Efficient recovering of operation tables of black box groups and rings

TL;DR

This paper investigates the minimal number of oracle queries needed to recover the entire operation table of black box groups and rings, providing bounds and algorithms especially for abelian groups.

Contribution

It establishes lower bounds for general binary operations and presents efficient algorithms for recovering operation tables in abelian groups and rings.

Findings

Lower bound on queries for general binary operations

Algorithm using |S| queries for abelian groups

Bounds for black box rings operation recovery

Abstract

People have been studying the following problem: Given a finite set S with a hidden (black box) binary operation * on S which might come from a group law, and suppose you have access to an oracle that you can ask for the operation x*y of single pairs (x,y) you choose. What is the minimal number of queries to the oracle until the whole binary operation is recovered, i.e. you know x*y for all x,y in S? This problem can trivially be solved by using |S|^2 queries to the oracle, so the question arises under which circumstances you can succeed with a significantly smaller number of queries. In this presentation we give a lower bound on the number of queries needed for general binary operations. On the other hand, we present algorithms solving this problem by using |S| queries, provided that * is an abelian group operation. We also investigate black box rings and give lower and upper bounds for the number of queries needed to solve product recovering in this case.