# Adaptive Security of Yao’s Garbled Circuits

## Abstract

A garbling scheme is used to garble a circuit *C* and an input *x* in a way that reveals the output *C*(*x*) but hides everything else. Yao’s construction from the 80’s is known to achieve *selective security*, where the adversary chooses the circuit *C* and the input *x* in one shot. It has remained as an open problem whether the construction also achieves adaptive security, where the adversary can choose the input *x* after seeing the garbled version of the circuit *C*.

A recent work of Hemenway et al. (CRYPTO’16) modifies Yao’s construction and shows that the resulting scheme is adaptively secure. This is done by encrypting the garbled circuit from Yao’s construction with a special type of “somewhere equivocal encryption” and giving the key together with the garbled input. The efficiency of the scheme and the security loss of the reduction is captured by a certain pebbling game over the circuit.

In this work we prove that Yao’s construction itself is already adaptively secure, where the security loss can be captured by the same pebbling game. For example, we show that for circuits of depth *d*, the security loss of our reduction is \(2^{O(d)}\), meaning that Yao’s construction is adaptively secure for NC1 circuits without requiring complexity leveraging. Our technique is inspired by the “nested hybrids” of Fuchsbauer et al. (Asiacrypt’14, CRYPTO’15) and relies on a careful sequence of hybrids where each hybrid involves some limited guessing about the adversary’s adaptive choices. Although it doesn’t match the parameters achieved by Hemenway et al. in their full generality, the main advantage of our work is to prove the security of Yao’s construction as is, without any additional encryption layer.

## References

- [AIKW13]Applebaum, B., Ishai, Y., Kushilevitz, E., Waters, B.: Encoding functions with constant online rate or how to compress garbled circuits keys. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 166–184. Springer, Heidelberg (2013)CrossRefGoogle Scholar
- [AS15]Ananth, P., Sahai, A.: Functional encryption for turing machines. Cryptology ePrint Archive, Report 2015/776 (2015). http://eprint.iacr.org/
- [BGG+14]Boneh, D., Gentry, C., Gorbunov, S., Halevi, S., Nikolaenko, V., Segev, G., Vaikuntanathan, V., Vinayagamurthy, D.: Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 533–556. Springer, Heidelberg (2014)CrossRefGoogle Scholar
- [BHK13]Bellare, M., Hoang, V.T., Keelveedhi, S.: Instantiating random oracles via UCEs. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part II. LNCS, vol. 8043, pp. 398–415. Springer, Heidelberg (2013)CrossRefGoogle Scholar
- [BHR12a]Bellare, M., Hoang, V.T., Rogaway, P.: Adaptively secure garbling with applications to one-time programs and secure outsourcing. In: Wang, X., Sako, K. (eds.) ASIACRYPT 2012. LNCS, vol. 7658, pp. 134–153. Springer, Heidelberg (2012)CrossRefGoogle Scholar
- [BHR12b]Bellare, M., Hoang, V.T., Rogaway, P.: Foundations of garbled circuits. In: Yu, T., Danezis, G., Gligor, V.D. (eds.) ACM CCS 12, pp. 784–796. ACM Press, October 2012Google Scholar
- [FJP15]Fuchsbauer, G., Jafargholi, Z., Pietrzak, K.: A quasipolynomial reduction for generalized selective decryption on trees. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9215, pp. 601–620. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-47989-6_29 CrossRefGoogle Scholar
- [FKPR14]Fuchsbauer, G., Konstantinov, M., Pietrzak, K., Rao, V.: Adaptive security of constrained PRFs. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014, Part II. LNCS, vol. 8874, pp. 82–101. Springer, Heidelberg (2014)Google Scholar
- [HJO+15]Hemenway, B., Jafargholi, Z., Ostrovsky, R., Scafuro, A., Wichs, D.: Adaptively secure garbled circuits from one-way functions. IACR Cryptology ePrint Archive, 2015, p. 1250 (2015)Google Scholar
- [LP09]Lindell, Y., Pinkas, B.: A proof of security of Yao’s protocol for two-party computation. J. Cryptol.
**22**(2), 161–188 (2009)MathSciNetCrossRefMATHGoogle Scholar - [Yao82]Yao, A.C.: Protocols for secure computations (extended abstract). In: 23rd FOCS, pp. 160–164. IEEE Computer Society Press, November (1982)Google Scholar
- [Yao86]Yao, A.C.: How to generate and exchange secrets (extended abstract). In: 27th FOCS, pp. 162–167. IEEE Computer Society Press, October 1986Google Scholar