NUMBER THEORY AND CRYPTOGRAPHY
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- PAOLO CODECA'
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- The aim of this course is to give an introduction to modern public-key cryptography.We present also the necessary mathematical tools.
The main acquired knowledge will be:
basic arithmetical theorems which are fundamental in public key cpyptography and applications.
The basic acquired abilities will be:
encryption and decryption of messages by means of the most important publick key cryptosystems (RSA. El-Gamal, Rabin, Blum-Goldwasser...) and relative digital signatures.
Prerequisites
- None
Course programme
- 1) Natural numbers and relative integers.Divisibility.Prime numbers.The fundamental theorem of Arithmetic.Greatest common divisor.Euclidean algorythm. (8 hours)
2)Linear congruences.Residue systems( complete and reduced).Abelian groups.Theorem of Eulero_Fermat.Quadratic congruences.Quadratic reciprocità law. (12 hours)
3) Primitive roots and indices.Running time of the euclidean algorythm and of modular exponentiation.Tests of primality. (10 hours)
4)Cryptographic applications.RSA,Rabin,El Gamal and Blum _ Goldwasser cryptosystems.Digital signatures.Diffie_ Hellman problem and Massey _ Omura protocol.Pseudo random Number generation. (30 hours) Didactic methods
- Lectures on theory and its applications.
In the academic year 21-22 the course will be held in a classroom but it will also be possible to follow the lessons in distance learning mode. Learning assessment procedures
- We have a written examination,divided in two parts.
- In the first part ( two hours of time),the candidates must solve problems strictly linked with the cryptosystems they have studied (for example how to cipher and decipher messages,how to use digital signatures or key exchange systems,etc..)
- In the second part ( two hours of time) the candidates must prove mathematical results which are fundamental in the applications (for example the running time of euclidean algorythm and of modular power,how cryptosystems like R.S.A.,El Gamal and Rabin work and why they are considered difficult to crack,etc...).
Passing the final exam is the proof that knowledge and abilities outlined in the training objectives of the course have been achieved.
The first part of the exam is passed if the score is greater than or equal to 18. Vote out of thirty. Reference texts
- -Stinson,Douglas R."Cryptography-Theory and practice"
Chapman & Hall/CRC.2002
-Koblitz,Neal "A course in number theory and Cryptography" Second edition;Springer,1994.
-Languasco,A.,Zaccagnini,A. "Introduzione alla crittografia" Hoepli(Informatica).2004
