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Combinatorially Homomorphic Encryption

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14370))

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Abstract

Homomorphic encryption enables public computation over encrypted data. In the past few decades, homomorphic encryption has become a staple of both the theory and practice of cryptography. Nevertheless, while there is a general loose understanding of what it means for a scheme to be homomorphic, to date there is no single unifying minimal definition that captures all schemes. In this work, we propose a new definition, which we refer to as combinatorially homomorphic encryption, which attempts to give a broad base that captures the intuitive meaning of homomorphic encryption.

Our notion relates the ability to accomplish some task when given a ciphertext, to accomplishing the same task without the ciphertext, in the context of communication complexity. Thus, we say that a scheme is combinatorially homomorphic if there exists a communication complexity problem f(x, y) (where x is Alice’s input and y is Bob’s input) which requires communication c, but can be solved with communication less than c when Alice is given in addition also an encryption \(E_k(y)\) of Bob’s input (using Bob’s key k).

We show that this definition indeed captures pre-existing notions of homomorphic encryption and (suitable variants are) sufficiently strong to derive prior known implications of homomorphic encryption in a conceptually appealing way. These include constructions of (lossy) public-key encryption from homomorphic private-key encryption, as well as collision-resistant hash functions and private information retrieval schemes.

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Notes

  1. 1.

    Several variants of the definition are possible depending on whether the similarity should be perfect, statistical or computational, and also whether it should hold even given additional information such as \(E_{pk}(m)\), or even given the corresponding secret-key. We ignore these subtleties here.

  2. 2.

    Indeed, this is the source of the term homomorphic encryption.

  3. 3.

    If compactness is not required, then the homomorphic evaluation can be trivially delegated to the decryptor (e.g., by appending the description of the circuit the ciphertext). Nevertheless, some homomorphic schemes such as [41] or constructions based on garbled circuits [12, 19, 25, 27] are not compact but are circuit private, meaning that the ciphertext does not reveal the evaluated circuit. In this work, we focus on compact homomorphic encryption, which is meaningful even without circuit privacy.

  4. 4.

    More precisely, we consider the VC dimension of the function family \(\big \{ f_x : \{0,1\}^n \rightarrow \{0,1\} \big \}_x\), where \(f_x(y)=f(x,y)\).

  5. 5.

    The homomorphic private-key to public-key transformation of Rothblum [40] can also be viewed as morally giving an abstraction of Regev’s scheme, but the actual formal definition of homomorphic encryption used in [40] is not technically achieved by the above private-key scheme.

  6. 6.

    In fact, Razborov [37] showed an input distribution on which the communication complexity of disjointness is \(\varOmega (n)\). However, since this input distribution is not a product distribution, using involves slightly more involved techniques (see further discussion in Sect. 3.1).

  7. 7.

    As usual in distributional communication complexity, this shared randomness can be eliminated by non-uniformly fixing the best choice.

  8. 8.

    The [30] construction is based on the Quadratic Residuosity assumption, but is easy to generalize to compact linearly homomorphic encryption (for a suitable notion of compactness), see [31, 42].

  9. 9.

    While a PIR scheme with communication, say, \(n-1\) does not seem directly useful, it is sufficient for deriving some important consequences of PIR such as CRH [26], oblivious transfer [14], lossy encryption [23] and SZK hardness [32].

References

  1. Alamati, N., Montgomery, H., Patranabis, S.: Symmetric primitives with structured secrets. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11692, pp. 650–679. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26948-7_23

    Chapter  Google Scholar 

  2. Alamati, N., Montgomery, H., Patranabis, S., Roy, A.: Minicrypt primitives with algebraic structure and applications. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11477, pp. 55–82. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17656-3_3

    Chapter  Google Scholar 

  3. Alekhnovich, M.: More on average case vs approximation complexity. In: Proceedings of the 44th Symposium on Foundations of Computer Science, FOCS 2003, 11–14 October 2003, Cambridge, MA, USA, pp. 298–307. IEEE Computer Society (2003). https://doi.org/10.1109/SFCS.2003.1238204

  4. Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: 27th Annual Symposium on Foundations of Computer Science, SFCS 1986, pp. 337–347. IEEE (1986)

    Google Scholar 

  5. Bellare, M., Hofheinz, D., Yilek, S.: Possibility and impossibility results for encryption and commitment secure under selective opening. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 1–35. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_1

    Chapter  Google Scholar 

  6. Benaloh, J.: Dense probabilistic encryption. In: Selected Areas of Cryptography, May 1994. https://www.microsoft.com/en-us/research/publication/dense-probabilistic-encryption/

  7. Bitansky, N., Freizeit, S.: Statistically sender-private OT from LPN and derandomization. In: Dodis, Y., Shrimpton, T. (eds.) Advances in Cryptology, CRYPTO 2022. LNCS, vol. 13509. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15982-4_21

  8. Bogdanov, A., Lee, C.H.: Limits of provable security for homomorphic encryption. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 111–128. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_7

    Chapter  Google Scholar 

  9. Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30576-7_18

    Chapter  Google Scholar 

  10. Brakerski, Z., Lyubashevsky, V., Vaikuntanathan, V., Wichs, D.: Worst-case hardness for LPN and cryptographic hashing via code smoothing. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 619–635. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_21

    Chapter  Google Scholar 

  11. Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. SIAM J. Comput. 43(2), 831–871 (2014). https://doi.org/10.1137/120868669

  12. Cachin, C., Camenisch, J., Kilian, J., Müller, J.: One-round secure computation and secure autonomous mobile agents. In: Montanari, U., Rolim, J.D.P., Welzl, E. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 512–523. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45022-X_43

    Chapter  MATH  Google Scholar 

  13. Cohen, S.P., Naor, M.: Low communication complexity protocols, collision resistant hash functions and secret key-agreement protocols. In: Dodis, Y., Shrimpton, T. (eds.) Advances in Cryptology, CRYPTO 2022. LNCS, vol. 13509, pp. 252–281. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15982-4_9

  14. Di Crescenzo, G., Malkin, T., Ostrovsky, R.: Single database private information retrieval implies oblivious transfer. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 122–138. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45539-6_10

    Chapter  Google Scholar 

  15. 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

    Chapter  MATH  Google Scholar 

  16. Dwork, C., Naor, M., Reingold, O.: Immunizing encryption schemes from decryption errors. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 342–360. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24676-3_21

    Chapter  Google Scholar 

  17. ElGamal, T.: A public key cryptosystem and a signature scheme based on discrete logarithms. In: Blakley, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985). https://doi.org/10.1007/3-540-39568-7_2

    Chapter  Google Scholar 

  18. Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University, USA (2009). https://searchworks.stanford.edu/view/8493082

  19. Gentry, C., Halevi, S., Vaikuntanathan, V.: i-hop homomorphic encryption and rerandomizable Yao circuits. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 155–172. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14623-7_9

    Chapter  Google Scholar 

  20. Gentry, C., Halevi, S., Vaikuntanathan, V.: A simple BGN-type cryptosystem from LWE. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 506–522. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_26

    Chapter  Google Scholar 

  21. Gentry, C., Sahai, A., Waters, B.: Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 75–92. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_5

    Chapter  Google Scholar 

  22. Goldwasser, S., Micali, S.: Probabilistic encryption. J. Comput. Syst. Sci. 28(2), 270–299 (1984). https://doi.org/10.1016/0022-0000(84)90070-9

    Article  MathSciNet  MATH  Google Scholar 

  23. Hemenway, B., Libert, B., Ostrovsky, R., Vergnaud, D.: Lossy encryption: constructions from general assumptions and efficient selective opening chosen ciphertext security. In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 70–88. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_4

    Chapter  Google Scholar 

  24. Holenstein, T., Renner, R.: One-way secret-key agreement and applications to circuit polarization and immunization of public-key encryption. In: Shoup, V. (ed.) CRYPTO 2005. LNCS, vol. 3621, pp. 478–493. Springer, Heidelberg (2005). https://doi.org/10.1007/11535218_29

    Chapter  Google Scholar 

  25. Horvitz, O., Katz, J.: Universally-composable two-party computation in two rounds. In: Menezes, A. (ed.) CRYPTO 2007. LNCS, vol. 4622, pp. 111–129. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74143-5_7

    Chapter  MATH  Google Scholar 

  26. Ishai, Y., Kushilevitz, E., Ostrovsky, R.: Sufficient conditions for collision-resistant hashing. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 445–456. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30576-7_24

    Chapter  Google Scholar 

  27. Ishai, Y., Kushilevitz, E., Ostrovsky, R., Prabhakaran, M., Sahai, A.: Efficient non-interactive secure computation. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 406–425. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20465-4_23

    Chapter  Google Scholar 

  28. 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

    Chapter  Google Scholar 

  29. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press (1997)

    Google Scholar 

  30. Kushilevitz, E., Ostrovsky, R.: Replication is NOT needed: SINGLE database, computationally-private information retrieval. In: 38th Annual Symposium on Foundations of Computer Science, FOCS 1997, Miami Beach, Florida, USA, 19–22 October 1997, pp. 364–373. IEEE Computer Society (1997). https://doi.org/10.1109/SFCS.1997.646125

  31. 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

    Chapter  Google Scholar 

  32. Liu, T., Vaikuntanathan, V.: On basing private information retrieval on NP-hardness. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016. LNCS, vol. 9562, pp. 372–386. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49096-9_16

    Chapter  Google Scholar 

  33. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994). https://doi.org/10.1007/3-540-48285-7_33

    Chapter  Google Scholar 

  34. Paillier, P.: Public-key cryptosystems based on composite degree residuosity classes. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 223–238. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48910-X_16

    Chapter  Google Scholar 

  35. Peikert, C., Vaikuntanathan, V., Waters, B.: A framework for efficient and composable oblivious transfer. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 554–571. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-85174-5_31

    Chapter  Google Scholar 

  36. Rao, A., Yehudayoff, A.: Communication Complexity: and Applications. Cambridge University Press (2020)

    Google Scholar 

  37. Razborov, A.A.: On the distributional complexity of disjointness. In: Paterson, M.S. (ed.) ICALP 1990. LNCS, vol. 443, pp. 249–253. Springer, Heidelberg (1990). https://doi.org/10.1007/BFb0032036

    Chapter  Google Scholar 

  38. Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: Gabow, H.N., Fagin, R. (eds.) Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 84–93. ACM (2005). https://doi.org/10.1145/1060590.1060603

  39. Rivest, R.L., Adleman, L., Dertouzos, M.L.: On data banks and privacy homomorphisms. Foundations of Secure Computation, pp. 169–179. Academia Press (1978)

    Google Scholar 

  40. Rothblum, R.: Homomorphic encryption: from private-key to public-key. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 219–234. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19571-6_14

    Chapter  Google Scholar 

  41. Sander, T., Young, A.L., Yung, M.: Non-interactive cryptocomputing for nc\({}^{\text{1}}\). In: 40th Annual Symposium on Foundations of Computer Science, FOCS 1999, 17–18 October, 1999, New York, NY, USA, pp. 554–567. IEEE Computer Society (1999). https://doi.org/10.1109/SFFCS.1999.814630

  42. 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

    Chapter  Google Scholar 

  43. Yao, A.C.C.: Some complexity questions related to distributive computing (preliminary report). In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, pp. 209–213 (1979)

    Google Scholar 

  44. Yu, Yu., Zhang, J., Weng, J., Guo, C., Li, X.: Collision resistant hashing from sub-exponential learning parity with noise. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11922, pp. 3–24. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34621-8_1

    Chapter  Google Scholar 

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Acknowledgments

We thank Aayush Jain and the TCC reviewers for their helpful comments. Y. Ishai was supported in part by ERC Project NTSC (742754), BSF grant 2018393, and ISF grant 2774/20. R. Rothblum is funded by the European Union (ERC, FASTPROOF, 101041208). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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  1. Technion, Haifa, Israel

    Yuval Ishai, Eyal Kushnir & Ron D. Rothblum

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  1. Yuval Ishai
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  2. Eyal Kushnir
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  3. Ron D. Rothblum
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Correspondence to Yuval Ishai .

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  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Ishai, Y., Kushnir, E., Rothblum, R.D. (2023). Combinatorially Homomorphic Encryption. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14370. Springer, Cham. https://doi.org/10.1007/978-3-031-48618-0_9

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