This paper introduces a new arithmetic group of n-tuples. Properties of circulant matrix and Galois field of prime order have been used in the construction. This group of n-tuples is highly useful in information security, cryptography, coding theory, communication and image processing.

Circulant Matrix: An n  n matrix of the form
      
is called a circulant matrix.

It has the following properties:

• Sum of two circulant matrices is a circulant matrix.
• Product of two circulant matrices is a circulant matrix
• Inverse of a circulant matrix is a circulant matrix.

1. Arithmetic Group: An arithmetic group is a nonempty set G of numbers together with a law of binary composition which is associative and has an identity element, such that every element of G has an inverse.
2. Abelian Group: An abelian group is a group whose law of composition is commutative.
3. Linear Congruences (Smarandache, 1980 and 1987): A number a is said to be linearly congruent to b modulus m if (a – b) is divisible by m. This
   
   fact is represented as a = b (mod m)

Computational Mathematics Journal, Logistic Regression Models, Business Management, Decision Theory, Logistic Regression Programs, SPSS Nonlinear Program, Proportional Reductions, Statistical Aanalysis Software Package, Statistical Software SPSS, Squared Pearson Correlation .