Computational aspects of rational residuosity

TL;DR

This paper explores the computational complexity of rational residuosity, extending Jacobi's symbol, and demonstrates how various hard problems relate to computing these symbols, impacting factoring and quadratic residuosity.

Contribution

It introduces a generalized rational $2^k$-th power residue symbol and proves a new version of Zolotarev's lemma, linking these symbols to complex computational problems.

Findings

Several hard problems are polynomial-time reducible to computing rational residue symbols

Derived new criteria related to the Quadratic Residuosity Problem

Extended Jacobi's symbol to rational $2^k$-th power residues

Abstract

In this paper, we consider an extension of Jacobi's symbol, the so called rational $2^k$-th power residue symbol. In Section 3, we prove a novel generalization of Zolotarev's lemma. In Sections 4, 5 and 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.