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We introduce a simple tool that can be used to reduce non-injective instancesof the hidden shift problem over arbitrary group to injective instances overthe same group. In particular, we show that the average-case non-injectivehidden shift problem admit this reduction. We show similar results for(non-injective) hidden shift problem for bent functions. We generalize thenotion of influence and show how it relates to applicability of this tool fordoing reductions. In particular, these results can be used to simplify the mainresults by Gavinsky, Roetteler, and Roland about the hidden shift problem forthe Boolean-valued functions and bent functions, and also to generalize theirresults to non-Boolean domains (thereby answering an open question that theypose).

Reduction from non-injective hidden shift problem to injective hidden shift problem