Efficient MILP Modelings for Sboxes and Linear Layers of SPN ciphers

DenseQMC: an efficient bit-slice implementation of the Quine-McCluskey algorithm

“…For a rough comparison, the work [10] reports 34 seconds for a function with n = 11, [5] reports 5 seconds for a very sparse function with n = 15, all of which are done instantly by any of our implementations; [14] gives mixed CPU/GPU timings such as 1000 seconds and 10 GiB RAM for the dense case of n = 20, 10 minutes for an n = 24-bit function of density 70%, 2000 seconds for an n = 28-bit function of density 30%, close to 10 6 seconds on an n = 32-bit function of density 42% using disk storage (due to ambiguous reports and absence of available implementation, it is difficult to provide a clear comparison). Note that the dense case is often occurring in practice when optimizing a CNF formula of a sparse function, for example, [2] report 2 hours of work for the case of n = 16 and 82%-dense function, appearing in cryptographic applications. Note that this work does not intend to compete with sparse or approximate methods such as ESPRESSO-Exact [3] or more recent "Consistency Cubes" method [4].…”