Subject: Computational Algebra	Code: MML0007	

Semester	 Hours: C+S+L	Type	 Section	
4	2+2+0	optional	Computer Science	

Teaching Staff in Charge	Lect. SACAREA Cristian, Ph.D., csacarea@math.ubbcluj.ro
Lect. SZANTO Csaba Lehel, Ph.D., szanto@math.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	
Selective 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