A computational technique for determining the fundamental unit in explicit types of real quadratic number fields

Abstract

In real quadratic number field Q (root d), integral basis element is denoted by w(d) = [a(0); a(1), a(2), ... , a(l)(d)(-1), a(l(d))] for the period length l(d). The fundamental unit epsilon(d) of real quadratic number field is also denoted by epsilon d = t(d)+u(d)root d/2 >1. The Unit Theorem for real quadratic fields says that every unit in the integer ring of a quadratic field is generated by the fundamental unit. Also, regulator in real quadratic cryptography is outstanding. We have seen that the regulator R = log epsilon(d) plays the role of a group order. The regulator problem is to find an integer R' satisfies vertical bar R' - R vertical bar < 1 where R' is an approximation of.. with any given precision can be computed in polynomial time for discriminant. However, some of the fundamental units can not be calculated by computer programme in short time because of the big numbers or long calculations of usual algorithm. This is also the main problem from the computing/informatics point of view. So, determining of the fundamental units is of great importance. In this paper, we construct a theorem to determine the some certain real quadratic fields Q(root d.) having specific form of continued fraction expansion of w(d) where d = 1(mod4) is a square-free integer. We also present the general context and obtain new certain parametric representation of fundamental unit epsilon(d) for such types of fields. By specialization, we get a fix on Yokoi's invariants and support all results with tables. (C) 2017 The Authors. Published by IASE.