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People have been studying the following problem: Given a finite set S with ahidden (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*yof single pairs (x,y) you choose. What is the minimal number of queries to theoracle until the whole binary operation is recovered, i.e. you know x*y for allx,y in S?  This problem can trivially be solved by using |S|^2 queries to the oracle, sothe question arises under which circumstances you can succeed with asignificantly smaller number of queries.  In this presentation we give a lower bound on the number of queries neededfor general binary operations. On the other hand, we present algorithms solvingthis problem by using |S| queries, provided that * is an abelian groupoperation. We also investigate black box rings and give lower and upper boundsfor the number of queries needed to solve product recovering in this case.

Efficient recovering of operation tables of black box groups and rings