1. Private Graph Algorithms in the Semi-Honest Model
Justin Brickell
November 24, 2004

2. Two Party Graph Algorithms
Parties P1 and P2 own graphs G1 and G2
f is a two-input graph algorithm
Compute f(G1,G2) without revealing “unnecessary information”

3. Unnecessary Information
Intuitively, the protocol should function as if a trusted third party computed the output
We use simulation to prove that a protocol is private

4. The Semi-Honest Model A malicious adversary can alter his input
A semi-honest adversary
adheres to protocol
tries to learn extra information from the message transcript

5. General Secure Two Party Computation
Any polynomial sized functionality can be made private (in the semi-honest model)
Yao’s Method
What are our goals?
Yao’s Method is inefficient
Efficient, private protocols to compute particular graph functionalities
Take advantage of information “leaked” by the result

6. Two-Input Graph Algorithms?
Graph Isomorphism
Comparison of graph statistics
f(G1) > f(G2) ?
max flow,diameter,average degree
Synthesized Graphs
f(G1 • G2)

7. Graph Synthesis
G1 and G2 are weighted complete graphs on the same vertex and edge set
G1 = (V,E,w1); G2 = (V,E,w2)
gmax(G1,G2) = (V,E,wmax)
wmax(e) = max(w1(e),w2(e))
gmin(G1,G2) = (V,E,wmin)
wmin(e) = min(w1(e),w2(e))

8. Graph Synthesis 1  2 1  2 4 7 G1 5 6 8 gmax(G1,G2) 8 2 5 4 3 4 3 7 7
gmin(G1,G2) 6 5 2 4 3 4 3 5 5

9. Graph Isomorphism Unlikely to find a private protocol
No known poly-time algorithm

10. Comparison of Graph Statistics
Compute statistic on own graph
Semi-honest participants can’t lie
Use a private comparison protocol
Yao’s Millionaire Protocol
Yao’s method (circuit protocol)

11. Synthesized Graphs This is the interesting case
All Pairs Shortest Distance and Single Source Shortest Distance both “leak” significant useful information
Solved: APSD(gmin),SSSD(gmin)
Solved with leaks: APSD(gmax),SSSD(gmax)

12. APSD(gmin(G1,G2))
Basic Idea: Add edges to the solution graph in order of smallest to largest
Private, because we can recover the order from the final solution graph

13. Run APSD on G1 and G2 7 1  2 1 2 4 4 G1 5 5 8
G1’ 8 2 2 4 3 4 3 7 6 2 1 2 2 1 2 7 7 6 6 G2 6
G2’ 6 5 5 4 3 4 3 5 5

14. Initialize G0’ 7 2  1 2 1 2 1 2 4 7  5 6  8 6 2 5   4 3 4 3 4 3 6 5 
G1’
G2’
G0’

15. Find shortest blue edge lengths7 2  1 2 1 2 1 2 4 7  5 6  8 6 2 5   4 3 4 3 4 3 6 5 
G1’
G2’
G0’
min1 = 2
min2 = 2
min0 = 

16. Privately find global shortest length7 2  1 2 1 2 1 2 4 7  5 6  8 6 2 5   4 3 4 3 4 3 6 5 
G1’
G2’
G0’
min1 = 2
min2 = 2
min0 = 
bluemin = min(min0,min1,min2) =2

17. Find edges of length bluemin7 2  1 2 1 2 1 2 4 7  5 6  8 6 2 5   4 3 4 3 4 3 6 5 
G1’
G2’
G0’
S1 = {e23}
S2 = {e12}
S0 = {}
bluemin = min(min0,min1,min2) =2

18. Privately find all edges of length bluemin7 2  1 2 1 2 1 2 4 7  5 6  8 6 2 5   4 3 4 3 4 3 6 5 
G1’
G2’
G0’
S1 = {e23}
S2 = {e12}
S0 = {}
S = S1  S2  S3 = {e12,e23}

19. Update S edges in G0’ 7 2 2 1 2 1 2 1 2 4 7  5 6  8 6 2 5  2 4 3 4 3 4 3 6 5 
G1’
G2’
G0’
S1 = {e23}
S2 = {e12}
S0 = {}
S = S1  S2  S3 = {e12,e23}

20. Run APSD on G0’ 7 2 2 1 2 1 2 1 2 4 7  5 6 4 8 6 2 5  2 4 3 4 3 4 3 6 5 
G1’
G2’
G0’

21. Repeat! 7 2 2 1 2 1 2 1 2 4 7  5 6 4 8 6 2 5  2 4 3 4 3 4 3 6 5 
G1’
G2’
G0’
bluemin = min(min0,min1,min2) =4
S = S1  S2  S3 = {e13,e24}

22. … 7 2 2 1 2 1 2 1 2 4 7 4 5 6 4 8 6 6 2 5 2 4 3 4 3 4 3 6 5 6
G1’
G2’
G0’
bluemin = min(min0,min1,min2) =5
S = S1  S2  S3 = {e34}

23. … 7 2 2 1 2 1 2 1 2 4 7 4 5 6 4 8 6 6 2 5 2 4 3 4 3 4 3 6 5 5
G1’
G2’
G0’
bluemin = min(min0,min1,min2) =6
S = S1  S2  S3 = {e14}

24. … until all edges are red7 2 2 1 2 1 2 1 2 4 7 4 5 6 4 8 6 6 2 5 2 4 3 4 3 4 3 6 5 5
G1’
G2’
G0’

25. The solution is correct!2 2 1 2 1 2 4 4 5 4 6 6 2 2 4 3 4 3 5 5
gmin(G1,G2)
G0’

26. Other Results A similar protocol for SSSD(gmin)
This isn’t free!
Protocol for special case of APSD(gmax) and SSSD(gmax)
Input graphs obey triangle inequality
“Leaky” protocol for APSD(gmax) and SSSD(gmax) in the general case

27. Final Thought Other graph algorithms don’t leak enough information
Questions?