Preimage resistance of LPmkr with r=m−1

In this paper, we give a concrete analysis of preimage resistance for a wide class of linearly-dependent permutation-based compression functions. Specifically, we prove the preimage resistance of LPmkr with r=m−1 up to queries. As a special case, the preimage resistance of LP362 is proved up to query complexity, closing the gap between the lower bound (=24n/5) and the upper bound (=25n/6) presented in Rogaway and Steinberger (2008) [9].