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arXiv · 0805.0514

Efficient recovering of operation tables of black box groups and rings

Jens Zumbragel·Gerard Maze·Joachim Rosenthal
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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.

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Jens Zumbragel, Gerard Maze, Joachim Rosenthal. 2008-05-05. Efficient recovering of operation tables of black box groups and rings. https://arxiv.org/abs/0805.0514

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