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Let be a real quadratic order, let Δ be the discriminant of , and let be the regulator of .

In this chapter we describe probabilistic algorithms for computing class numbers and regulators. We also show how they can be extended to solve the equivalence problem. Those algorithms are much faster than the deterministic algorithms presented in Chapter 9. They use an approach dubbed which originated in work by Kraichik [Kra22] and seemingly independent work by Western and Miller [WM68]. The first proposals to apply this approach in the context of imaginary quadratic class groups came from Seysen [Sey87] and Hafner/McCurley [HM89]. In this chapter we let Δ be a fundamental discriminant. It is rather straightforward, however, to extend the algorithms of this chapter to the case of nonfundamental discriminants.

In this chapter, we will discuss several ways in which the theory of binary quadratic forms can be employed for cryptographic applications. Goals of cryptography encompass the maintenance of confidentiality, authenticity, integrity and non-reputability of electronic documents.