\textit{D. Doichinov} [C. R. Acad. Bulg. Sci. 41, No. 7, 5-8 (1988; Zbl 0649.54015); Topology Appl. 38, No. 3, 205-217 (1991; Zbl 0723.54030)] developed a completeness theory for the (introduced by him) class of quiet quasi-uniform spaces. This theory is a natural extension of the completeness theory of uniform spaces. In 1993 \textit{H.-P. A. Künzi} [Bolyai Soc. Math. Stud. 4, 303-338 (1995; Zbl 0888.54029), Problem 4], asked whether there exist natural applications of this theory to topological algebra. The paper under review shows that this is indeed the case. The obtained results generalize some important results about topological groups to regular paratopological groups. It is shown that the two-sided quasi-uniformity of a regular \(T_0\)-paratopological group is quiet and that its Doichinov completion yields a paratopological group whenever the product of any two Cauchy filter pairs is a Cauchy filter pair. The latter condition holds in any Abelian paratopological group. It is shown by an example that this condition is not satisfied in paratopological groups in general. Furthermore, some conditions, depending on quasi-uniform completeness properties, under which a paratopological group is a topological group, are obtained.