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.
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