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Adaptive Oblivious Transfer And Generalization Olivier Blazy, C eline Chevalier, Paul Germouty December 5, 2016 O. Blazy, C. Chevalier, P . Germouty December 5, 2016 1 / 31 Oblivious Transfer 1 OLBE: A Natural Generalization 2 Adaptive

  1. Adaptive Oblivious Transfer And Generalization Olivier Blazy, C´ eline Chevalier, Paul Germouty December 5, 2016 O. Blazy, C. Chevalier, P . Germouty December 5, 2016 1 / 31

  2. Oblivious Transfer 1 OLBE: A Natural Generalization 2 Adaptive Oblivious Transfer 3 What To Remember 4 O. Blazy, C. Chevalier, P . Germouty December 5, 2016 2 / 31

  3. Oblivious Transfer 1 OLBE: A Natural Generalization 2 Adaptive Oblivious Transfer 3 What To Remember 4 O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 3 / 31

  4. Oblivious Transfer O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 4 / 31

  5. Oblivious Transfer Server DB 1 DB 2 ... DB n O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 4 / 31

  6. Oblivious Transfer Recipient Server Request( i ) DB 1 DB 2 ... DB n O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 4 / 31

  7. Oblivious Transfer Recipient Server Request( i ) DB 1 DB i DB 2 ... DB n O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 4 / 31

  8. Oblivious Transfer Recipient Server Request( i ) DB 1 DB i DB 2 ... DB n Privacy: S shouldn’t know i and R shouldn’t have any information about other lines. O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 4 / 31

  9. Identity Based Encryption O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 5 / 31

  10. Identity Based Encryption Alice O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 5 / 31

  11. Identity Based Encryption Bob Alice O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 5 / 31

  12. Identity Based Encryption mpk , Bob, m → C C Bob Alice O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 5 / 31

  13. Identity Based Encryption mpk , Bob, m → C C Bob Alice usk [ Bob ] , C → m O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 5 / 31

  14. Identity Based Key Encapsulation Mechanism Gen ( param ) : generates ( mpk , msk ) USKGen ( msk , id ) : computes usk [ id ] Enc ( mpk , id ) : encrypts a key K into C Dec ( usk [ id ] , C ) : decrypts C into K O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 6 / 31

  15. UC-framework and Security Model O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 7 / 31

  16. UC-framework and Security Model Ideal functionality vs real world O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 7 / 31

  17. UC-framework and Security Model Ideal functionality vs real world Adaptive corruptions: the adversary can ask for internal state of the recipient at any moment and then play his role. O. Blazy, C. Chevalier, P . Germouty Oblivious Transfer December 5, 2016 7 / 31

  18. Oblivious Transfer 1 OLBE: A Natural Generalization 2 Adaptive Oblivious Transfer 3 What To Remember 4 O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 8 / 31

  19. The Oblivious Signature Based Envelope Protocol Server Info O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 9 / 31

  20. The Oblivious Signature Based Envelope Protocol Recipient Server C =Commit( σ ; ρ ) Info O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 9 / 31

  21. The Oblivious Signature Based Envelope Protocol Recipient Server C =Commit( σ ; ρ ) ⊕ Mask Info Info ⊕ Mask O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 9 / 31

  22. The Oblivious Signature Based Envelope Protocol Recipient Server C =Commit( σ ; ρ ) ⊕ Mask Info Info ⊕ Mask ⊕ Mask Info O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 9 / 31

  23. The Oblivious Signature Based Envelope Protocol Recipient Server C =Commit( σ ; ρ ) ⊕ Mask Info Info ⊕ Mask ⊕ Mask Info Mask computable for the user if and only if C is a commitment of σ O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 9 / 31

  24. How To Do So: Commitment O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 10 / 31

  25. How To Do So: Commitment Commitment: Setup KeyGen Commit Decommit O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 10 / 31

  26. How To Do So: Commitment Commitment: Properties: Setup Extractable KeyGen Equivocable Commit Decommit O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 10 / 31

  27. How To Do So: Commitment Commitment: Properties: Setup Extractable KeyGen Equivocable Commit Decommit Example: Encryption, Chameleon Hash Function: ( KeyGen , CH , Coll ) O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 10 / 31

  28. How To Do So: Commitment Commitment: Properties: Setup Extractable KeyGen Equivocable Commit Decommit Example: Encryption, Chameleon Hash Function: ( KeyGen , CH , Coll ) If CH ( ck , m ; r ) = H then coll ( ck , tk , H ; m ′ ) = r ′ s. t. CH ( ck , m ′ ; r ′ ) = H O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 10 / 31

  29. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  30. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X HashKG ( L , param ) → hk O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  31. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X HashKG ( L , param ) → hk ProjKG ( hk , ( L , param ) , W ) → hp O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  32. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X HashKG ( L , param ) → hk ProjKG ( hk , ( L , param ) , W ) → hp Hash ( hk , ( L , param ) , W ) → H ∈ G O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  33. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X HashKG ( L , param ) → hk ProjKG ( hk , ( L , param ) , W ) → hp Hash ( hk , ( L , param ) , W ) → H ∈ G ProjHash ( hp , ( L , param ) , W, w ) → H ′ ∈ G O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  34. How To Do So: Smooth Projective Hash Function Functions over a set X and L ⊂ X HashKG ( L , param ) → hk ProjKG ( hk , ( L , param ) , W ) → hp Hash ( hk , ( L , param ) , W ) → H ∈ G ProjHash ( hp , ( L , param ) , W, w ) → H ′ ∈ G Hash ( hk , ( L , param ) , W ) = ProjHash ( hp , ( L , param ) , W, w ) . If w is a witness for W ∈ L . O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 11 / 31

  35. Properties Of SPHF Smoothness: If W / ∈ L nobody can distinguish a hashed value from a random one. O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 12 / 31

  36. Properties Of SPHF Smoothness: If W / ∈ L nobody can distinguish a hashed value from a random one. Pseudo-Randomness: Without w , if W ∈ L it is hard to distinguish a hashed value from a random one. O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 12 / 31

  37. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  38. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G L = { ( g 1 , h 1 ) |∃ α ∈ Z p , g 1 = g α ∧ h 1 = h α } , w = α . O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  39. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G L = { ( g 1 , h 1 ) |∃ α ∈ Z p , g 1 = g α ∧ h 1 = h α } , w = α . hk = ( λ, µ ) ∈ Z 2 p O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  40. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G L = { ( g 1 , h 1 ) |∃ α ∈ Z p , g 1 = g α ∧ h 1 = h α } , w = α . hk = ( λ, µ ) ∈ Z 2 p hp = g λ h µ O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  41. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G L = { ( g 1 , h 1 ) |∃ α ∈ Z p , g 1 = g α ∧ h 1 = h α } , w = α . hk = ( λ, µ ) ∈ Z 2 p hp = g λ h µ 1 h µ H = g λ 1 O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  42. A Simple Example Of SPHF Here param contains ( g, h ) ∈ G L = { ( g 1 , h 1 ) |∃ α ∈ Z p , g 1 = g α ∧ h 1 = h α } , w = α . hk = ( λ, µ ) ∈ Z 2 p hp = g λ h µ 1 h µ H = g λ 1 H ′ = hp α O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 13 / 31

  43. SPHF And Implicit Decommitment Achieving OSBE Server Info O. Blazy, C. Chevalier, P . Germouty OLBE: A Natural Generalization December 5, 2016 14 / 31

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