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# Short Transitive Signatures For Directed Trees

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### Short Transitive Signatures For Directed Trees

1. 1. San Francisco, Feb 27-Mar 2 2012Short Transitive Signatures for Directed TreesPhilippe Camacho and Alejandro Hevia University of Chile
2. 2. How do we sign a graph? a b
3. 3. Trivial solutions
4. 4. Transitive signature schemes [MR02,BN05,SMJ05] Combiner
5. 5. Landscape
6. 6. Transitive Signatures for Directed Trees
7. 7. Previous Work
8. 8. Our ResultsExamplesTime tosignedge /verifypathsignatureTime to compute apathsignatureSize of pathsignature
9. 9. Security [MR02]σBC E
10. 10. BASIC CONSTRUCTION
11. 11. Pre/Post Order Tree Traversal a b h d i j k c e f g Pre order: a b c d e f g h i j k Post order: c e f g d b i j k h a
12. 12. Property of Pre/Post order Traversal• Proposition [Dietz82] a b h d c i j k e f g Pre order: a b c d e f g h i j k Post order: c e f g d b i j k h a
13. 13. Idea a b h d c i j k e f g How do we avoidPosition 1 2 3 4 5 6 7 8 9 10 11 recomputing a lot ofPre a b c d e f g h h i i j j k k signatures when an element is inserted?Post c e f g d d b b i i j j k k h h a a
14. 14. Order Data Structure
15. 15. Trivial Order Data Structure A Toy ExampleElements -∞ d a e b c +∞ 0 250 1000 500 675 750 875Labels
16. 16. a b dc Pre a b c d +∞-∞ 0 500 750 875 937 1000 Post c d b a +∞-∞ 0 125 187 250 500 1000
17. 17. Trivial Order Data Structure-∞ d a b c +∞0 250 500 750 875 1000 New CRHF! It allows to: • compress the strings • efficiently compare them from their hashes
18. 18. HASHING WITH COMMON PREFIXPROOFS
19. 19. The Idea
20. 20. Security
21. 21. n-BDHI assumption [BB04]
22. 22. The hash function
23. 23. Generating & Verifying Proofs
24. 24. Generating & Verifying Proofs
25. 25. Security• Proposition: If the n-BDHI assumption holds then the previous construction is a secure HCPP family.• Proof (idea)
26. 26. CRHF is incremental
27. 27. Comparing strings
28. 28. FULL CONSTRUCTION
29. 29. Trivial Order Data Structure d a b c0 250 500 750 875 1000
30. 30. New Order Data Structure d e a b c Use a binary tree to obtain a 1an «incremental» order data structure d 0 b 1 1 e c
31. 31. a b dc ODPre ODPost a a 1 0 b b 1 0 c c 1 1 d d