Roots of Unity in ℤ/mℤ

Abstract

In this chapter we develop the information needed to prove Rabin’s theorem. We first count the number of nth roots of 1 or of −1 modulo m for any n and m. This allows us to count, for any odd composite number m, the number of false witnesses for m—that is, the number of numbers a modulo m such that m is a strong a-pseudoprime. These techniques yield a proof of Rabin’s theorem. We conclude the chapter with some observations about designing RSA codes related to strong a-pseudoprime testing.