Compared to full feedback of the CSI in the SVD design in [1], the partial information, which between the RS and the MSs and is fed back to the BS, is a Gaussian integer matrix bounded by a modulo operation.

At the RS, we use the CLLL algorithm, which is a suboptimum solution of the shortest vector problem with a polynomial-time computational complexity [21], to reduce the lattice basis of and obtain H = U, where is the right pseudoinverse of [H.sub.MR] (i.e., [H.sup.[dagger].sub.MR] = [H.sup.H.sub.MR] [([H.sub.MR][H.sup.H.sub.MR]).sup.-1]; U = [[[u.sub.i,j]].sub.1[less than or equal to]t[less than or equal to]k,1[less than or equal to]j[less than or equal to]k] is a Kx K unimodular matrix, i.e., a square matrix with Gaussian integer entries, such that det(U) = [+ or -]1).

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.

A Gaussian integer is a complex number a + bi with both a and b in Z.

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. Combining [12, Theorem 4] and the discussion in [17], it follows that the family of 4-valent first-kind Frobenius circulants is precisely the family of Gaussian graphs [12, Definition 3] of odd orders (see Lemma 5 and Remark 6).

Newer cryptography applications use discrete logarithms in multiplicative group [G.sup.*.sub.[beta]] = [[].sup.*.sub.[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.

[1] Huber K., "Codes Over Gaussian Integers" IEEE Trans.

and Dymacek W.M., "Finding Factors of Factor Rings Over The Gaussian Integers" The Mathematical Association of America, Monthly Aug-Sep.2005.

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.

4.4 The RSA on the quotient ring of

Gaussian integers