1. On the Security of the “Free-XOR” Technique Ranjit Kumaresan Joint work with Seung Geol Choi, Jonathan Katz, and Hong-Sheng Zhou (UMD)

2. Research in Secure Two-party Computation (2PC) Generic protocols [Yao86, GMW87] “Tailored” protocols for specific applications [FNP04,HL08,KO97,…] Fairplay [MNPS04]: Implemented generic protocols – Hope for practicality

3. Research in Secure Two-party Computation (2PC) Active research improving concrete efficiency of generic protocols – Garbled circuit approach [PSSW09,HEKM11,KM11,LP07,LP11,…] – GMW approach [NNOB11, CHKMR12,...] Moving secure computation from theory to practice

4. Talk Outline Background on Yao GC & the Free-XOR technique [KS08] – Description in the random oracle (RO) model – Replacing RO with correlation robust hash functions? Sufficient assumptions on the hash function – Why correlation robust hash functions are not enough – New notion: Circular correlation robust hash functions – Security of the Free-XOR technique Conclusions

5. Yao Garbled Circuit (GC) [Yao86] Generic secure computation protocol Constant round solution Mostly symmetric-key operations Popular choice for efficient 2PC

6. Yao Garbled Circuit u v w AND u u v v u v v u uv XOR Credit: V. Kolesnikov

7. Yao Garbled Circuit AND XOR u0u0 u1u1 v0v0 v1v1 w0w0 w1w1 H(u 0,v 0,g) ⊕ w 0 H(u 0,v 1,g) ⊕ w 0 H(u 1,v 0,g) ⊕ w 0 H(u 1,v 1,g) ⊕ w 1 x0x0 x1x1 y0y0 y1y1 H(w 0,x 0,g’) ⊕ y 0 H(w 0,x 1,g’) ⊕ y 1 H(w 1,x 0,g’) ⊕ y 1 H(w 1,x 1,g’) ⊕ y 0 g,g’: gate indices H: hash function

8. …. GC GC Based Semi-Honest 2PC [Yao86] Alice input keys OT Bob input keys GC …. input bits Bob keys Evaluate GC using received input keys

9. Efficiency Improvements to Yao GC Garbled row reduction [NPS99,PSSW09] – Just 3 entries per garbled table Point-and-permute [MNPS04] – Decrypt only one entry Free-XOR technique [KS08] – No garbled table for XOR gates

10. Free-XOR Technique [KS08] Idea: XOR gates evaluated for “free” – No cryptographic operations or communication (like [Kol05,GMW87]) – GC based 2PC in the semi-honest setting Gains in practice? – 40% improvement for “typical” circuits – 300% improvement for universal circuits Impact – All recent implementations use Free-XOR technique [PSSW09, SS11,…] – Efforts to minimize #non-XOR gates in circuit [KS08, KSS09, PSSW09]

11. Free-XOR Technique [KS08] AND XOR u0u0 u1u1 v0v0 v1v1 w0w0 w1w1 H(u 0,v 0,g) ⊕ w 0 H(u 0,v 1,g) ⊕ w 0 H(u 1,v 0,g) ⊕ w 0 H(u 1,v 1,g) ⊕ w 1 x0x0 x1x1 y0y0 y1y1 H(w 0,x 0,g’) ⊕ y 0 H(w 0,x 1,g’) ⊕ y 1 H(w 1,x 0,g’) ⊕ y 1 H(w 1,x 1,g’) ⊕ y 0

12. AND XOR u0u0 v0v0 w0w0 x0x0 u 1 = u 0 ⊕ R v 1 = v 0 ⊕ R w 1 = w 0 ⊕ R x 1 = x 0 ⊕ R y 1 = y 0 ⊕ R y 0 = w 0 ⊕ x 0 Free-XOR Technique [KS08] H(u 0,v 0,g) ⊕ w 0 H(u 0,v 1,g) ⊕ w 0 H(u 1,v 0,g) ⊕ w 0 H(u 1,v 1,g) ⊕ w 1 H(w 0,x 0,g’) ⊕ y 0 H(w 0,x 1,g’) ⊕ y 1 H(w 1,x 0,g’) ⊕ y 1 H(w 1,x 1,g’) ⊕ y 0 R : hidden global parameter

13. Free-XOR Technique [KS08] AND XOR u v w x Set y = w ⊕ x y H(u 0,v 0,g) ⊕ w 0 H(u 0,v 1,g) ⊕ w 0 H(u 1,v 0,g) ⊕ w 0 H(u 1,v 1,g) ⊕ w 1 H(w 0,x 0,g’) ⊕ y 0 H(w 0,x 1,g’) ⊕ y 1 H(w 1,x 0,g’) ⊕ y 1 H(w 1,x 1,g’) ⊕ y 0 R : hidden global parameter Use H(u,v,g) to recover w

14. Proof in the RO Model [KS08] Corrupt Alice: Trivial Corrupt Bob: – Sim creates a fake garbled circuit whose output is always correct – Intuitively, security reduces to proving R is completely hidden – Indistinguishability proved by induction on topological ordering of gates H(u,v,g) ⊕ w H(u,v ⊕ R,g) ⊕ w H(u ⊕ R,v,g) ⊕ w H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R)  By induction, known input keys: u, v  Only w is recovered  Except with negl. prob., all other values are hidden H(u,v,g) ⊕ w random 1 random 2 random 3 Real table Simulated table

15. Proof in the Standard Model? RO is not programmed Can RO be replaced by a suitable hash function? – [KS08]: a variant of correlation robust hash functions (CorRHF) works – Repeated wherever Free-XOR is used [PSSW09,SS11,AHI11,NO09,…] Our contributions Specify variant of CorRHF that is sufficient “Natural” variant of CorRHF is NOT sufficient

16. Proof in the Standard Model? Main issue is circularity [BK03,BRS03, HK07, …] – H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R) – CorRHF does not capture circularity Specify variant of CorRHF that is sufficient “Natural” variant of CorRHF is NOT sufficient H(u,v,g) ⊕ w H(u,v ⊕ R,g) ⊕ w H(u ⊕ R,v,g) ⊕ w H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R) Circular Correlation Robust Hash Functions – Captures circularity – Security proof for the Free-XOR technique

17. Why is this important? Implementors happy with RO… In theory, RO methodology is inherently flawed [CGH04] – Want precise formulation of concrete properties required by RO “Natural” variant of CorRHF used in other contexts [AHI11,NO09] “CorRHF is sufficient for Free-XOR technique” claimed in several works [PSSW09,SS11, AHI11,…] Assumptions required for Free-XOR tech. in Yao GC? – Free-XOR in [GMW87, Kol05] with no other assumptions

18. Correlation Robust Hash Functions [IKNP03] Proposed by [IKNP03] for removing RO in OT extension Definition: (CorRHF) H is CorRHF if for randomly chosen u 1,…, u p, the following two distributions are comp. indistinguishable – (u 1,…, u p, H(u 1 ⊕ R), …, H(u p ⊕ R)) where R is chosen uniformly – (u 1,…, u p, w 1,…, w p ) where each w i is chosen uniformly (Arithmetic variant) realized under PDH assumption [AHI11] [KS08]: Variant can replace RO in Free-XOR – Use of hidden off-set in both [KS08] and [IKNP03]

19. “Natural” Variant of CorRHF Definition: (weak 2-CorRHF) H is weakly 2-CorRHF if for given u 1,…, u p, v 1,…, v p, the following two distributions are comp. indistinguishable –. – ` where R is chosen uniformly – (w 1,…, w 3p ) where each w i is chosen uniformly H(u 1 ⊕ R,v 1,1), H(u 1,v 1 ⊕ R,1), H(u 1 ⊕ R,v 1 ⊕ R,1) H(u p ⊕ R,v p,p), H(u p,v p ⊕ R,p), H(u p ⊕ R,v p ⊕ R,p)......

20. Our Working Definition of 2-CorRHF Oracle based – Cor R (u,v,g): output H(u,v ⊕ R,g), H(u ⊕ R,v,g), H(u ⊕ R,v ⊕ R,g) – Rand(u,v,g): if input was queried before then output answer given previously, else output a uniformly chosen string Definition: (2-CorRHF) H is 2-CorRHF if every non-uniform PPT adversary A with oracle access to O (either Cor R or Rand) cannot tell whether O is Cor R or Rand except with negligible advantage Stronger than previous definition – Oracle queries can be adaptive

21. 2-CorRHF and Free-XOR technique  Reduction adversary B for 2-CorRHF  Given O (either Cor R or Rand)  How to create garbled table?  Choose random u,v,w  Query O (u,v,g) to get h 1, h 2, h 3  First 3 entries can be set  How to obtain fourth entry using h 3 ?  Unclear how to complete reduction Reduction Table H(u,v,g) ⊕ w H(u,v ⊕ R,g) ⊕ w H(u ⊕ R,v,g) ⊕ w H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R) H(u,v,g) ⊕ w random 1 random 2 random 3 Real table Simulated table H(u,v,g) ⊕ w h 1 ⊕ w h 2 ⊕ w ?

22. Counterexample Rule out fully black-box reduction using two oracles H and Break H is 2-CorRHF even if A has oracle access to H and Break Free-XOR technique is insecure when A has access to H and Break H(u,v,g)  Random function Break(u,v,g,z 1,z 2,z 3 )  Output r when  z 1 = H(u,v ⊕ r,g)  z 2 = H(u ⊕ r,v,g)  z 3 = H(u ⊕ r,v ⊕ r,g) ⊕ r  Else output nothing

23. H is 2-CorRHF against A H, Break O = Rand: uniform, independent of A ’s view O = Cor R : uniform, independent of A ’s view unless A queries O (u,v,g) & – O (u’,v’,g) with u’ ⊕ u = R or v’ ⊕ v = R, or – H(u’,v’,g) with u’ ⊕ u = R or v’ ⊕ v = R, or – Break(u,v,g,z 1,z 2,z 3 ) with z 3 ⊕ H(u ⊕ R,v ⊕ R,g) = R Happens with negligible prob. H(u,v,g)  Random function Break(u,v,g,z 1,z 2,z 3 )  Output r when  z 1 = H(u,v ⊕ r,g)  z 2 = H(u ⊕ r,v,g)  z 3 = H(u ⊕ r,v ⊕ r,g) ⊕ r  Else output nothing

24. Insecurity of Free-XOR Tech.: A H, Break Attack: A acting as Bob recovers R Recover w from gate g using H(u,v,g) – z 1 = c 1 ⊕ w – z 2 = c 2 ⊕ w – z 3 = c 3 ⊕ w Query Break(u,v,g,z 1,z 2,z 3 ) to get R H(u,v,g) ⊕ w H(u,v ⊕ R,g) ⊕ w H(u ⊕ R,v,g) ⊕ w H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R) AND gate g c1c1 c3c3 c2c2 H(u,v,g)  Random function Break(u,v,g,z 1,z 2,z 3 )  Output r when  z 1 = H(u,v ⊕ r,g)  z 2 = H(u ⊕ r,v,g)  z 3 = H(u ⊕ r,v ⊕ r,g) ⊕ r  Else output nothing

25. Capturing Circularity: Circular 2-CorRHF Recall indistinguishable oracles in 2-CorRHF – Cor R (u,v,g): output H(u,v ⊕ R,g), H(u ⊕ R,v,g), H(u ⊕ R,v ⊕ R,g) – Rand(u,v,g): if input was queried before then output answer given previously, else output uniformly chosen Oracles for Circular 2-CorRHF – Circ R (u,v,g,b 1,b 2,b 3 ): output H(u ⊕ b 1 R, v ⊕ b 2 R, g) ⊕ b 3 R – Rand(u,v,g,b 1,b 2,b 3 ): same as before bR = 0 when b=0 bR = R when b=1

26. Capturing Circularity: Circular 2-CorRHF Recall indistinguishable oracles in 2-CorRHF – Cor R (u,v,g): output H(u,v ⊕ R,g), H(u ⊕ R,v,g), H(u ⊕ R,v ⊕ R,g) – Rand(u,v,g): if input was queried before then output answer given previously, else output uniformly chosen Oracles for Circular 2-CorRHF – Circ R (u,v,g,b 1,b 2,b 3 ): output H(u ⊕ b 1 R, v ⊕ b 2 R, g) ⊕ b 3 R – Rand(u,v,g,b 1,b 2,b 3 ): same as before Allowing b 3 = 1 captures circularity

27. Circular 2-CorRHF Oracles for Circular 2-CorRHF – Circ R (u,v,g,b 1,b 2,b 3 ): output H(u ⊕ b 1 R, v ⊕ b 2 R, g) ⊕ b 3 R – Rand(u,v,g,b 1,b 2,b 3 ): same as before Indistinguishability conditioned on restricted queries to Circ R – No queries of the form (u,v,g,0,0,b 3 ) – No queries on both (u,v,g,b 1,b 2,0) and (u,v,g,b 1,b 2,1) Definition: (Circular 2-CorRHF) H is circular 2-CorRHF if every non-uniform PPT adversary A making legal queries to oracle O cannot tell whether O is Circ R or Rand except with negligible advantage

28. Proof of Security for the Free-XOR Tech. Corrupt Alice: Trivial Corrupt Bob: Sim creates a fake garbled circuit AND XOR u v w x y = w ⊕ x  Choose random key for all wires except output wires of XOR gates  XOR chosen keys for input wires to get key for output wire of XOR gate  Populate unknown values in non- XOR gate table with random values  Set output garbled table to give correct output z H(u,v,g) ⊕ w random 1 random 2 random 3 Simulated table......

29. Reduction to Circular 2-CorRHF Reduction adversary B for Circular 2-CorRHF B given access to O (either Circ R or Rand) & real inputs for both parties AND XOR u v w x y = w ⊕ x H(u,v,g) ⊕ w O (u,v,g,0,1,0) ⊕ w O (u,v,g,1,0,0) ⊕ w O (u,v,g,1,1,1) ⊕ w Reduction Table......  Choose random key for all wires except output wires of XOR gates  XOR chosen keys for input wires to get key for output wire of XOR gate  Populate unknown values in non- XOR gate table using O  Set output garbled table to give correct output z

30. Circular 2-CorRHF & Free-XOR technique Recall Circ R (u,v,g,b 1,b 2,b 3 ):  output H(u ⊕ b 1 R, v ⊕ b 2 R, g) ⊕ b 3 R Reduction Table H(u,v,g) ⊕ w H(u,v ⊕ R,g) ⊕ w H(u ⊕ R,v,g) ⊕ w H(u ⊕ R,v ⊕ R,g) ⊕ (w ⊕ R) H(u,v,g) ⊕ w random 1 random 2 random 3 Real table Simulated table H(u,v,g) ⊕ w O (u,v,g,0,1,0) ⊕ w O (u,v,g,1,0,0) ⊕ w O (u,v,g,1,1,1) ⊕ w O = Rand O = Circ R

31. Conclusions & Open Questions Free-XOR technique extremely influential – Used in all Yao GC implementations Secure in the random oracle model “Natural” variant of 2-CorRHF is not sufficient – Circularity Stronger notion of 2-CorRHF: Circular 2-CorRHF – Security proof for the Free-XOR technique “Free” gate evaluation under OWF? Realize Circular 2-CorRHF from standard crypto assumptions?

32. Thank You!