Public-Key Cryptography and Pepin’s Test for the Primality of Fermat Numbers

Abstract

In this article, we have proved the correctness of the PublicKey Cryptography and the Pepin’s Test for the Primality of Fermat Numbers (F (n) = 2 n + 1). It is a very important result in the IDEA Cryptography that F(4) is a prime number. At first, we prepared some useful theorems. Then, we proved the correctness of the Public-Key Cryptography. Next, we defined the Order’s function and proved some properties. This function is very important in the proof of the Pepin’s Test. Next, we proved some theorems about the Fermat Number. And finally, we proved the Pepin’s Test using some properties of the Order’s Function. And using the obtained result we have proved that F(1), F(2), F(3) and F(4) are prime number.