Gauss Ring, Quaternions And RSA Arithmetic

With the development of internet and electronic commerce, secure productionhas become more and more important. Encryption is the most useful method thatensures the security of electronic commerce, and is widely used in Cryptography. Onexchanging a message, the sender encodes the data, and then sends it to the receiver,the receiver decodes the data, and he gets the original message. Encryption includestwo elements: arithmetic and key. The arithmetic is used to combine plain-text whithsome numbers (key) to create cryptograph that no one will understand. The key isused to encode and decode message. The arithmetic is the kernel of Encryption. Agood arithmetic will speed up the development of Encryption. We can useEncryption to the safety of internet communication. In recent years, the burst ofcomputer hardware has a great impact on cryptograph. So we should be devoted tothe researching of Encryption.In this paper, We discussed the RSA arithmetic, Gauss integer ring andquaternions, especially we discussed the multiplicative groups of Gauss quotientring and the group of congruence classes of integral quaternions modulo n, andapplied it into RSA arithmetic, which expanded the proclaimed space of RSAarithmetic greatly and enhanced the security of RSA arithmetic greatly.