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

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).

Quantum algorithms for abelian difference sets and applications to dihedral hidden subgroups

“…This leads to the question whether it is possible to relate instances of hidden shift problems where the hiding function f : A → S is not injective to the injective case. Thankfully, as shown in [17] such a connection indeed exists. We briefly review this construction.…”

“…The study of hidden shift problems has resulted in quantum algorithms that are of independent interest and have even inspired cryptographic schemes that might be candidates for post-quantum cryptography [39]. Besides the mentioned works, problems of hidden shift type were also studied in [42,16,17], in the rejection sampling [36] framework, and in the context of multiregister PGM algorithms for Boolean hidden shift problems [7]. The main result of this paper is Theorem 4 which asserts that there exist instances of the hidden subgroup problem over the dihedral groups D N that can be solved in O(log N ) queries to the hiding function, O(polylog(N )) quantum time, O(log N ) quantum space, and trivial classical post-processing.…”

“…, f (x + v m )). Gharibi showed in [17] that if the set V is chosen uniformly at random, then the probability that the function f V is not injective can be upper bounded as…”