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

TL;DR

This paper presents a reduction technique that transforms non-injective hidden shift problems into injective ones over the same group, simplifying analysis and extending previous results to broader function classes.

Contribution

It introduces a general reduction method for non-injective hidden shift problems, applicable to arbitrary groups and functions like bent functions, and relates influence to reduction applicability.

Findings

Reduction applies to average-case non-injective hidden shift problems

Extends results to non-Boolean domains and bent functions

Simplifies and generalizes previous hidden shift problem results

Abstract

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