1. The Many Faces of Garbled Circuits MIT Vinod Vaikuntanathan

2. Garbled Circuits [Yao’86, refined by BMR’90, AIK’04, BHR’12] Garble Circuit C: Encode Input x: Garble(C; R) Encode(x; R) Simplicity: Garbling and Encoding are “simple” (e.g., Garble: small depth, Encode: affine)

3. The Swiss-army Knife of Crypto Secure Two-party Computation [Yao86] (Constant Round) MPC [BMR90, IK00] Parallel Cryptography [AIK05] One-time Programs [GKR08] KDM-Security [BHHI09, A11] Verifiable Outsourcing [GGP10,AIK10] Circuit-private Homomorphic Encryption [GHV10] Functional Encryption [SS10, GVW12] And many others: [AF90, FKN94, NPS99, KO04, FM06, AL07, LP07, GKR08, GMS08, BFK+09, PSS09, BHHI10, GGP10, HS10, KM10, SS10, A11, KMR11, LP11, GVW12, …]

4. Powerful Theorems Theorem: [Yao’86, LP’04] Assuming one-way functions, there is a garbling scheme for the class of all poly-size circuits. Theorem: [IK’00, AIK’04] Assuming nothing, there is a garbling scheme for the class of logspace computable functions.

5. Much Work, Many Constructions Better Efficiency: Variants [BMR90, NPS99] Free XOR [KS08, CKKZ12, A13], Garbled Circuits with Short Input Encoding [AIKW13] Several practical efficiency improvements [MNPS04, BNP08, HSEKS11, KSS12, BHKR13] Better Security: Adaptive vs. Static Security [BHR12a, BHR12b, JSW16] New Goals, New Models: Re-randomizable Garbled Circuits [GHV10], Arithmetic Garbled Circuits [AIK11], Garbling RAM machines [LO13, GHLORW14, GLOS15, GLO15] and so on…

6. This Talk: “Cryptopia through the Garbled Circuit Lens” Fully Homomorphic Encryption Functional Encryption Indistinguishability Obfuscation Attribute-based encryption

7. Yao’s Garbled Circuits [A. Yao 1986] AND-filter

8. Yao’s Garbled Circuits [A. Yao 1986] garbled gate

9. The Reusability Problem No Reusability! “Mix-and-match” attack

10. This Talk: Fully Homomorphic Encryption Functional Encryption Indistinguishability Obfuscation Attribute-based encryption “Cryptopia through the reusable Garbled Circuit Lens”

11. Functional Encryption (FE) [Sahai-Waters’05, refined by BSW’12, O’neill’12] Secret Key for Circuit C: SK C Encrypt Input x: CT SK C, C, CT → C(x) Decrypt: KeyGen(SK, C) Enc(PK, x)  No circuit hiding: SK C does not hide the circuit C  Public-key: many-input security for free  Many-key Security: Can release many SK C_i revealing only C i (x)  Succinctness: Encryption time (and size) independent of |C|

12. Attribute-based Encryption (ABE) [Sahai-Waters’05, Goyal-Pandey-Sahai-Waters’06] Secret Key for Circuit C: SK C Encrypt Input x, Message M: CT SK C, C, CT, x → M C(x) KeyGen(SK, C) Enc(PK, x, M0, M1) Decrypt:  No circuit or input hiding: only messages M0 and M1 hidden  Public-key: many-input security for free  Many-key Security: If C i (x) = b for all i, M 1-b hidden  Succinctness: Encryption time (and size) independent of |C| (either M0 or M1 not both)

13. Garbled Circuits, FE & Friends Size non-succinct succinct Garbled circuits + Public-key encryption Single-key Security Many-key [Sahai-Seyalioglu’11] (Sub-exp.) LWE [Gorbunov-V-Wee’13, Goldwasser-KPVZ’13, Boneh-GGHNSVV’14] → “Reusable” GC for single-bit fns. = [Bitansky-V’15, Ananth-Jain-Sahai’15] [Bitansky-V’15, Ananth-Jain’15, Lin-Pass-Seth-Telang’16] Many-key FE for NC1 → Reusable GC for many-bit fns. → Indistinguishability Obfuscation [Bitansky-V’15, Ananth-Jain’15]

14. First Try: FE from Garbled Circuits Many key, single input, secret key FE Many input, single key, secret key FE: use the universal circuit (and thus, lose succinctness) Single key (public key) FE: use public-key encryption and the “decomposability” of Yao’s garbled circuits Secret Key for Circuit C = Garbled Input for C Secret Key for Circuit C = SK i,Ci [Sahai-Seyalioglu’11]

15. Garbled Circuits, FE & Friends Size non-succinct succinct Garbled circuits + Public-key encryption Single-key Security Many-key [Sahai-Seyalioglu’11] (Sub-exp.) LWE [Gorbunov-V-Wee’13, Goldwasser-KPVZ’13, Boneh-GGHNSVV’14] → “Reusable” GC for single-bit fns. = [Bitansky-V’15, Ananth-Jain-Sahai’15] [Bitansky-V’15, Ananth-Jain’15, Lin-Pass-Seth-Telang’16] Many-key FE for NC1 → Reusable GC for many-bit fns. → Indistinguishability Obfuscation [Bitansky-V’15, Ananth-Jain’15]

16. Theorem 1.1: [Gorbunov-V.-Wee’13, Boneh-Gentry- Gorbunov-Halevi-Nikolaenko-Segev-V.-Vinayagamurthy’14]] Assuming “sub-exponential LWE”, there is an ABE scheme for the class of all poly-size circuits (of a-priori bounded depth). Theorem 1.2: [Goldwasser-Kalai-Popa-V.-Zeldovich’13] 1.FHE for P 2.ABE for (bounded depth) P 3.(One-time) garbling FE for (bounded depth) P Compiler from ABE to (single key, succinct) FE +

17. Theorem 1.1 + 1.2 = FE from Subexp. LWE ABE Subexp. LWE FHE Yao garbling + + LWE Theorem 1.1 [GSW13, BV14] Single-key Succinct FE and LWE Reusable Garbled Circuits Theorem 1.2

18. (Recall Yao’s garbled circuits) Labels = Strings: single-use Labels = Functions: many-use KEY IDEA

19. ABE Construction (e.g., x=0101) ? NEED: Family of trapdoor functions

20. ABE Construction (e.g., x=0011) Reusable filter

21. ABE Construction (e.g., x=0011) Reusable filter NO MIX-and-MATCH

22. ABE Construction (e.g., x=0011) = “Two-to-one Recoding” Keys

23. What are these Trapdoor Functions? Learning with errors [BFKL’93, Regev’05] [Ajtai’99,Micciancio-Peikert’13] Trapdoor function (Sample uniformly random A with trapdoor)

24. How to “Recode”? reusable AND-filter A1A1 A2A2 A3A3 (Let’s start with no noise) SUCH THAT A 3 = R 1 A 1 + R 2 A 2 FIND matrices (R 1, R 2 ) R1R1 R2R2 Recoding key: Matrices (R 1,R 2 ) Key Idea: Linearity! With noise: need R’s to be low-weight Use “GPV Theorem”: Find low-weight R s.t. R A = B, given trapdoor for A

25. Theorem 1.1: [Gorbunov-V.-Wee’13, Boneh-Gentry- Gorbunov-Halevi-Nikolaenko-Segev-V.-Vinayagamurthy’14]] Assuming “sub-exponential LWE”, there is an ABE scheme for the class of all poly-size circuits (of a-priori bounded depth). Theorem 1.2: [Goldwasser-Kalai-Popa-V.-Zeldovich’13] 1.FHE for P 2.ABE for (bounded depth) P 3.(One-time) garbling FE for (bounded depth) P Compiler from ABE to (single key, succinct) FE +

26. ABE + FHE + Yao = Single-key Succinct FE FE Secret Key for C: ABE.SK C FE Encryption of x: ABE.Enc(x, L 0, L 1 ) Idea 1. To hide x, encrypt it with FHE. Generate keys for the FHE evaluation circuit for C. FE Secret Key for C: ABE.SK EvalC FE Encryption of x: ABE.Enc(FHE.Enc(x), L 0, L 1 ) FE Encryption of x: ABE.Enc(FHE.Enc(x), L i,0, L i,1 ) ABE decryption results in L i,fhe.ct_i where fhe.ct is the encryption of C(x). Idea 2. Yao-Garble the FHE decryption circuit with input labels L i,b. Yao = single-use → FE = single-key. + Yao.Garble(FHE.Dec SK ) w/ input labels L i,b

27. Reusable Garbled Circuits Garble C: FE.SK C Encode x: FE.Enc(x) Problem. Hides x but not C. Solution. Use a universal circuit and encrypt C (using a simple secret-key encryption) Garble C: FE.SK for U(SymEnc(symsk,C), ∙, ∙) Encode x: FE.Enc(symsk, x) Garble once. Encode many times (using the same symsk) One-time garblingABE (single key) FE Reusable Garbling

28. Garbled Circuits, FE & Friends Size non-succinct succinct Garbled circuits + Public-key encryption Single-key Security Many-key [Sahai-Seyalioglu’11] (Sub-exp.) LWE [Gorbunov-V-Wee’13, Goldwasser-KPVZ’13, Boneh-GGHNSVV’14] → “Reusable” GC for single-bit fns. [Bitansky-V’15, Ananth-Jain’15, Lin-Pass-Seth-Telang’16] Many-key FE for NC1 → Reusable GC for many-bit fns. → Indistinguishability Obfuscation

29. Obfuscation = Public-key Garbling Obfuscation of Circuit C: Obf(C) Eval on Input x: C(x) Same as reusable garbling except for public evaluation. = Public-key (and therefore, reusable) garbling Indistinguishability obfuscation: for C 0 ≣ C 1, Obf(C 0 ) ≈ Obf(C 1 ). Obfuscation [BGIRSVY’01, GR’07, GGHRSW’13, SW’14]

30. Theorem: Reusable Garbling++ to IO [Bitansky-V.’15, Ananth-Jain’15, simplified by Lin-Pass-Seth-Telang’16] If there is a compact reusable garbled circuit + sub- exponentially secure OWF, then there is an IO scheme. Many-key FE Compact Reusable GC IO easy [GGHRSW’13] this theorem Succinct: garbled input size ind. of |C| for one-bit functions ++ = Compact: garbled input size ind. of |C| for many-bit functions

31. Theorem: Reusable Garbling++ to IO Idea in a nutshell: Encodings that output (two) encodings Garble(П n ) Enc(0 n-1 ) Enc(0 n-2 1) Enc(1 n-1 ) П n takes as input an n-1 bit string x and outputs encodings of x0 and x1 Enc(0 n-2 ) Enc(0 n-3 )Enc(1 n-3 ) Enc(ε) Enc(0 n )Enc(0 n-1 1) Enc(1 n ) Garble(C) … Need compactness to avoid exponential blowup. Obf(C) = Enc(1 n-2 ) Garble(П n-1 ) Garble(П n-2 ) Garble(П 1 ) (more in Rafael’s talk Wed.)

32. “Cryptopia” through the Garbling Lens Fully Homomorphic Encryption Functional Encryption Indistinguishability Obfuscation Attribute-based encryption (Key Property: Reusability)

33. Many Open Questions Many-key FE (and thus, IO) from LWE. ($300 from Amit + $100 from me) Unconditional Garbled Circuits for all of P. Yao: one-way functions Applebaum-Ishai-Kushilevitz’04: unconditional for Logspace Ishai-Kushilevitz-Paskin’12: “degree-2” impossible ($100 to resolve this one way or the other)

34. Thank You!