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

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).

💡 Research Summary

The paper addresses a fundamental obstacle in quantum algorithms for the hidden shift problem: the difficulty of handling non‑injective functions. In the hidden shift setting, one is given two functions f and g defined on a finite group G such that g(x)=f(x·s) for an unknown shift s∈G, and the goal is to recover s. When f is injective, the shift is uniquely determined and standard quantum techniques—most notably Fourier sampling—yield efficient algorithms. However, if f is non‑injective, multiple inputs may map to the same output, causing collisions that invalidate the usual Fourier‑based approach and dramatically increase the query complexity.

The authors introduce a remarkably simple reduction that transforms any non‑injective instance into an injective one on the same group. The core construction is a “multiple‑shift transformation.” One selects a set T={t₁,…,t_k} of k independent, uniformly random group elements and defines a new vector‑valued function
 F(x) = (f(x·t₁), f(x·t₂), …, f(x·t_k)).
Similarly, G(x) = (g(x·t₁), …, g(x·t_k)). By construction, the hidden shift s still satisfies G(x)=F(x·s) for all x. The crucial observation is that, for a sufficiently large k, the function F becomes injective with high probability, even though each coordinate individually may be highly non‑injective.

To quantify “sufficiently large,” the paper generalizes the notion of influence from Boolean analysis to arbitrary value sets. For each group element e_i that generates a one‑step shift, the influence Inf_i(f) is defined as the probability (over a uniform x) that f(x)≠f(x·e_i). Let γ = min_i Inf_i(f) be the smallest influence. The authors prove a concentration bound: if k ≥ (2/γ)·ln(|G|/ε), then the probability that F fails to be injective is at most ε. The proof uses a union bound over all possible collisions and exploits the independence of the random shifts. Consequently, the required number of auxiliary shifts grows only logarithmically with the group size and inversely with the minimal influence.

The reduction works in the average‑case setting as well. When f and g are drawn uniformly at random from the set of all functions G→S, the minimal influence is typically a constant, so a modest k = O(log|G|) suffices. After the transformation, any quantum algorithm that solves the injective hidden shift problem (e.g., the standard O(√|G|) query algorithm) can be applied directly to F and G, yielding the same asymptotic performance for the original non‑injective instance.

A particularly important special case is that of bent functions—maximally nonlinear Boolean functions defined on {0,1}ⁿ. Bent functions have the property that every input bit has influence exactly ½. Plugging γ=½ into the bound shows that k = O(log|G|) is enough to guarantee injectivity of the transformed function. This observation allows the authors to recover and simplify the results of Gavinsky, Roetteler, and Roland (GRR) on hidden shifts of bent functions, and to extend those results to non‑Boolean domains, thereby answering an open question posed by GRR.

Beyond reproducing known results, the reduction provides a unifying framework. It eliminates the need for problem‑specific structural analyses that were previously required for non‑injective functions. By reducing to the injective case, any existing quantum algorithm for hidden shifts—whether based on Fourier sampling, amplitude amplification, or more recent techniques—can be leveraged unchanged. The authors also discuss potential extensions to related problems such as hidden subgroup or hidden coset problems, where a similar “vector‑valued lifting” could convert ambiguous instances into uniquely identifiable ones.

In summary, the paper contributes a powerful, conceptually simple tool: a randomized multi‑shift embedding that turns non‑injective hidden shift instances into injective ones with provably high probability. The analysis hinges on a generalized influence measure, yielding explicit bounds on the number of auxiliary shifts needed. This reduction not only streamlines the proofs of earlier works on Boolean and bent functions but also broadens the applicability of quantum hidden‑shift algorithms to arbitrary value sets and average‑case scenarios, thereby advancing our understanding of the quantum complexity landscape for shift‑based problems.