In this paper we describe symbolic side-channel analysis techniques for detecting and quantifying information leakage, given in terms of Shannon and min-entropy. Measuring the precise leakage is challenging due to the randomness and noise often present in program executions and side-channel observations. We account for this noise by introducing additional (symbolic) program inputs which are interpreted probabilistically, using symbolic execution with parametrized model counting. We also explore a sampling approach for increased scalability. In contrast to typical Monte Carlo techniques, our approach works by sampling symbolic paths, representing multiple concrete paths, and uses pruning to accelerate computation and guarantee convergence to the optimal results. A key novelty of our approach is to provide bounds on the leakage that are provably under- and over-approximating the exact leakage. We implemented the techniques in the Symbolic PathFinder tool and demonstrate them on Java programs.

We present symbolic analysis techniques for detecting vulnerabilities that are due to adaptive side-channel attacks, and synthesizing inputs that exploit the identified vulnerabilities. We start with a symbolic attack model that encodes succinctly all the side-channel attacks that an adversary can make. Using symbolic execution over this model, we generate a set of mathematical constraints, where each constraint characterizes the set of secret values that lead to the same sequence of side-channel measurements. We then compute the optimal attack, i.e, the attack that yields maximum leakage over the secret, by solving an optimization problem over the computed constraints. We use information-theoretic concepts such as channel capacity and Shannon entropy to quantify the leakage over multiple runs in the attack, where the measurements over the side channels form the observations that an adversary can use to try to infer the secret. We also propose greedy heuristics that generate the attack by exploring a portion of the symbolic attack model in each step. We implemented the techniques in Symbolic PathFinder and applied them to Java programs encoding web services, string manipulations and cryptographic functions, demonstrating how to synthesize optimal side-channel attacks.

Side-channel attacks recover confidential information from non-functional characteristics of computations, such as time or memory consumption. We describe a program analysis that uses symbolic execution to quantify the information that is leaked to an attacker who makes multiple side-channel measurements. The analysis also synthesizes the concrete public inputs (the "attack") that lead to maximum leakage, via a novel reduction to Max-SMT solving over the constraints collected with symbolic execution. Furthermore model counting and information-theoretic metrics are used to compute an attacker's remaining uncertainty about a secret after a certain number of side-channel measurements are made. We have implemented the analysis in the Symbolic PathFinder tool and applied it in the context of password checking and cryptographic functions, showing how to obtain tight bounds on information leakage under a small number of attack steps.

We present an automated approach for detecting and quantifying side channels in Java programs, which uses symbolic execution, string analysis and model counting to compute information leakage for a single run of a program. We further extend this approach to compute information leakage for multiple runs for a type of side channels called segmented oracles, where the attacker is able to explore each segment of a secret (for example each character of a password) independently. We present an efficient technique for segmented oracles that computes information leakage for multiple runs using only the path constraints generated from a single run symbolic execution. Our implementation uses the symbolic execution tool Symbolic PathFinder (SPF), SMT solver Z3, and two model counting constraint solvers LattE and ABC. Although LattE has been used before for analyzing numeric constraints, in this paper, we present an approach for using LattE for analyzing string constraints. We also extend the string constraint solver ABC for analysis of both numeric and string constraints, and we integrate ABC in SPF, enabling quantitative symbolic string analysis.

Satisfiability Modulo Theories (SMT) is a decision problem for logical formulas over one or more first-order theories. In this paper, we study the problem of finding all solutions of an SMT problem with respect to a set of Boolean variables, henceforth All-SMT. First, we show how an All-SMT solver can benefit various domains of application: Bounded Model Checking, Automated Test Generation, Reliability analysis, and Quantitative Information Flow. Secondly, we then propose algorithms to design an All-SMT solver on top of an existing SMT solver, and implement it into a prototype tool, called aZ3. Thirdly, we create a set of benchmarks for All-SMT in the theory of linear integer arithmetic QF_LIA and the theory of bit vectors with arrays and uninterpreted functions QF_AUFBV. We compare aZ3 against Math SAT, the only existing All-SMT solver, on our benchmarks. Experimental results show that aZ3 is more precise than Math SAT.

We present a novel method for Quantitative Information Flow analysis. We show how the problem of computing information leakage can be viewed as an extension of the Satisfiability Modulo Theories (SMT) problem. This view enables us to develop a framework for QIF analysis based on the framework DPLL(T) used in SMT solvers. We then show that the methodology of Symbolic Execution (SE) also fits our framework. Based on these ideas, we build two QIF analysis tools: the first one employs CBMC, a bounded model checker for ANSI C, and the second one is built on top of Symbolic PathFinder, a Symbolic Executor for Java. We use these tools to quantify leaks in industrial code such as C programs from the Linux kernel, a Java tax program from the European project HATS, and anonymity protocols.

We report on our work-in-progress into the use of reliability analysis to quantify information leaks. In recent work we have proposed a software reliability analysis technique that uses symbolic execution and model counting to quantify the probability of reaching designated program states, e.g. assert violations, under uncertainty conditions in the environment. The technique has many applications beyond reliability analysis, ranging from program understanding and debugging to analysis of cyber-physical systems. In this paper we report on a novel application of the technique, namely Quantitative Information Flow analysis (QIF). The goal of QIF is to measure information leakage of a program by using information-theoretic metrics such as Shannon entropy or Renyi entropy. We exploit the model counting engine of the reliability analyzer over symbolic program paths, to compute an upper bound of the maximum leakage over all possible distributions of the confidential data.