Computational aspects of rational residuosity

In this paper, we consider an extension of Jacobi’s symbol, the so-called rational [Formula: see text]th power residue symbol. In Sec. 3, we prove a novel generalization of Zolotarev’s lemma. In Secs. 4–6, we show that several hard computational problems are polynomial-time reducible to computing these residue symbols, such as getting nontrivial information about factors of semiprime numbers. We also derive criteria concerning the Quadratic Residuosity Problem.