On the mean value of the total length of chains computed during the additional checkings in the tradeoff method with distinguished points

Time-memory balancing methods are used to solving the problem of inverting a one-way function. Balancing with distinguished points and non-perfect tables is well-known variant of such methods. Recently Hong and Moon had presented an estimate of the mean total length of chains calculated during additional checks. Here we generalize this result. We describe a theoretical probabilistic model within which a two-sided estimates of the specified value are obtained. The results obtained are applicable under any conditions on the length of the main chain, under any rules for breaking an additional chain, and for one-way functions with atypical properties. Also our results allow to identify the set of values of the method parameters, where the resulting estimates remain fairly accurate.