Abstract
We introduce a new primitive, called trapdoor hash functions (TDH), which are hash functions \(\mathsf {H}: \{0,1\}^n \rightarrow \{0,1\}^\lambda \) with additional trapdoor function-like properties. Specifically, given an index \(i\in [n]\), TDHs allow for sampling an encoding key \(\mathsf {ek}\) (that hides i) along with a corresponding trapdoor. Furthermore, given \(\mathsf {H}(x)\), a hint value \(\mathsf {E}(\mathsf {ek},x)\), and the trapdoor corresponding to \(\mathsf {ek}\), the \(i^{th}\) bit of x can be efficiently recovered. In this setting, one of our main questions is: How small can the hint value \(\mathsf {E}(\mathsf {ek},x)\) be? We obtain constructions where the hint is only one bit long based on DDH, QR, DCR, or LWE.
This primitive opens a floodgate of applications for low-communication secure computation. We mainly focus on two-message protocols between a receiver and a sender, with private inputs x and y, resp., where the receiver should learn f(x, y). We wish to optimize the (download) rate of such protocols, namely the asymptotic ratio between the size of the output and the sender’s message. Using TDHs, we obtain:
-
1.
The first protocols for (two-message) rate-1 string OT based on DDH, QR, or LWE. This has several useful consequences, such as:
-
(a)
The first constructions of PIR with communication cost poly-logarithmic in the database size based on DDH or QR. These protocols are in fact rate-1 when considering block PIR.
-
(b)
The first constructions of a semi-compact homomorphic encryption scheme for branching programs, where the encrypted output grows only with the program length, based on DDH or QR.
-
(c)
The first constructions of lossy trapdoor functions with input to output ratio approaching 1 based on DDH, QR or LWE.
-
(d)
The first constant-rate LWE-based construction of a 2-message “statistically sender-private” OT protocol in the plain model.
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(a)
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2.
The first rate-1 protocols (under any assumption) for n parallel OTs and matrix-vector products from DDH, QR or LWE.
We further consider the setting where f evaluates a RAM program y with running time \(T\ll |x|\) on x. We obtain the first protocols with communication sublinear in the size of x, namely \(T\cdot \sqrt{|x|}\) or \(T\cdot \root 3 \of {|x|}\), based on DDH or, resp., pairings (and correlated-input secure hash functions).
G. Malavolta—Part of the work done while at Friedrich-Alexander-Universität Erlangen-Nürnberg.
T. Mour—Part of the work done while at Technion.
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Notes
- 1.
It seems plausible that these gaps can be closed using indistinguishability obfuscation [25]. However, the focus of this work is on constructions that can be based on more traditional assumptions.
- 2.
A simple “hybrid FHE” technique [28] can generically convert any FHE scheme into one whose encrypted (long) input is roughly as long as the input. However, no such generic technique is known for compressing the encrypted output.
- 3.
Our protocols can be efficiently extended to provide security against malicious parties (under the same assumptions) using sublinear arguments [46]. This increases the number of rounds, but does not affect the asymptotic communication.
- 4.
The work of Cho et al. [16] on laconic OT gives a batch OT protocol where \(|\mathsf {msg}_1|\) is independent of |x|. This generalizes to arbitrary functions f; however, even in the simple case of batch OT the download rate is sub-constant. The same applies to the more recent work of Quach et al. [52] on laconic function evaluation.
- 5.
In this work, we consider this question in the more stringent two-message setting. However, we note that no other protocols with rate \(> \frac{1}{2}\) are known even when additional rounds of communication are allowed.
- 6.
- 7.
Our protocols achieve asymptotic download rate 1, which is clearly optimal. However, the (additive) difference between the sender’s message length and the output length grows with the security parameter \(\lambda \). In the full version we show that that this gap is necessary even in the more liberal setting of secure computation with preprocessing.
- 8.
More specifically, as our group-based constructions are black-box in the underlying group, we can count the communication complexity in terms of the number of group elements, which in our case is \(\log n\cdot \mathsf {poly}(\lambda )\), where n is the size of the database and \(\lambda \) is the security parameter.
- 9.
For this application, we insist on the two-message setting. Allowing O(T) rounds of interaction, similar protocols can be based on any single-server PIR scheme [46].
- 10.
We can ensure that both sender and receiver run in strict polynomial time by introducing a suitable polynomial upper bound for the number of trials in the exhaustive search step of \(\mathsf {Dist}(\cdot )\). For small erasure probabilities, a near-quadratic improvement in the running time can be obtained via the recent optimal “distributed discrete log” algorithm of Dinur, Keller, and Klein [19].
- 11.
In fact, string OT can be thought of as a special case of batch OT, where all the choice bits \(i_j\) are equal.
- 12.
In fact they require an OT protocol with a strong notion of sender privacy, which is satisfied by all of our constructions.
- 13.
References
Aiello, B., Ishai, Y., Reingold, O.: Priced oblivious transfer: how to sell digital goods. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 119–135. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_8
Applebaum, B., Damgård, I., Ishai, Y., Nielsen, M., Zichron, L.: Secure arithmetic computation with constant computational overhead. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 223–254. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_8
Attrapadung, N., Matsuda, T., Nishimaki, R., Yamada, S., Yamakawa, T.: Constrained PRFs for \(\rm NC^1\) in traditional groups. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 543–574. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_19
Badrinarayanan, S., Garg, S., Ishai, Y., Sahai, A., Wadia, A.: Two-message witness indistinguishability and secure computation in the plain model from new assumptions. In: Takagi, T., Peyrin, T. (eds.) ASIACRYPT 2017, Part III. LNCS, vol. 10626, pp. 275–303. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70700-6_10
Banerjee, A., Peikert, C., Rosen, A.: Pseudorandom functions and lattices. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 719–737. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_42
Beaver, D.: Precomputing oblivious transfer. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 97–109. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-44750-4_8
Bellare, M., Cash, D.: Pseudorandom functions and permutations provably secure against related-key attacks. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 666–684. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_36
Block, A.R., Maji, H.K., Nguyen, H.H.: Secure computation with constant communication overhead using multiplication embeddings. In: Chakraborty, D., Iwata, T. (eds.) INDOCRYPT 2018. LNCS, vol. 11356, pp. 375–398. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-05378-9_20
Boyle, E., Gilboa, N., Ishai, Y.: Breaking the circuit size barrier for secure computation under DDH. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part I. LNCS, vol. 9814, pp. 509–539. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53018-4_19
Boyle, E., Gilboa, N., Ishai, Y.: Group-based secure computation: optimizing rounds, communication, and computation. In: Coron, J.-S., Nielsen, J.B. (eds.) EUROCRYPT 2017, Part II. LNCS, vol. 10211, pp. 163–193. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56614-6_6
Brakerski, Z., Döttling, N.: Two-message statistically sender-private OT from LWE. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018, Part II. LNCS, vol. 11240, pp. 370–390. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03810-6_14
Brakerski, Z., Goldwasser, S.: Circular and leakage resilient public-key encryption under subgroup indistinguishability. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 1–20. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_1
Brakerski, Z., Lombardi, A., Segev, G., Vaikuntanathan, V.: Anonymous IBE, leakage resilience and circular security from new assumptions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part I. LNCS, vol. 10820, pp. 535–564. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78381-9_20
Cachin, C., Micali, S., Stadler, M.: Computationally private information retrieval with polylogarithmic communication. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 402–414. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_28
Chang, Y.-C.: Single database private information retrieval with logarithmic communication. In: Wang, H., Pieprzyk, J., Varadharajan, V. (eds.) ACISP 2004. LNCS, vol. 3108, pp. 50–61. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-27800-9_5
Cho, C., Döttling, N., Garg, S., Gupta, D., Miao, P., Polychroniadou, A.: Laconic oblivious transfer and its applications. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part II. LNCS, vol. 10402, pp. 33–65. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63715-0_2
Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private information retrieval. In: 36th Annual Symposium on Foundations of Computer Science, Milwaukee, Wisconsin, 23–25 October 1995, pp. 41–50. IEEE Computer Society Press (1995)
Damgård, I., Jurik, M.: A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Kim, K. (ed.) PKC 2001. LNCS, vol. 1992, pp. 119–136. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44586-2_9
Dinur, I., Keller, N., Klein, O.: An optimal distributed discrete log protocol with applications to homomorphic secret sharing. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part III. LNCS, vol. 10993, pp. 213–242. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96878-0_8
Dodis, Y., Halevi, S., Rothblum, R.D., Wichs, D.: Spooky encryption and its applications. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016, Part III. LNCS, vol. 9816, pp. 93–122. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53015-3_4
Döttling, N., Garg, S.: Identity-based encryption from the Diffie-Hellman assumption. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017, Part I. LNCS, vol. 10401, pp. 537–569. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63688-7_18
Döttling, N., Garg, S., Hajiabadi, M., Masny, D.: New constructions of identity-based and key-dependent message secure encryption schemes. In: Abdalla, M., Dahab, R. (eds.) PKC 2018, Part I. LNCS, vol. 10769, pp. 3–31. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-76578-5_1
Fazio, N., Gennaro, R., Jafarikhah, T., Skeith, W.E.: Homomorphic secret sharing from Paillier encryption. In: Okamoto, T., Yu, Y., Au, M.H., Li, Y. (eds.) ProvSec 2017. LNCS, vol. 10592, pp. 381–399. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-68637-0_23
Garg, S., Gay, R., Hajiabadi, M.: New techniques for efficient trapdoor functions and applications. Cryptology ePrint Archive, Report 2018/872 (2018). https://eprint.iacr.org/2018/872
Garg, S., Gentry, C., Halevi, S., Raykova, M., Sahai, A., Waters, B.: Candidate indistinguishability obfuscation and functional encryption for all circuits. In: 54th Annual Symposium on Foundations of Computer Science, Berkeley, CA, USA, 26–29 October 2013, pp. 40–49. IEEE Computer Society Press (2013)
Garg, S., Hajiabadi, M.: Trapdoor functions from the computational Diffie-Hellman assumption. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018, Part II. LNCS, vol. 10992, pp. 362–391. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96881-0_13
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) 41st Annual ACM Symposium on Theory of Computing, Bethesda, MD, USA, 31 May–2 June 2009, pp. 169–178. ACM Press (2009)
Gentry, C., Groth, J., Ishai, Y., Peikert, C., Sahai, A., Smith, A.D.: Using fully homomorphic hybrid encryption to minimize non-interative zero-knowledge proofs. J. Cryptol. 28(4), 820–843 (2015)
Gentry, C., Halevi, S.: Compressible FHE with applications to PIR. Technical report (2019). (Personal communication)
Goldenberg, D., Liskov, M.: On related-secret pseudorandomness. In: Micciancio, D. (ed.) TCC 2010. LNCS, vol. 5978, pp. 255–272. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11799-2_16
Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: Aho, A. (ed.) 19th Annual ACM Symposium on Theory of Computing, New York City, NY, USA, pp. 218–229, 25–27 May 1987. ACM Press (1987)
Goyal, V., O’Neill, A., Rao, V.: Correlated-input secure hash functions. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 182–200. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19571-6_12
Groth, J., Kiayias, A., Lipmaa, H.: Multi-query computationally-private information retrieval with constant communication rate. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 107–123. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13013-7_7
Halevi, S., Kalai, Y.T.: Smooth projective hashing and two-message oblivious transfer. J. Cryptol. 25(1), 158–193 (2012)
Hemenway, B., Ostrovsky, R.: Extended-DDH and lossy trapdoor functions. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 627–643. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-30057-8_37
Hubacek, P., Wichs, D.: On the communication complexity of secure function evaluation with long output. Cryptology ePrint Archive, Report 2014/669 (2014). http://eprint.iacr.org/2014/669
Ishai, Y., Kilian, J., Nissim, K., Petrank, E.: Extending oblivious transfers efficiently. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 145–161. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_9
Ishai, Y., Kushilevitz, E., Meldgaard, S., Orlandi, C., Paskin-Cherniavsky, A.: On the power of correlated randomness in secure computation. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 600–620. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-36594-2_34
Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Cryptography with constant computational overhead. In: STOC 2008, pp. 433–442 (2008)
Ishai, Y., Paskin, A.: Evaluating branching programs on encrypted data. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 575–594. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70936-7_31
Ishai, Y., Prabhakaran, M., Sahai, A.: Secure arithmetic computation with no honest majority. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 294–314. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-00457-5_18
Juvekar, C., Vaikuntanathan, V., Chandrakasan, A.: GAZELLE: a low latency framework for secure neural network inference. In: USENIX Security Symposium, pp. 1651–1669 (2018)
Koppula, V., Lewko, A.B., Waters, B.: Indistinguishability obfuscation for turing machines with unbounded memory. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th Annual ACM Symposium on Theory of Computing, Portland, OR, USA, 14–17 June 2015, pp. 419–428. ACM Press (2015)
Kushilevitz, E., Ostrovsky, R.: Replication is NOT needed: SINGLE database, computationally-private information retrieval. In: 38th Annual Symposium on Foundations of Computer Science, Miami Beach, Florida, 19–22 October 1997, pp. 364–373. IEEE Computer Society Press (1997)
Lipmaa, H.: An oblivious transfer protocol with log-squared communication. In: Zhou, J., Lopez, J., Deng, R.H., Bao, F. (eds.) ISC 2005. LNCS, vol. 3650, pp. 314–328. Springer, Heidelberg (2005). https://doi.org/10.1007/11556992_23
Naor, M., Nissim, K.: Communication preserving protocols for secure function evaluation. In: 33rd Annual ACM Symposium on Theory of Computing, Crete, Greece, 6–8 July 2001, pp. 590–599. ACM Press (2001)
Naor, M., Pinkas, B.: Oblivious transfer and polynomial evaluation. In: 31st Annual ACM Symposium on Theory of Computing, Atlanta, GA, USA, 1–4 May 1999, pp. 245–254. ACM Press (1999)
Naor, M., Pinkas, B.: Efficient oblivious transfer protocols. In: SODA 2001, pp. 448–457 (2001)
Okamoto, T., Pietrzak, K., Waters, B., Wichs, D.: New realizations of somewhere statistically binding hashing and positional accumulators. In: Iwata, T., Cheon, J.H. (eds.) ASIACRYPT 2015, Part I. LNCS, vol. 9452, pp. 121–145. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48797-6_6
Ostrovsky, R., Skeith, W.E.: A survey of single-database private information retrieval: techniques and applications. In: Okamoto, T., Wang, X. (eds.) PKC 2007. LNCS, vol. 4450, pp. 393–411. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71677-8_26
Peikert, C., Waters, B.: Lossy trapdoor functions and their applications. In: Ladner, R.E., Dwork, C. (eds.) 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, 17–20 May 2008, pp. 187–196. ACM Press (2008)
Quach, W., Wee, H., Wichs, D.: Laconic function evaluation and applications. In: 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 859–870 (2018)
Stern, J.P.: A new and efficient all-or-nothing disclosure of secrets protocol. In: Ohta, K., Pei, D. (eds.) ASIACRYPT 1998. LNCS, vol. 1514, pp. 357–371. Springer, Heidelberg (1998). https://doi.org/10.1007/3-540-49649-1_28
Winkler, S., Wullschleger, J.: On the efficiency of classical and quantum oblivious transfer reductions. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 707–723. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_38
Yao, A.C.-C.: How to generate and exchange secrets (extended abstract). In: 27th Annual Symposium on Foundations of Computer Science, Toronto, Ontario, Canada, 27–29 October 1986, pp. 162–167. IEEE Computer Society Press (1986)
Acknowledgments
We thank Craig Gentry, Shai Halevi, Srinath Setty, and Vinod Vaikuntanathan for helpful discussions and pointers.
S. Garg supported by DARPA/ARL SAFEWARE Award W911NF15C0210, AFOSR Award FA9550-15-1-0274, AFOSR YIP Award, DARPA and SPAWAR under contract N66001-15-C-4065, a Hellman Award and research grants by the Okawa Foundation, Visa Inc., and Center for Long-Term Cybersecurity (CLTC, UC Berkeley). The views expressed are those of the author and do not reflect the official policy or position of the funding agencies.
Y. Ishai supported by ERC Project NTSC (742754), ISF grant 1709/14, NSF-BSF grant 2015782, and a grant from the Ministry of Science and Technology, Israel and Department of Science and Technology, Government of India.
G. Malavolta supported by a gift from Ripple, a gift from DoS Networks, a grant from Northrop Grumman, a Cylab seed funding award, and a JP Morgan Faculty Fellowship.
T. Mour supported by BSF grant 2012378, and NSF-BSF grant 2015782.
R. Ostrovsky supported by NSF grant 1619348, BSF grant 2015782, DARPA SafeWare subcontract to Galois Inc., DARPA SPAWAR contract N66001-15-C-4065, JP Morgan Faculty Research Award, OKAWA Foundation Research Award, IBM Faculty Research Award, Xerox Faculty Research Award, B. John Garrick Foundation Award, Teradata Research Award, and Lockheed-Martin Corporation Research Award. The views expressed are those of the authors and do not reflect position of the Department of Defense or the U.S. Government.
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Döttling, N., Garg, S., Ishai, Y., Malavolta, G., Mour, T., Ostrovsky, R. (2019). Trapdoor Hash Functions and Their Applications. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_1
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