Algorithm Designs and Complexity Analyses for Information Security

Modular exponentiation is composed of repetition of modular multiplication and is often the dominant part of modern cryptographic method. Taking the RSA cryptosystem for example, both the encryption and decryption operations are accomplished by modular exponentiation. It is performed by using successive modular multiplications. Exponentiation is to compute M^E for a positive integer E and modular exponentiation is to compute M^E mod N for positive integers E and N. When the lengths of the operators are at least 512 up to 2048 binary representations, modular exponentiation can be time-consuming and is often the dominant part of the computation in many computer systems. In this paper, I will describe some methods, which use some software. Most importantly, I will analyze the computational complexities for these methods respectively.