Not Distributive Lattice Over Residue Classes Polynomial Ring

Abstract

In the era of soft skills and almost every area of physical activity and space, the shied ankle role player is data security. However safe the data may be today by creating the firewall, in few hours or days, those firewalls are pierced through and the data will be hacked/stolen. So, every moment, there is a need for creating new firewalls or new techniques in security systems. Ciphering through algebraic techniques is my view point and taking a lattice based on algebra and failure of distributive property may be considered as a productive approach that enciphers a code and deciphering may be difficult while there is no regularity/balance in the system. The polynomial ring defined over a finite field of residue classes modulo p  where p is a prime, is a commutative ring with unity. A principal ideal generated by an irreducible polynomial is a maximal ideal in that ring. So, the theorem `if R is a commutative ring with unity, M is an ideal of R, then M is maximal if and only if the quotient ring of R by M is a field'', helps us to construct a field. Defining the join' denoted by `$\vee $' and meet' , `$\wedge $' operations on this field using the modulo p operation both on the coefficient and exponent of each monomial of each polynomial will allow the closedness under these operations and thus the formed lattice is a closed algebra. In the present discussion, we restrict our view to the elements of $\mathbb{Z}_{5}^{5} \left[x\right]$ up to distributive property.