The Rsa Algorithm, Prime Numbers To Determine The Problem

With the rapid development of Internet, resource-sharing is widely used in the fields such as politics, military, economy, electronic commerce and etc. A great deal of data is saving and transmitting on Internet. However, the data may be interrupted, intercepted, captured, tampered with and counterfeited during the process of saving, using and transmitting in network. So, safeguarding measures should be adopted to protect data during the transmission process in network.This paper is analyzing the demand of the safe RSA public-key Cryptosystem to prime number and the relevant algorithm of the current judging prime number. In the foundation in its a various and each aspect for discussing Primality, we mainly study some properties of the generalized Carmichael numbers proposed by Bhattacharjee and Pandey, and the polynomial time testing algorithm suggested by Kayal and Saxena, which guarantees that any composite n which manages to pass the test is square-free. The study of the above two problems led Agrawal, Kayal and Saxena to the final solution of a worldwide problem for testing whether a number is prime or not in deterministic polynomial time on August 2002. The testing is closely related to RSA algorithm, therefore, the study of the above two problems is a great help to perfect and deepen RSA algorithm more closely.The article, by studying the properties of the generalized Carmichael numbers, has obtained the necessary and sufficient conditions for a composite number n to be a k-th Carmichael number, for a composite number n to be a 2-th or a 3-th Carmichael number, and for a Carmichael number (the usual Carmichael number) to be a 2-th Carmichael number. Then, we design a testing algorithm, and by using the testing algorithm and the computation, we prove that the set of Carmichael numbers and the set of 2-th Carmichael numbers do not contain each other, which answer an open unresolved question suggested by Bhattacharjee and Pandey.This article has also designed a fast polynomial time testing algorithm, which is based on Fermat little theorem, to decide whether a given integeris square-free or not. The testing algorithm given here is better than that of Kayal and Sanexa by analyzing their computational time.