The collision-resistant hash function is an early cryptographic primitive that finds extensive use in various applications. Remarkably, the Merkle-Damgård and Merkle tree hash structures possess the collision-resistance preserving property, meaning the hash function remains collision-resistant when the underlying compression function is collision-resistant. This raises the intriguing question of whether reducing the number of underlying compression function calls with the collision-resistance preserving property is possible. In pursuit of addressing these inquiries, we prove that for an ${\ell}n$-to-$sn$-bit collision-resistance preserving hash function designed using $r$ $tn$-to-$n$-bit compression function calls, we must have $r \geq \lceil (\ell-s)/(t-1) \rceil $. Throughout the paper, all operations other than the compression function are assumed to be linear (which we call linear hash mode).

We introduce InspectorGadget, an Open-Source Python-based software for assessing and comparing the complexity of masking gadgets. By providing a limited set of characteristics of a hardware platform, our tool allows to estimate the cost of a masking gadget in terms of cycle count equivalent and memory footprint. InspectorGadget is highly flexible. It enables the user to define her own estimation functions, as well as to expand the set of gadgets and predefined microcontrollers. As a case-study, we produce a fair comparison of several masked versions of Kyber compression function from the literature, together with novel alternatives automatically generated by our tool. Our results confirm that an interesting middle ground exists between theoretical performance measures (asymptotic complexity or operations count) and real implementations benchmarks (clock cycle accurate evaluations). InspectorGadget offers both simplicity and genericity while capturing the main performance-related parameters of a hardware platform.