# Reconciling Non-malleability with Homomorphic Encryption

- 393 Downloads

## Abstract

Homomorphic encryption schemes are useful in designing conceptually simple protocols that operate on encrypted inputs. On the other hand, non-malleable encryption schemes are vital for designing protocols with robust security against malicious parties, in a composable setting. In this paper, we address the problem of constructing public-key encryption schemes that meaningfully combine these two opposing demands. The intuitive tradeoff we desire in an encryption scheme is that anyone should be able to change encryptions of unknown messages \(m_1, \ldots , m_k\) into a (fresh) encryption of \(T(m_1, \ldots , m_k)\) for a specific set of allowed functions *T*, but the scheme should be otherwise “non-malleable.” That is, no adversary should be able to construct a ciphertext whose value is related to that of other ciphertexts in any other way. For the case where the allowed functions *T* are all *unary*, we formulate precise definitions that capture our intuitive requirements and show relationships among these new definitions and other more standard ones (IND-CCA, gCCA, and RCCA). We further justify these new definitions by showing their equivalence to a natural formulation of security in the framework of Universally Composable security. Next, we describe a new family of encryption schemes that satisfy our definitions for a wide variety of allowed transformations *T* and prove their security under the Decisional Diffie-Hellman (DDH) assumption in two groups with related sizes. Finally, we demonstrate how encryption schemes that satisfy our definitions can be used to implement conceptually simple protocols for non-trivial computation on encrypted data, which are secure against malicious adversaries in the UC framework without resorting to general-purpose multi-party computation or zero-knowledge proofs. For the case where the allowed functions *T* are *binary*, we show that a natural generalization of our definitions is unattainable if some *T* is a group operation. On the positive side, we show that if one of our security requirements is relaxed in a natural way, we can in fact obtain a scheme that is homomorphic with respect to (binary) group operations, and non-malleable otherwise.

## Notes

### Acknowledgments

We thank Josh Benaloh, Ran Canetti, Anna Lisa Ferrara, Rui Xue, and many anonymous referees for helpful suggestions on earlier versions of these results.

## References

- 1.J. H. Ahn, D. Boneh, J. Camenisch, S. Hohenberger, abhi shelat, and B. Waters. Computing on authenticated data. In R. Cramer, editor,
*TCC*, volume 7194 of*Lecture Notes in Computer Science*, pp. 1–20. Springer, 2012.Google Scholar - 2.J. H. An, Y. Dodis, and T. Rabin. On the security of joint signature and encryption. In Knudsen [46], pp. 83–107.Google Scholar
- 3.J. K. Andersen and E. W. Weisstein. Cunningham chain. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CunninghamChain.html, 2005.
- 4.M. Bellare, A. Boldyreva, A. Desai, and D. Pointcheval. Key-privacy in public-key encryption. In C. Boyd, editor,
*ASIACRYPT*, volume 2248 of*Lecture Notes in Computer Science*, pp. 566–582. Springer, 2001.Google Scholar - 5.M. Bellare and A. Sahai. Non-malleable encryption: Equivalence between two notions, and an indistinguishability-based characterization. In M. J. Wiener, editor,
*CRYPTO*, volume 1666 of*Lecture Notes in Computer Science*, pp. 519–536. Springer, 1999.Google Scholar - 6.J. Benaloh.
*Verifiable Secret-Ballot Elections*. PhD thesis, Department of Computer Science, Yale University, 1987.Google Scholar - 7.M. Blaze, G. Bleumer, and M. Strauss. Divertible protocols and atomic proxy cryptography. In K. Nyberg, editor,
*EUROCRYPT*, volume 1403 of*Lecture Notes in Computer Science*, pp. 127–144. Springer, 1998.Google Scholar - 8.D. Boneh. The decision Diffie-Hellman problem. In J. Buhler, editor,
*ANTS*, volume 1423 of*Lecture Notes in Computer Science*, pp. 48–63. Springer, 1998.Google Scholar - 9.D. Boneh, editor.
*Advances in Cryptology - CRYPTO 2003, 23rd Annual International Cryptology Conference, Santa Barbara, California, USA, August 17-21, 2003, Proceedings*, volume 2729 of*Lecture Notes in Computer Science*. Springer, 2003.Google Scholar - 10.D. Boneh, E.-J. Goh, and K. Nissim. Evaluating 2-DNF formulas on ciphertexts. In Kilian [44], pp. 325–341.Google Scholar
- 11.D. Boneh, G. Segev, and B. Waters. Targeted malleability: homomorphic encryption for restricted computations. In S. Goldwasser, editor,
*ITCS*, pp. 350–366. ACM, 2012.Google Scholar - 12.D. Boneh and B. Waters. Conjunctive, subset, and range queries on encrypted data. In Vadhan [65], pp. 535–554.Google Scholar
- 13.A. Broadbent and A. Tapp. Information-theoretic security without an honest majority. In Kurosawa [48], pp. 410–426.Google Scholar
- 14.R. Canetti. Universally composable security: A new paradigm for cryptographic protocols. Cryptology ePrint Archive, Report 2000/067, 2005.Google Scholar
- 15.R. Canetti, S. Halevi, and J. Katz. Chosen-ciphertext security from identity-based encryption. In C. Cachin and J. Camenisch, editors,
*EUROCRYPT*, volume 3027 of*Lecture Notes in Computer Science*, pp. 207–222. Springer, 2004.Google Scholar - 16.R. Canetti and J. Herzog. Universally composable symbolic analysis of mutual authentication and key-exchange protocols. In Halevi and Rabin [39], pp. 380–403.Google Scholar
- 17.R. Canetti and S. Hohenberger. Chosen-ciphertext secure proxy re-encryption. In P. Ning, S. D. C. di Vimercati, and P. F. Syverson, editors,
*ACM Conference on Computer and Communications Security*, pp. 185–194. ACM, 2007.Google Scholar - 18.R. Canetti, H. Krawczyk, and J. B. Nielsen. Relaxing chosen-ciphertext security. In Boneh [9], pp. 565–582.Google Scholar
- 19.M. Chase, M. Kohlweiss, A. Lysyanskaya, and S. Meiklejohn. Malleable proof systems and applications. In D. Pointcheval and T. Johansson, editors,
*EUROCRYPT*, volume 7237 of*Lecture Notes in Computer Science*, pp. 281–300. Springer, 2012.Google Scholar - 20.D. Chaum. Untraceable electronic mail, return addresses, and digital pseudonyms.
*Commun. ACM*, 24(2):84–88, 1981.CrossRefGoogle Scholar - 21.B. Chor, N. Gilboa, and M. Naor. Private information retrieval by keywords. TR CS0917, Department of Computer Science, Technion, 1997.Google Scholar
- 22.R. Cramer, M. K. Franklin, B. Schoenmakers, and M. Yung. Multi-authority secret-ballot elections with linear work. In U. M. Maurer, editor,
*EUROCRYPT*, volume 1070 of*Lecture Notes in Computer Science*, pp. 72–83. Springer, 1996.Google Scholar - 23.R. Cramer and V. Shoup. A practical public key cryptosystem provably secure against adaptive chosen ciphertext attack. In H. Krawczyk, editor,
*CRYPTO*, volume 1462 of*Lecture Notes in Computer Science*, pp. 13–25. Springer, 1998.Google Scholar - 24.R. Cramer and V. Shoup. Universal hash proofs and a paradigm for adaptive chosen ciphertext secure public-key encryption. In Knudsen [46], pp. 45–64.Google Scholar
- 25.I. Damgård, N. Fazio, and A. Nicolosi. Non-interactive zero-knowledge from homomorphic encryption. In Halevi and Rabin [39], pp. 41–59.Google Scholar
- 26.I. Damgård and J. B. Nielsen. Universally composable efficient multiparty computation from threshold homomorphic encryption. In Boneh [9], pp. 247–264.Google Scholar
- 27.G. Danezis. Breaking four mix-related schemes based on universal re-encryption.
*Int. J. Inf. Sec.*, 6(6):393–402, 2007.CrossRefzbMATHGoogle Scholar - 28.D. Dolev, C. Dwork, and M. Naor. Non-malleable cryptography (extended abstract). In C. Koutsougeras and J. S. Vitter, editors,
*STOC*, pp. 542–552. ACM, 1991.Google Scholar - 29.T. El Gamal. A public key cryptosystem and a signature scheme based on discrete logarithms. In G. R. Blakley and D. Chaum, editors,
*CRYPTO*, volume 196 of*Lecture Notes in Computer Science*, pp. 10–18. Springer, 1984.Google Scholar - 30.Free Haven Project. Anonymity bibliography. http://freehaven.net/anonbib/, 2006.
- 31.C. Gentry. Fully homomorphic encryption using ideal lattices. In M. Mitzenmacher, editor,
*STOC*, pp. 169–178. ACM, 2009.Google Scholar - 32.Y. Gertner, T. Malkin, and S. Myers. Towards a separation of semantic and CCA security for public key encryption. In Vadhan [65], pp. 434–455.Google Scholar
- 33.S. Goldwasser and S. Micali. Probabilistic encryption.
*J. Comput. Syst. Sci.*, 28(2):270–299, Apr. 1984. Preliminary version appeared in STOC’ 82.Google Scholar - 34.P. Golle, M. Jakobsson, A. Juels, and P. F. Syverson. Universal re-encryption for mixnets. In T. Okamoto, editor,
*CT-RSA*, volume 2964 of*Lecture Notes in Computer Science*, pp. 163–178. Springer, 2004.Google Scholar - 35.J. Groth. A verifiable secret shuffle of homomorphic encryptions. In Y. Desmedt, editor,
*Public Key Cryptography*, volume 2567 of*Lecture Notes in Computer Science*, pp. 145–160. Springer, 2003.Google Scholar - 36.J. Groth. Rerandomizable and replayable adaptive chosen ciphertext attack secure cryptosystems. In Naor [50], pp. 152–170.Google Scholar
- 37.J. Groth and S. Lu. A non-interactive shuffle with pairing based verifiability. In Kurosawa [48], pp. 51–67.Google Scholar
- 38.J. Groth and S. Lu. Verifiable shuffle of large size ciphertexts. In T. Okamoto and X. Wang, editors,
*Public Key Cryptography*, volume 4450 of*Lecture Notes in Computer Science*, pp. 377–392. Springer, 2007.Google Scholar - 39.S. Halevi and T. Rabin, editors.
*Theory of Cryptography, Third Theory of Cryptography Conference, TCC 2006, New York, NY, USA, March 4-7, 2006, Proceedings*, volume 3876 of*Lecture Notes in Computer Science*. Springer, 2006.Google Scholar - 40.M. Hirt and K. Sako. Efficient receipt-free voting based on homomorphic encryption. In B. Preneel, editor,
*EUROCRYPT*, volume 1807 of*Lecture Notes in Computer Science*, pp. 539–556. Springer, 2000.Google Scholar - 41.Y. Ishai, E. Kushilevitz, and R. Ostrovsky. Sufficient conditions for collision-resistant hashing. In Kilian [44], pp. 445–456.Google Scholar
- 42.M. J. Jurik.
*Extensions to the Paillier Cryptosystem with Applications to Cryptological Protocols*. PhD thesis, BRICS, 2003.Google Scholar - 43.A. Kiayias and M. Yung. Non-interactive zero-sharing with applications to private distributed decision making. In R. N. Wright, editor,
*Financial Cryptography*, volume 2742 of*Lecture Notes in Computer Science*, pp. 303–320. Springer, 2003.Google Scholar - 44.J. Kilian, editor.
*Theory of Cryptography, Second Theory of Cryptography Conference, TCC 2005, Cambridge, MA, USA, February 10-12, 2005, Proceedings*, volume 3378 of*Lecture Notes in Computer Science*. Springer, 2005.Google Scholar - 45.M. Klonowski, M. Kutylowski, A. Lauks, and F. Zagórski. Universal re-encryption of signatures and controlling anonymous information flow. In
*WARTACRYPT ’04 Conference on Cryptology*. Bedlewo/Poznan, 2006.Google Scholar - 46.L. R. Knudsen, editor.
*Advances in Cryptology - EUROCRYPT 2002, International Conference on the Theory and Applications of Cryptographic Techniques, Amsterdam, The Netherlands, April 28 - May 2, 2002, Proceedings*, volume 2332 of*Lecture Notes in Computer Science*. Springer, 2002.Google Scholar - 47.T. Koshy.
*Elementary Number Theory with Applications*. Academic Press, 2001.Google Scholar - 48.K. Kurosawa, editor.
*Advances in Cryptology - ASIACRYPT 2007, 13th International Conference on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia, December 2-6, 2007, Proceedings*, volume 4833 of*Lecture Notes in Computer Science*. Springer, 2007.Google Scholar - 49.P. D. MacKenzie, M. K. Reiter, and K. Yang. Alternatives to non-malleability: Definitions, constructions, and applications (extended abstract). In Naor [50], pp. 171–190.Google Scholar
- 50.M. Naor, editor.
*Theory of Cryptography, First Theory of Cryptography Conference, TCC 2004, Cambridge, MA, USA, February 19-21, 2004, Proceedings*, volume 2951 of*Lecture Notes in Computer Science*. Springer, 2004.Google Scholar - 51.M. Naor and M. Yung. Public-key cryptosystems provably secure against chosen ciphertext attacks. In H. Ortiz, editor,
*STOC*, pp. 427–437. ACM, 1990.Google Scholar - 52.P. Paillier. Public-key cryptosystems based on composite degree residuosity classes. In J. Stern, editor,
*EUROCRYPT*, volume 1592 of*Lecture Notes in Computer Science*, pp. 223–238. Springer, 1999.Google Scholar - 53.A. Patil. On symbolic analysis of cryptographic protocols. Master’s thesis, Massachusetts Institute of Technology, 2005.Google Scholar
- 54.M. Prabhakaran and M. Rosulek. Rerandomizable RCCA encryption. In A. Menezes, editor,
*CRYPTO*, volume 4622 of*Lecture Notes in Computer Science*, pp. 517–584. Springer, 2007. Full version available from http://eprint.iacr.org/2007/119. - 55.M. Prabhakaran and M. Rosulek. Cryptographic complexity of multi-party computation problems: Classifications and separations. In D. Wagner, editor,
*CRYPTO*, volume 5157 of*Lecture Notes in Computer Science*, pp. 262–279. Springer, 2008.Google Scholar - 56.M. Prabhakaran and M. Rosulek. Homomorphic encryption with CCA security. In L. Aceto, I. Damgård, L. A. Goldberg, M. M. Halldórsson, A. Ingólfsdóttir, and I. Walukiewicz, editors,
*ICALP (2)*, volume 5126 of*Lecture Notes in Computer Science*, pp. 667–678. Springer, 2008. Full version available from http://eprint.iacr.org/2008/079. - 57.M. Prabhakaran and M. Rosulek. Towards robust computation on encrypted data. In J. Pieprzyk, editor,
*ASIACRYPT*, volume 5350 of*Lecture Notes in Computer Science*, pp. 216–233. Springer, 2008.Google Scholar - 58.C. Rackoff and D. R. Simon. Non-interactive zero-knowledge proof of knowledge and chosen ciphertext attack. In J. Feigenbaum, editor,
*CRYPTO*, volume 576 of*Lecture Notes in Computer Science*, pp. 433–444. Springer, 1991.Google Scholar - 59.M. Rosulek.
*The Structure of Secure Multi-Party Computation*. PhD thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, 2009.Google Scholar - 60.A. Sahai. Non-malleable non-interactive zero knowledge and adaptive chosen-ciphertext security. In P. Beame, editor,
*FOCS*, pp. 543–553, 1999.Google Scholar - 61.K. Sako and J. Kilian. Secure voting using partially compatible homomorphisms. In Y. Desmedt, editor,
*CRYPTO*, volume 839 of*Lecture Notes in Computer Science*, pp. 411–424. Springer, 1994.Google Scholar - 62.T. Sander, A. Young, and M. Yung. Non-interactive cryptocomputing for NC\(^{1}\). In P. Beame, editor,
*FOCS*, pp. 554–567, 1999.Google Scholar - 63.V. Shoup. A proposal for an ISO standard for public key encryption. Cryptology ePrint Archive, Report 2001/112, 2001. http://eprint.iacr.org/.
- 64.D. X. Song, D. Wagner, and A. Perrig. Practical techniques for searches on encrypted data. In
*IEEE Symposium on Security and Privacy*, pp. 44–55, 2000.Google Scholar - 65.S. P. Vadhan, editor.
*Theory of Cryptography, 4th Theory of Cryptography Conference, TCC 2007, Amsterdam, The Netherlands, February 21-24, 2007, Proceedings*, volume 4392 of*Lecture Notes in Computer Science*. Springer, 2007.Google Scholar - 66.D. Wikström. A note on the malleability of the El Gamal cryptosystem. In A. Menezes and P. Sarkar, editors,
*INDOCRYPT*, volume 2551 of*Lecture Notes in Computer Science*, pp. 176–184. Springer, 2002.Google Scholar