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A hash-and-sign signature based on preimage-sampleable function (PSF) (Gentry et al. [STOC 2008]) is secure in the Quantum Random Oracle Model (QROM) if the PSF is collision-resistant (Boneh et al. [ASIACRYPT 2011]) or one-way (Zhandry [CRYPTO 2012]). However, trapdoor functions (TDFs) in code-based and multivariate-quadratic-based (MQ-based) signatures are not PSFs; for example, underlying TDFs of the Courtois-Finiasz-Sendrier (CFS), Unbalanced Oil and Vinegar (UOV), and Hidden Field Equations (HFE) signatures are not surjection. Thus, such signature schemes adopt probabilistic hash-and-sign with retry. This paradigm is secure in the (classical) Random Oracle Model (ROM), assuming that the underlying TDF is non-invertible; that is, it is hard to find a preimage of a given random value in the range (e.g., Sakumoto et al. [PQCRYPTO 2011] for the modified UOV/HFE signatures). Unfortunately, there is no known security proof for the probabilistic hash-and-sign with retry in the QROM.
We give the first security proof for the probabilistic hash-and-sign with retry in the QROM, assuming that the underlying non-PSF TDF is non-invertible. Our reduction from the non-invertibility is tighter than the existing ones that apply to only signature schemes based on PSFs. We apply the security proof to code-based and MQ-based signatures. Moreover, we extend the proof into the multi-key setting by using prefix hashing (Duman et al.[ACM CCS 2021]).