  In the design of software and cyber-physical systems, security is often
perceived as a qualitative need, but can only be attained quantitatively.
Especially when distributed components are involved, it is hard to predict and
confront all possible attacks. A main challenge in the development of complex
systems is therefore to discover attacks, quantify them to comprehend their
likelihood, and communicate them to non-experts for facilitating the decision
process. To address this three-sided challenge we propose a protection analysis
over the Quality Calculus that (i) computes all the sets of data required by an
attacker to reach a given location in a system, (ii) determines the cheapest
set of such attacks for a given notion of cost, and (iii) derives an attack
tree that displays the attacks graphically. The protection analysis is first
developed in a qualitative setting, and then extended to quantitative settings
following an approach applicable to a great many contexts. The quantitative
formulation is implemented as an optimisation problem encoded into
Satisfiability Modulo Theories, allowing us to deal with complex cost
structures. The usefulness of the framework is demonstrated on a national-scale
authentication system, studied through a Java implementation of the framework.
