Weakened random oracle models (WROMs) are variants of the random oracle model
(ROM). The WROMs have the random oracle and the additional oracle which breaks
some property of a hash function. Analyzing the security of cryptographic
schemes in WROMs, we can specify the property of a hash function on which the
security of cryptographic schemes depends. Liskov (SAC 2006) proposed WROMs and
later Numayama et al. (PKC 2008) formalized them as CT-ROM, SPT-ROM, and
FPT-ROM. In each model, there is the additional oracle to break collision
resistance, second preimage resistance, preimage resistance respectively. Tan
and Wong (ACISP 2012) proposed the generalized FPT-ROM (GFPT-ROM) which
intended to capture the chosen prefix collision attack suggested by Stevens et
al. (EUROCRYPT 2007). In this paper, in order to analyze the security of
cryptographic schemes more precisely, we formalize GFPT-ROM and propose
additional three WROMs which capture the chosen prefix collision attack and its
variants. In particular, we focus on signature schemes such as RSA-FDH, its
variants, and DSA, in order to understand essential roles of WROMs in their
security proofs