A Gaussian integer is any complex number of the form a + bi where a and b are integers.

In later secondary years students may also study prime numbers as part of enrichment and extension of the curriculum in the area of number theory, or as a component of school-based assessment, for example exploration of Gaussian integers as part of work on complex numbers in advanced mathematics.

Mathematica includes an option for number theoretic functionality to apply for Gaussian integers, for example:

In our work we bigining with a study of the classical diffie-hellman key exchange that use the integer number, then we extend it using the domain of gaussian integer.

beta]][i] the group of units gaussian integer modula a gaussian prime [beta], with the multiplication binary operation.

In this work when we extend the cryptosystem to the domain of gaussian integer, this extension make the cryptosystem more secure and very difficult to be broken bu at the same time this extension need to provide potential and need a more divelopment computer to accomplishment this work.

In this paper, we extended the computational procedures behind the elgamal algorithm using arithmetics module gaussian integers.

p] and then we will modify these cryptosystems to the domain of gaussian integers.

The alogrithm here is similar to the algorithm in classical case but we choose the prime [beta] in this case from the domain of gaussian integers.

To be able to generate a common private key using Diffie-Hellman key exchange over the domain of gaussian integers, Ali and Basem must follow these steps:

2 ElGamal public-key cryptosystem in the domain of gaussian integers [][i]

After a fine introduction to basic notions, he covers unique factorization, the

Gaussian integers, and Pell's equation, and moves on to algebraic number theory.