Università Roma Tre

1. Theory of Elliptic Curves
Weierstrass Equation, The structure of the group on rational points, formulas for the addition and duplication. Generalities on the intersections between lines and curves in P2(K)
Preparatory results for the proof of the associativity of points on elliptic curves. Proof of the associativity of the sum for the points of an elliptic curve. Other equations for elliptic curves, Legendre's equation, Cubic equations, Quartic equations, intersections of two cubic surfaces. The j-invariant, elliptic curve in characteristic 2, Endomorphisms, singular curves, elliptic curves module n.

2. Torsion points
Torsion points, Division polynomials. Weil's pairing

3. Elliptic curve on finite fields
Frobenius endomorphism. The problem of determining the order of the group. Curves on subfields, Legendre's symbols, Point orders, Shanks's "Baby Step, Giant Step" algorithm. Particular families of elliptic curves. Schoof's algorithm.

4. Cryptosystems on Elliptical Curves.
The Discrete Logarithm Problem. Algorithms for calculating the discrete logarithm: Baby-Step Giant-Step and Polig-Hellman. MOV attack. Attack on anomalous curves. Diffie-Hellman Key Exchange. Cryptosystems by Massey Omura and ElGamal. El Gamal Signature Scheme. Cryptosystems on elliptic curves analysis on the factorization problem. A cryptosystem based on Weil coupling. Factorization of internal numbers using elliptic curves. Using Pari.

Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003.