"Babes-Bolyai" University of Cluj-Napoca
Faculty of Mathematics and Computer Science

Computational Algebra
Code
Semes-
ter
Hours: C+S+L
Type
Section
MML0007
4
2+2+0
optional
Informatică
Teaching Staff in Charge
Lect. SACAREA Cristian, Ph.D.,  csacareamath.ubbcluj.ro
Lect. SZANTO Csaba Lehel, Ph.D.,  szantomath.ubbcluj.ro
Aims
An introduction to Computational Algebra by a survey of applications of algebraic algorithms in Cryptography, Coding Theory and Conceptual Knowledge Processing and Representation.
Content
1. Notions of algorithms complexity. The O notation, classes of complexity.
2. Congruences and residue classes. Euclid's algorithm, Euler's function, Chinese remainder theorem, quadratic residues, Legendre's symbol.
3. Primality tests. Fermat, Solovay-Strassen and Miller-Rabin primality tests, deterministic tests.
4. Factorization methods. Elementary methods, Rho method, factor bases method, continued fractions method, quadratic sieve method. Applications to Cryptography.
5. Polynomials over finite fields. Finite fields, discret logarithm, irreducible polynomials, Berlekamp's algorithm of polynomial factorization. Applications to Cryptography.
6. Other algorithms. Fast addition, fast Fourier transform.
References
1. W. Bosma, A. van der Porten, Computational Algebra and Number Theory, Kluwer 1995.
2. D. Bressoud, S. Wagon, A Course in Computational Number Theory, Springer-Verlag 2000.
3. H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, 2000.
3. H. Cohen, A.M. Cuypers, H. Sterk, Some Tapas of Computer Algebra, Springer-Verlag, 1999.
4. R. Crandall, C. Pomerance, Prime Numbers. A Computational Perspective, Springer-Verlag, 2001.
5. K. Ireland, M. Rosen, A Classical Introduction to Number Theory, Springer-Verlag, 1990.
6. N. Koblitz, A Course in Number Theory and Cryptography, Springer-Verlag, 1994.
7. R. Lidl, G. Pilz, Applied Abstract Algebra, Springer-Verlag, 1998.
8. A.J. Menezes, P.C. van Oorschot, S.A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, 1997.
9. B. Schneier, Applied Cryptography, John Wiley & Sons, New York, 1996.
10. H.S. Wilf, Algorithmes et complexite, Masson, Paris, 1989.
Assessment
Exam.
Links: Syllabus for all subjects
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