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Dec 07, 2022 •104 likes •2k views

Hashing Garbled Circuits for Free Xiong Fan, Chaya Ganesh and Vladimir Kolesnikov Motivation Garbled circuits (GC) main technique for secure computation Motivation Garbled circuits (GC) main technique for secure computation Primitive in

  1. GC hash definition Take advantage of the input to hash being a Garbled Circuit GC hash definition weaker than standard collision resistance Given a correctly generated garbled circuit and hash ( GC , h ) If A finds � GC such that H ( � GC ) = H ( GC ) Then, w.h.p, the garbled circuit property of � GC is broken

  2. GC hash definition Take advantage of the input to hash being a Garbled Circuit GC hash definition weaker than standard collision resistance Given a correctly generated garbled circuit and hash ( GC , h ) If A finds � GC such that H ( � GC ) = H ( GC ) Then, w.h.p, the garbled circuit property of � GC is broken � GC will fail to evaluate

  3. C

  4. GC, GC , e, e , d, h C

  5. GC, GC , e, e , d, h C H(GC) = H( GC ) = h

  6. GC, GC , e, e , d, h C H(GC) = H( GC ) = h Ve(C, GC, d, e ) = accept

  7. GC, GC , e, e , d, h GC, GC , e, e , d, h C H(GC) = H( GC ) = h Ve(C, GC, d, e ) = accept De( Eval( GC , En( e , x), d) ) = 丄 for all x , w.h.p

  8. GC, GC , e, e , d, h C H(GC) = H( GC ) = h Ve(C, GC, d, e ) = accept De( Eval( GC , En( e , x), d) ) = 丄 for all x , w.h.p

  9. GC, GC , e, e , d, h C H(GC) = H( GC ) = h Same decoding information d Ve(C, GC, d, e ) = accept De( Eval( GC , En( e , x), d) ) = 丄 for all x , w.h.p

  10. Overview Definition of GC hash security Hashed garbling constructions – standard garbling and half-gates [ZRE’15] Implementation and evaluation Impact – Applications of free hash

  11. Overview Definition of GC hash security Hashed garbling constructions – standard garbling and half-gates [ZRE’15] Implementation and evaluation Impact – Applications of free hash

  12. GC hash construction Intertwine hash generation and verification with GC generation and evaluation

  13. GC hash construction Intertwine hash generation and verification with GC generation and evaluation Attempt 1: H ( GC ) = ⊕ i GR i

  14. a c e b d

  15. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1

  16. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 E A0, B0 ( C 0 ) E A0, B1 ( C 0 ) E A1, B0 ( C 0 ) E A1, B1 ( C 1 )

  17. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) E A1, B1 ( C 1 ) E C1, D1 ( E 1 )

  18. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) E A1, B1 ( C 1 ) E C1, D1 ( E 1 )

  19. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) E A1, B1 ( C 1 ) E C1, D1 ( E 1 )

  20. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) E A1, B1 ( C 1 ) E C1, D1 ( E 1 )

  21. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) E A1, B1 ( C 0 ) E C1, D1 ( E 1 )

  22. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 )

  23. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ⊕ Δ

  24. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ⊕ Δ

  25. A 0 , A 1 C 0 , C 1 Inactive row E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ⊕ Δ

  26. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E C0, D0 ( E 0 ) ⊕ Δ E A0, B0 ( C 0 ) GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ⊕ Δ

  27. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) ⊕ Δ GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ⊕ Δ ⊕ Δ

  28. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( C 0 ) E C0, D0 ( E 0 ) ⊕ Δ GC = (GT1, GT2) E A0, B1 ( C 0 ) E C0, D1 ( E 1 ) ⊕ h = H(GC) = h E A1, B0 ( C 0 ) E C1, D0 ( E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( C 0 ) E C1, D1 ( E 1 ) H(ĜC) = h ✔

  29. GC hash construction Make each gate’s output wire label depend on all entries of GT

  30. GC hash construction Make each gate’s output wire label depend on all entries of GT XOR hash correction involves modifying an active GT entry

  31. GC hash construction Make each gate’s output wire label depend on all entries of GT XOR hash correction involves modifying an active GT entry This affects the computed output wire label of the gate

  32. GC hash construction Make each gate’s output wire label depend on all entries of GT XOR hash correction involves modifying an active GT entry This affects the computed output wire label of the gate Does this suffice?

  33. A 0 , A 1 C 0 , C 1 E 0 , E 1 B 0 , B 1 D 0 , D 1

  34. C 0 , C 1 E 0 , E 1 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1

  35. Temporary wire labels C 0 , C 1 E 0 , E 1 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1

  36. C 0 , C 1 E 0 , E 1 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) E A1, B1 ( tC 1 ) E C1, D1 ( t E 1 )

  37. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) E A1, B1 ( tC 1 ) E C1, D1 ( t E 1 )

  38. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) E A1, B1 ( tC 1 ) E C1, D1 ( t E 1 )

  39. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ

  40. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 Fixes Δ for h But tC 0 ⊕ GT1 = C b ? GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ

  41. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ

  42. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ

  43. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) ⊕ Δ E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ

  44. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) ⊕ Δ E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ⊕ Δ ⊕ Δ

  45. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) ⊕ Δ E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ✔

  46. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) ⊕ Δ E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ✔ tC 0 ⊕ GT1 = C 0

  47. C b = tC b ⊕ GT1 C 0 , C 1 E 0 , E 1 E b = tE b ⊕ GT2 A 0 , A 1 tC 0 , tC 1 tE 0 , tE 1 B 0 , B 1 D 0 , D 1 GT1 GT2 E A0, B0 ( tC 0 ) ⊕ Δ E C0, D0 ( t E 0 ) GC = (GT1, GT2) E A0, B1 ( tC 0 ) E C0, D1 ( t E 1 ) ⊕ h = H(GC) = h E A1, B0 ( tC 0 ) E C1, D0 ( t E 1 ) ĜC = (ĜT1, GT2) E A1, B1 ( tC 0 ) E C1, D1 ( t E 1 ) H(ĜC) = h ✔ tC 0 ⊕ GT1 = C 0 ✔

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