Orders and Euler’s Theorem

Abstract

This chapter concludes the theory needed to describe the cryptography in Chap. 9. The key concept is that of the order of a unit b modulo m, and Euler’s Theorem, which places a constraint on the possible values of the order of b. When m is prime, Euler’s Theorem is the same as Fermat’s Theorem, which is given a proof using the Binomial Theorem. The final section describes an efficient algorithm for computing a high power of a number modulo m. This algorithm will have both an obvious use in using the cryptosystems presented in Chaps. 9 and 13 and a less obvious use to help construct cryptosystems in the last section of Chap. 9.