Compression function iterated via many kinds of extension transforms plays an important role in the construction of the hash function, and the analysis on its security properties seems to have significant meaning. Some variants of second preimage resistance (SPR) [1] assumptions of the keyless compression function have been put forth by Halevi and Krawczyk [2], in order to obtain higher security properties for the hash function. In this paper, we compare one variant, called r-SPR, with  which has been raised by Mironov [3]. We give several different schemes of the dedicated-key compression functions firstly, in which the  property implies r-SPR property in common. In the second part of this paper, we prove that this specific relation does not only hold in these four schemes, but in any reversible-key-transform whose inverse transform is target collision resistance used dedicated-key compression functions.

r-SPR,  compression function, hash function.