Black-box separations on fiat-shamir-type signatures in the non-programmable random oracle model

In recent years, Fischlin and Fleischhacker showed the impossibility of proving the security of specific types of FS-type signatures, the signatures constructed by the Fiat-Shamir transformation, via a single-instance reduction in the non-programmable random oracle model (NPROM, for short). In this paper, we pose a question whether or not the impossibility of proving the security of any FS-type signature can be shown in the NPROM. For this question, we show that each FS-type signature cannot be proven to be secure via a key-preserving reduction in the NPROM from the security against the impersonation of the underlying identification scheme under the passive attack, as long as the identification scheme is secure against the impersonation under the active attack. We also show the security incompatibility between the discrete logarithm assumption and the security of the Schnorr signature via a singleinstance key-preserving reduction, whereas Fischlin and Fleischhacker showed that such an incompatibility cannot be proven via a non-keypreserving reduction.