Lucas Numbers and Cryptography

Abstract

We know that the Fibonacci numbers are the numbers from Fibonacci sequence. It was discovered by Leonardo de Fibonacci de Pisa. The Fibonacci series was derived from the solution to a problem about rabbits. The problem
is: If a newborn pair of rabbits requires one month to mature and at the end of the second month and every month thereafter reproduce itself, how many pairs will one have at the end of n months? Lucas numbers are the numbers from the Lucas sequence. Lucas se-quence is dened by the same recurrence relations as Fibonacci sequence with dierent initial values. We are considering general Lucas sequence dened by second-order relation i.e, fTng = PTn􀀀1 􀀀 QTn􀀀2. where gcd(P;Q) = 1 and P;Q 2 Z. The general solution of the sequence is given by fc1n + c2ng,
where and are the roots of the corresponding polynomial equation of fTng. If we put the particular values of c1 and c2 in the general solu-tion, then we get two particular solutions Un(P;Q) = n 􀀀 n 􀀀 where where (c1 = 1 􀀀 = 􀀀c2) and Vn(P;Q) = n+n where (c1 = 1 = c2). This Un(P;Q) gives the Fibonacci sequence and Vn(P;Q) gives the Lucas sequence. The later will use in the LUC cryptosystem. We also know that Cryptography is very important in security problems.Nowadays, Everybody want secure information.In the rst chapter we give some denations, theorems, lemmas, on el-ementary number theory. This chapter is useful for next discussion on the main topic. In the second chapter, we shall discuss the properties of Fi-bonacci numbers and related numbers call Lucas numbers. And also, we shall give few applications of Fibonacci numbers. In the third chapter, we shall discuss basic things of cryptosystem including RSA Public-key system. Ronald Rivest, Adi Shamir, and Leonard Adleman developed the RSA sys-1 tem in 1977. RSA stands for the rst letter in each of its inventors' last names. Finally, we shall also discuss about new LUC cryptosystem which is based on Lucas functions. enables you to recognize a mistake when you make it again" By :FRANKLIN P. JONES.