Nonadaptive One-Way to Hiding Implies Adaptive Quantum Reprogramming

TL;DR

The paper shows that adaptive quantum reprogramming results can be derived from the nonadaptive AHU one-way-to-hiding theorem by employing a Fixed-Permutation O2H framework. It develops a distinguisher-based technique that interleaves the attacker and the security game using fixed permutations, together with sparse-function representations of quantum random oracles, to obtain tight bounds for adaptive reprogramming scenarios. The authors then demonstrate that the Pan-Zeng adaptive reprogramming framework, the GHHM tight adaptive reprogramming theorem, and Unruh's adaptive O2H lemmas all follow from this Fixed-Permutation O2H approach, often with improved square-root type bounds. This unifies several prior adaptive analyses under a single, largely classical reasoning paradigm while highlighting certain limitations of current sparse-permutation methods for random-permutation reprogramming.

Abstract

An important proof technique in the random oracle model involves reprogramming it on hard to predict inputs and arguing that an attacker cannot detect that this occurred. In the quantum setting, a particularly challenging version of this considers adaptive reprogramming wherein the points to be reprogrammed (or the output values they should be programmed to) are dependent on choices made by the adversary. Some quantum frameworks for analyzing adaptive reprogramming were given by Unruh (CRYPTO 2014, EUROCRYPT 2015), Grilo-Hövelmanns-Hülsing-Majenz (ASIACRYPT 2021), and Pan-Zeng (PKC 2024). We show, counterintuitively, that these adaptive results follow from the \emph{nonadaptive} one-way to hiding theorem of Ambainis-Hamburg-Unruh (CRYPTO 2019). These implications contradict beliefs (whether stated explicitly or implicitly) that some properties of the adaptive frameworks cannot be provided by the Ambainis-Hamburg-Unruh result.