World Journal of Engineering Research and Technology

Ahmed Asimi*

ABSTRACT

In this digital age, modern cryptographic techniques have many uses, such as to digitally sign documents, access control, implement electronic money, elliptic curves, IT security and network security for example design and validation of authentication and trust architectures. Because of these important uses it is necessary that users be able to estimate the efficiency and security of cryptographic techniques. It is ot sufficient for them to know only how the techniques work. One of the most useful of these structures is that of finite fields which are perfectly connected to these primitive elements. Indeed, every finite field is commutative and admits a primitive element. In this paper, we effectively determinate the primitive elements of finite fields where is a prime number. We show that 1) if is a prime number then is a Fermat prime number; 2) is a primitive element of if and only if is not a quadratic residue modulo ; 3) the elements 32n+1 modulo for all ?? are the primitive elements of with ; and 4) 2 is a primitive element of if and only if (ie ).

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