  Attack trees (ATs) are an important tool in security analysis, and an
important part of AT analysis is computing metrics. However, metric computation
is NP-complete in general. In this paper, we showcase the use of mixed integer
linear programming (MILP) as a tool for quantitative analysis. Specifically, we
use MILP to solve the open problem of calculating the min time metric of
dynamic ATs, i.e., the minimal time to attack a system. We also present two
other tools to further improve our MILP method: First, we show how the
computation can be sped up by identifying the modules of an AT, i.e. subtrees
connected to the rest of the AT via only one node. Second, we define a general
semantics for dynamic ATs that significantly relaxes the restrictions on attack
trees compared to earlier work, allowing us to apply our methods to a wide
variety of ATs. Experiments on a synthetic testing set of large ATs verify that
both the integer linear programming approach and modular analysis considerably
decrease the computation time of attack time analysis.
