices over Z. The shortest vector problem and successive minima. The LLL reduction algorithm. Factorization of polynomials over global fields.
- Elliptic curves over finite fields. Basic properties. Counting algorithms. Factorization algorithms. Pairings. Elliptic curve cryptography.