So far, there have been many variants of RSA constructed in this manner: In 1985, Varadharajan and Odoni constructed an extension of RSA to matrix rings [15]; In 1993, Demytko, proposed an elliptic curve-based RSA variant at EUROCRYPT [16]; In 2004, El-Kassar, Hatary, and Awad developed a modified RSA in the domains of Gaussian integers and polynomials over finite fields [17].

4 The RSA on the quotient ring of Gaussian integers

The RSA on the quotient ring of Gaussian integers can be regarded as an instance of the proposed model described in Section 4.

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.

Clearly the natural numbers are Gaussian integers where a [member of] N and b = 0.

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

Such graphs are also useful in coding theory, and they were studied independently in [12] from a coding-theoretic point of view by using the language of Gaussian integers.

To this end we will use an equivalent definition of a 4-valent Frobenius circulant in terms of Gaussian integers.

The set Z[i] of all Gaussian integers is a ring under the usual addition and multiplication of complex numbers, the ring of Gaussian integers.

It describe the elgamal public-key cryptosystem and the diffehellman key exchange and the then extends these cryptosystem over the domain of gaussian integers.

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

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.