Higher order eTCR hash functions

We study higher order eTCR (enhanced target collision resistance) hash functions, where rrth-order eTCR is denoted by eTCR(r)(r). We prove that a few rounds of the MD (Merkle–Damgård) scheme and a few levels of the TR (tree) scheme can be eTCR under the compression function is eTCR(r)(r) for some positive integer rr. Additionally, we prove that the TH (tree hash) scheme also preserves eTCR.