Not Distributive Lattice Over Cartesian Residue Classes Modulo Primes

Abstract: The concept of almost distributive lattices (ADL) 6 and perfect distributive lattices 10 have been utilized in various fields starting from mathematical logic and the concept of not distributive lattices can be used in coding. Even though the ring of residue classes modulo prime p has no divisors of zero, the polynomial ring that has polynomials with coefficients as n - tuples formed by the Cartesian product of finite fields residue classes modulo primes and this algebra is a polynomial ring using direct sum and direct product, forms a ring with divisors of zero. This helps to create a not distributive lattice over a polynomial ring. The formed lattice will be encrypted using a specific key and with the help of parity digit that will be included into the encrypted message that will be sent through three messages and with a question among these three messages, one message will have the correct parity digit and that will allow the receiver to decrypt the original message. Any hacker will face difficulty due to the not distributive nature of the original algebra abbreviated as NDLOCRCMP. Key words: Divisors of zero, Euclidean Ring, Direct Sum, Direct Product, degree of the polynomial modulo prime, field, Polynomial ring, partial ordering, ciphering