Applied Number Theory and Algebra

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Omschrijving

• Multi-precision arithmetic: Representation of integers, Efficient integer arithmetic, Efficient polynomial arithmetic
• Modular algorithms: Euclidean algorithm, Efficient modular arithmetic (exponentiation), Montgomery multiplication
• Multiplicative structure of Zn*: Order of an element, Primitive roots, Algorithms
• Prime number distribution, probabilistic primality testing, prime number generation
• arithmetic and multiplicative functions, convolution, principle of inclusion/exclusion, counting necklaces. Moebius inversion
• applications to cryptography: schoolbook RSA and Diffie-Hellman
• Polynomials: division, Euclid’s algorithm, factorization of polynomials, irreducible polynomials
• Finite fields: inverse elements, finite fields, characteristic, order of field, order of element, primitive elements, multiplicative group, Fermat’s Theorem
• Structure and classification of finite fields, uniqueness, subfields, minimal polynomials, automorphisms
• applications to coding theory: Error detection and correction, minimum distance, Repetition and product codes, Hamming codes, Linear codes, generator matrix, parity check matrix, cyclic codes, decoding

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