Mathematical Cryptography

B242 - Summer 24/25

B4M01MKR

Mathematical Cryptography - B4M01MKR

Credits	6
Semesters	Summer
Completion	Assessment + Examination
Language of teaching	Czech
Extent of teaching	4P+2S

Annotation

The lecture sets mathematical foundations of modern cryptography (RSA, El-Gamal, elliptic curve cryptography). Related algorithms for primality testing, number factorisation and discrete logarithm are treated as well.

Study targets

None

Course outlines

1. Introduction into cryptography. Basic notions of number theory.
2. Counting modulo n. Complexity of operations in Z_n.
3. RSA cryptosystem and attacks on it.
4. Abelian groups.
5. Element order in a group, cyclic groups.
6. Structure of Z_n^* groups.
7. Discrete logarithm, Diffie-Hellman protocol.
8. Elliptic curves, discrete logarithm on an elliptic curve.
9. Generating of random primes, probability algoritms.
10. Primality tests, Carmichael numbers.
11. Factorisation using the Euler function.
12. Subexponential algorithms for discrete logarithm.
13. Subexponential algorithms for factorisation, a quadratic sieve.
13. Quantum computing and satefy of cryptosystems.

Exercises outlines

None

Literature

[1] V.Shoup, A Computational introduction to number theory and algebra, Cambridge University Press, 2008, http://shoup.net/ntb/
[2] D.Boneh, Twenty Years of Attacks on the RSA Cryptosystem. https://crypto.stanford.edu/~dabo/papers/RSA-survey.pdf
[3] D.Hankerson, A.J.Menezes, S.Vanstone, Guide to elliptic curve cryptography, Springer, 2004.

Requirements

None