This document presents methods for computing information flow and quantifying information leakage in non-probabilistic programs using symbolic model checking. It discusses using binary decision diagrams (BDDs) and algebraic decision diagrams (ADDs) to represent program states and calculate fixed points. Algorithms are provided for symbolically computing min-entropy and Shannon entropy leakage by constructing ADDs representing the program summary and sets of possible outputs. The methods were implemented in a tool called Moped-QLeak and evaluated on example programs. Future work includes supporting recursive programs and using other symbolic verification approaches.

1. Computing Information Flow Using
Symbolic Model-Checking
10 avril 2018

2. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

3. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

4. Introduction
Information Leakage :
I Information about the secret inputs using publicly observable
outputs
I Outputs independent of inputs =⇒ No leakage
I Unique input corresponding to given output =⇒ Full leakage
The quantitative information flow bounding problem and the
quantitative information flow comparison problem for Boolean
programs are PSPACE-complete .

5. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

6. Literature review
I .[M. Backes , al] - S&P 2009 . DisQuant :
Automatic discovery and quantification of information leaks
I .[Heusser , al] - ACSAC 2010 . selfcomp :
Quantifying Information Leaks in Software
I .[Quoc-Sang Phan , al ] - ACM SIGSOFT 2012. jpf-qif :
Symbolic Quantitative Information Flow
I .[Quoc-Sang Phan , al ] - ASIACCS 2014. sqifc :
Abstract model counting : a novel approach for quantification
of information leaks.
I .[Rohit Chadha , al] - FSTTCS 2014. Moped-QLeak :
Computing Information Flow Using Symbolic Model-Checking

7. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

8. Symbolic Model-Checking
Model Checking :
I Exhaustive search of the state space of the system .
I Disadvantage : State Explosion Problem
Symbolic Model Checking :
I uses boolean encoding for state machine and sets of states.
I BDDs used to represent boolean functions .

9. Boolean programs
I Global variables G : Input and output
I Local variables : Internal calculations
I Program statements : transform global and local variables
I For Program P, Fp : 2G 7−→ 2G ∪ {⊥}
I Fp( ¯
g0) = ⊥ : P does not terminate ¯
g0

10. Binary decision diagrams (BDDs)
I Set of variables V = {x, y, z} with the order x < y < z
I f (x, y, z) = x → (y ↔ z)
I BDDs are data structures for storing elements : 2G −→ {0, 1}
Unreduced decision diagram (left) and corresponding BDD (right)

11. Algebraic Decision Diagrams (ADDs)
I Set of variables V
I ADD : 2G −→ M (M = R)
I BDDs + real values on the terminal nodes
I BDDs as 0/1-ADDs
ADD (right) and its reduced form (left)

12. Fixed-point
Consider the complete lattice (2G , ⊆)
Consider a function f : 2S 7−→ 2S
X ∈ 2S is fixed point of f ⇐⇒ f (X) = X
X is least fixed point of f if for every fixed point Y , X ⊆ Y
Theorem (Tarski-knaster)
Consider a function f : 2S 7−→ 2S that is monotonic
X ⊆ Y then f (X) ⊆ f (Y )
Consider the sequence :
X0 = ∅
X1 = f (X0)
X2 = f (X1)... This eventually converge to the least fixed point

13. Information leakage in programs
1. Min-entropy leakage measures vulnerability of the secret
inputs to being guessed correctly in a single attempt of the
adversary
MEU(P) = log
P
o∈O maxs∈S µ(S = s|O = o)
2. Shannon entropy leakage measures expected number of
guesses required to correctly guess the secret input
SEU(P) = log|S| − 1
|S|
P
o∈O |P−1(o)|log(|P−1(o)|)

14. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

15. Definition (Summary of a Program)
Let P be a program with G = {x1, ..., xn} as the set of global
variables. Let G0 = {x0
1, ..., x0
n} and G ∩ G0 = ∅.
The summary of P , denoted TP , is a function
TP : 2(G∪G0) → {0, 1}
such that for every z1, ..., zn, z0
1, ..., z0
n ∈ {true, false}, we have
TP(z1, ..., zn, z0
1, ..., z0
n) = 1 ⇐⇒ P(z1, ..., zn) = (z0
1, ..., z0
n).
Example : Boolean program Pex with global variables : s1, s2, o1
and o2.
o1 = false ; o2 = false ;
while s1 ;
o1 = false ; o2 = s2 ;
s1 = false ; s2 = false ;

16. Assuming the order s1 < s0
1 < s2 < s0
2 < o1 < o0
1 < o2 < o0
2 :
Pex is shown as a 0/1-ADD :

17. Leakage measured using min-entropy
Let post(2G ) = { ¯
g0 ∈ 2G |∃ḡ ∈ 2G .P(ḡ) = ¯
g0}.
If P terminates on all inputs then the min-entropy leakage : is
log(|post(2G )|) .
otherwise it is log(|post(2G )| + 1)
Lemma
Let Tout,P = orAbstract(G, TP) ∧ Tterm,P = orAbstract(G0, TP).
1. |post(2G )| = val(abstract(+, G0, Tout,P)).
2. P terminates on every input iff
isConst(Tterm,P) and val(Tterm,P) = 1.

18. Leakage measured using min-entropy
The ADD representing Tout,P : the set of all possible outputs of P
.

19. Algorithm : Symbolic computation of min-entropy leakage

20. Leakage measured using Shannon entropy
Two possible outputs =⇒ We need to compute :
P
¯
g0∈2G0 |P−1( ¯
g0)|log|P−1( ¯
g0)| + |P−1(⊥)|log|P−1(⊥)|
.
The ADD Teq−size,P : The ADD Tnon−term,P :

21. Algorithm :Symbolic computation of Shannon-entropy
leakage

22. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

24. Example : Sum Query
Input language : Remopla

25. Instructions for running

26. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

27. Examples used for evaluation

29. Introduction
Literature review
Background
Leakage in non-probabilistic programs
Demo
Experimental results
Conclusion and future work

30. Conclusion and future work
I Symbolic algorithms for measuring information leakage
I Integrable in any BDD based reachability analysis tool
I Summary calculation is the overhead - BDD size (algebraic
operations) and variable orderings
I future work :
I ProPed :Support recursive programs
I Other symbolic verification approaches : CEGAR
(Counterexample Guided Abstraction Refinement )

31. References
R. Chadha, U. Mathur, S. Schwoon
Computing Information Flow Using Symbolic Model Checking
Foundations of Software Technology and Theoretical
Computer Science (FSTTCS), 2014
.
Moped QLeak : https://sites.google.com/site/mopedqleak
Presented By : LOTFI LARBAOUI