  Hybrid testing approaches that involve fuzz testing and symbolic execution
have shown promising results in achieving high code coverage, uncovering subtle
errors and vulnerabilities in a variety of software applications. In this paper
we describe Badger - a new hybrid approach for complexity analysis, with the
goal of discovering vulnerabilities which occur when the worst-case time or
space complexity of an application is significantly higher than the average
case. Badger uses fuzz testing to generate a diverse set of inputs that aim to
increase not only coverage but also a resource-related cost associated with
each path. Since fuzzing may fail to execute deep program paths due to its
limited knowledge about the conditions that influence these paths, we
complement the analysis with a symbolic execution, which is also customized to
search for paths that increase the resource-related cost. Symbolic execution is
particularly good at generating inputs that satisfy various program conditions
but by itself suffers from path explosion. Therefore, Badger uses fuzzing and
symbolic execution in tandem, to leverage their benefits and overcome their
weaknesses. We implemented our approach for the analysis of Java programs,
based on Kelinci and Symbolic PathFinder. We evaluated Badger on Java
applications, showing that our approach is significantly faster in generating
worst-case executions compared to fuzzing or symbolic execution on their own.
