The course is also offered to the students of the fields other than those the course is directly associated with.
Fields of study the course is directly associated with
there are 16 fields of study the course is directly associated with, display
At the end of this course, students should be able to:
understand and use methods of number theory to solve moderately difficult tasks;
understand how results of number theory are applied in cryptography:
understand basic computational context;
understand algebraic notions and explain general implications and context;
model and solve combinatorial problems and use generating functions during their solutions.
The fourth part of the block of four courses in Mathematics in its extended version. In the entire course, the fundamentals of general algebra, linear algebra and mathematical analysis, including their applications in probability, statistics are presented. This semester introduces elemetns of number theory, algebra and some combinatorial methods, including numerical and applied aspects.
1. Number theory (4 weeks) – divisiblity (gcd, extended Euclid algorithm, Bezout); numerics of big numbers (gcd, modular exponential); prime numbers (properties, basic theorems of arithmetics, factorization, prime number testing (Rabin-Miller, Mersenneho prime numbers); congruences (basic properties, small Fermat theorem; Euler theorem; linear congruences; binomial congruences a primitiv roots; discrete logarithm; prime numbers - testing up to AKS, divisors, eliptic curves (introduction); Legendre symbol and the quadratic reciprocity law.
2. Number theory applications (1 week) – short introduction to asymetric cryptography (RSA, DH, ElGamal, DSA, ECC); basic coding theory (linear and polynomial codes); aplication of fast Fourier transform for quick computations (e.g. Schönhage-Strassen)
3. Computer algebra introduction (3 týdny) – groups, permutations, symetries, modular groups, homomorfisms and factorization, group actions (Burnside lemma); rings and fields (polynomials and their roots, divisibility in integers and in polynomial rings, ideals; finite fields and their basic properties (including applications in computer science; polynomials of more variables (Gröbner basis).
4. Combinatorics (4 weeks) – reminder of basics of combinatorics; generalized binomial theorem; combinatorial identities; Catalan numbers; formal power series; (ordinary) generating functions; exponential generating functions; probabilistic generating functions; solving combinatorial problems with the help of generating functions; solving basic reccurences (Fibonacci); complexity of reccurent algorithm; generating functions in computer science (graph applications, complexity, hashing analysis)
J. Slovák, M. Panák a kolektiv, Matematika drsně a svižně, učebnice v přípravě
RILEY, K.F., M.P. HOBSON and S.J. BENCE. Mathematical Methods for Physics and Engineering. second edition. Cambridge: Cambridge University Press, 2004. 1232 pp. ISBN 0 521 89067 5. info
Four hours of lectures combining theory with problem solving. Seminar groups devoted to solving problems.
During the semester, two obligatory mid-term exams are evaluated (each for max 10 points). In the seminar groups there are tests during the semester being written. The seminars are evaluated in total by max 5 points. The final written test for max 20 points is followed by the oral examination. For successful examination (the grade at least E) the student needs to obtain 20 points or more and to succcessfully pass the oral exam.
Language of instruction
The course is taught annually.
The course is taught: every week.