In the permutation inversion problem, the task is to find the preimage of some challenge value, given oracle access to the permutation. This is a fundamental problem in query complexity, and appears in many contexts, particularly cryptography. In this work, we examine the setting in which the oracle allows for quantum queries to both the forward and the inverse direction of the permutation -- except that the challenge value cannot be submitted to the latter. Within that setting, we consider two options for the inversion algorithm: whether it can get quantum advice about the permutation, and whether it must produce the entire preimage (search) or only the first bit (decision). We prove several theorems connecting the hardness of the resulting variations of the inversion problem, and establish a number of lower bounds. Our results indicate that, perhaps surprisingly, the inversion problem does not become significantly easier when the adversary is granted oracle access to the inverse, provided it cannot query the challenge itself.

翻译：在置换求逆问题中，目标是在具有预言机访问置换权限的条件下，找到某个挑战值的前像。这是查询复杂度中的一个基本问题，出现在许多场景中，尤其是密码学领域。在本工作中，我们研究了一种设定：预言机允许对置换的正向和逆向方向进行量子查询——但挑战值不能提交给后者。在此设定下，我们考虑了求逆算法的两种选择：是否可以获得关于置换的量子建议，以及算法需要输出整个前像（搜索）还是仅输出第一位（判定）。我们证明了关于该求逆问题变体难度的若干定理，并建立了多个下界。我们的结果表明，或许令人惊讶的是，当敌手被授予对逆向方向的预言机访问权限（前提是其不能查询挑战值本身）时，求逆问题并未变得显著更容易。