


Politecnico di Torino  
Anno Accademico 2007/08  
02ISNIT Curve ellittiche e crittografia 

Dottorato di ricerca in Ingegneria Elettronica E Delle Comunicazioni  Torino 





Obiettivi dell'insegnamento
This course is an introduction to elliptic curves over finite fields and their arithmetic properties that relevant in their cryptographic applications. The aspects of computational complexity are analyzed in some details as they are the main choice criterion of encryption schemes.
The presentation is set in the historical perspective of the development of elliptic functions which were discovered by an Italian mathematician and brilliantly extended by Euler. 
Programma
1. Elliptic Function, Abelian integrals, and Elliptic curves.
2. Elliptic curves over Q and Diophantine Equations. 3. Structure and properties of the additive point group of an elliptic curve. 4. Elliptic curves over finite fields. 5. Structure of the additive point group of an elliptic curve over finite fields. 6. Iterated sums and duplications over elliptic curves in both affine and homogeneous coordinates, and related complexity. 7. Discrete logarithm over elliptic curves. 8. DiffieHellman and El Gamal publickey cryptographic schemes via elliptic curves 9. Computational complexity 10. Arithmetical complexity of sums, multiplications, and powers of elements in finite fields. 11. Arithmetical complexity of point sums, duplications, and 'powers' on elliptic curves over finite fields. 12. Good choices of elliptic curves for cryptographic applications. 13. Public and secret parameters. Comparison with classic publickey systems. 14. Elliptic curves and factoring. 
Orario delle lezioni 
Statistiche superamento esami 
