Abstract not found

Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...

The demands of secured electronic transactions are increasing rapidly. Prime Field Elliptic curve cryptosystems (PFECC) are becoming most popular because of the reduced number of key bits required in comparison to other cryptosystems. PFECC is emerging as an attractive alternative to traditional public-key cryptosystems. PFECC offers equivalent security with smaller key sizes resulting in faster computations, lower power consumption, as well as memory and bandwidth savings. While these characteristics make PFECC especially appealing for small devices, they can also alleviate the computational burden on secure web servers.