Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate

SUBJECT

Code
Subject
MML0007 Computational Algebra
Section
Semester
Hours: C+S+L
Category
Type
Computer Science - in Romanian
Computer Science - in English
4
2+1+1
speciality
optional
Information engineering - in English
Information engineering - in Romanian
4
2+2+0
optional
Teaching Staff in Charge
Assoc.Prof. CRIVEI Septimiu, Ph.D.,  criveimath.ubbcluj.ro
Assoc.Prof. SZANTO Csaba Lehel, Ph.D.,  szantomath.ubbcluj.ro
Lect. SACAREA Cristian, Ph.D.,  csacareamath.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. Public key encryption algorithms: RSA, ElGamal. Electronic signatures. Hash functions.
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 and Coding Theory. Linear codes, cyclic codes, Reed-Muller codes.
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
Romanian version for this subject
Rtf format for this subject