Non-commutative finite associative algebras of 2-dimensional vectors

Abstract

In this paper properties of the non-commutative finite associative algebra of two-dimensional vectors are presented. Interesting features of algebra are mutual associativity of all modifications of the defined parameterized multiplication operation and existing of a large set of single-side unit elements. In the ordinary case one unique two-side unit element is connected with each element of the algebra, except the elements that are square roots from zero element. There are also presented four different variants of defining commutative associative algebras of 2-dimension vectors. For the case of commutativity the algebra has common unit element for all its elements.

Keywords

• finite algebra
• ring
• Galois field
• vector
• associative multiplication
• parameterized multiplication
• cryptoscheme