On Finite Cyclic Groups And Applications:- Obi Ikenna G

ABSTRACT x Cyclic groups are not only common in pure mathematics, but also in patterns, shapes, music, and chaos. Cyclic groups are an imperative part of number theory used with the Chinese remainder theorem. In this research project we have emphasized on the different applications of cyclic groups, such as the application of cyclic groups in number theory, clock arithmetic and cryptography. On a clock the numbers cycle from one to twelve. The set ofnumbers on a clock are C= {0,1,2,3,4,5,6,7,8,9,10,11}. This set ofnumbers forms a group with the binary operation of addition mod 12. The identity element is 0 which is equivalent to 12. If we add 12 hours to anywhere on the clock, we will end up in the same position. We then proceed to construct the 12-hours clock addition table which then shows clearly the cyclic group properties ofthe clock system. In cryptography, especially within the encryption of data transmission, prime numbers play an important role as most encryption systems use two very large prime numbers to r thv encryption key. We find that cyclic groups come into play here in the encryptions process. The system works in where n is the product oftwo large primes. represents the set of integers modulo n which is an additive cyclic group, and I/(Z„) represents the units of Zn - which are the invertible elements of The security of the system rests on the difficulty of factorizing | |, which is difficult because ofthe magnitude ofthe prime factors.