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Private Graph Algorithms in the Semi-Honest Model

Private Graph Algorithms in the Semi-Honest Model. Justin Brickell November 24, 2004. Two Party Graph Algorithms. Parties P 1 and P 2 own graphs G 1 and G 2 f is a two-input graph algorithm Compute f ( G 1 , G 2 ) without revealing “unnecessary information”.

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Private Graph Algorithms in the Semi-Honest Model

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  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 7 4 G1 6 5 8 gmax(G1,G2) 8 5 2 4 3 4 3 7 7 2 2 1 2 1 2 7 4 6 5 6 6 gmin(G1,G2) G2 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 6 7 2 2 1 2 1 2 7 7 6 6 6 6 G2’ G2 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 lengths 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’ min1 = 2 min2 = 2 min0 = 

  16. Privately find global shortest length 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’ min1 = 2 min2 = 2 min0 =  bluemin = min(min0,min1,min2) =2

  17. Find edges of length bluemin 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’ S1 = {e23} S2 = {e12} S0 = {} bluemin = min(min0,min1,min2) =2

  18. Privately find all edges of length bluemin 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’ 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 4 7 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 4 7 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 red 7 2 2 1 2 1 2 1 2 4 4 7 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 4 5 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?

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