Some characteristics of cyclic prime, weakly prime and almost prime submodule of Gaussian integer modulo over integer

Abstract

Cryptography is a branch of mathematics used in digital security systems. Some cryptographic algorithms, such as RSA, depending on the prime factorization of integers. However, quantum computers might be a threat to some algorithms in cryptography. One of the mathematicians efforts in finding alternatives to create new security technologies in cryptography is to study the abstraction of prime numbers. The prime submodule is one of the prime numbers abstraction, which was introduced by Dauns in 1978. Lately, the prime submodules were generalized into weakly prime submodules and almost prime submodules. This study will examine the characteristics of prime, weakly prime, and almost prime of cyclic submodules on ℤ−module ℤn[i]. The results are if n is a prime number, then the cyclic submodule is prime submodule on ℤ−module ℤn[i] and the cyclic almost prime submodule equivalent with cyclic weakly prime submodule on ℤ−module ℤn[i].