Abstract
Homomorphic cryptosystems are fundamental and highly effective cryptographic primitives for addressing security problems arising in information processing, data analysis and data applications, particularly in secure cloud computing and secure multiparty computation. To privately compute functions such as E(x1 ∧ ⋯ ∧ xt), E(x1 ∨ ⋯ ∨ xt) and E(x11 ∧ ⋯ ∧ xm1) ∨ ⋯ ∨ (x1n ∧ ⋯ ∧ xmn)] (m disjunctive normal form (mDNF)) on E(x1), …, E(xt) and E(x11), …, E(xmn) without knowing the decryption key, Boolean homomorphic cryptosystems are necessary. Exploring new homomorphic cryptosystems to solve these problems is appealing, and is of high theoretical and practical significance. To solve problems such as these, we propose a polynomial AND homomorphic cryptosystem based on the ideal theory of abstract algebra; the scheme can be obtained from available multiplicatively homomorphic cryptosystems such as the ElGamal. We prove that the cryptosystem is semantically secure. This polynomial AND homomorphic cryptosystem is a highly effective tool for designing various cryptographic protocols. As examples, we demonstrate its applications to privately compute any DNF (i.e., P(X1, …, Xm) = E[(x11 ∧ ⋯ ∧ xm1) ∨ ⋯ ∨ (x1n ∧ ⋯ ∧ xmn)] on ciphertexts E(xij) of xij without knowing the decryption key) and the intersection and union of certain private sets.
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References
Rivest R L, Adleman L, Dertouzos M L. On data banks and privacy homomorphisms. Found Secure Comput, 1978, 4: 169–180
López-Alt A, Tromer E, Vaikuntanathan V. On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Proceedings of the 44th ACM Symposium on Theory of Computing, 2012. 1219–1234
Atayero A A, Feyisetan O. Security issues in cloud computing: the potentials of homomorphic encryption. J Emerg Trends Comput Inf Sci, 2011, 2: 546–552
Zhang R, Ma H, Lu Y, et al. Provably secure cloud storage for mobile networks with less computation and smaller overhead. Sci China Inf Sci, 2017, 60: 122104
Kuang L W, Yang L T, Feng J, et al. Secure tensor decomposition using fully homomorphic encryption scheme. IEEE Trans Cloud Comput, 2018, 6: 868–878
Anunay K, Akshay R, Matthew D, et al. Cryptographically secure multiparty computation and distributed auctions using homomorphic encryption. Cryptography, 2017, 1: 25
Lin H Y, Tzeng W G. An efficient solution to the millionaires’ problem based on homomorphic encryption. In: Proceedings of the 3rd International Conference Applied Cryptography and Network Security, 2005. 456–466
Jiang B B, Zhang Y. Securely min and k-th min computations with fully homomorphic encryption. Sci China Inf Sci, 2018, 61: 058103
Wang W, Hu Y, Chen L, et al. Exploring the feasibility of fully homomorphic encryption. IEEE Trans Comput, 2015, 64: 698–706
Cramer R, Damgard I B, Nielsen J B. Secure Multiparty Computation. Cambridge: Cambridge University Press, 2015
Rivest R L, Shamir A, Adleman L. A method for obtaining digital signatures and public-key cryptosystems. Commun ACM, 1978, 21: 120–126
ElGamal T. A public key cryptosystem and a signature scheme based on discrete logarithms. In: Proceedings of Annual International Cryptology Conference, Santa Barbara, 1984. 10–18
Rabin M O. Digitalized Signatures and Public-key Functions as Intractable as Factorization. Massachusetts INST of Tech Cambridge Lab For Computer Science, Technical Report, No. ADA078415. 1979
Okamoto T, Uchiyama S. A new public-key cryptosystem as secure as factoring. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Espoo, 1998. 308–318
Paillier P. Public-key cryptosystems based on composite degree residuosity classes. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Prague, 1999. 223–238
Miller V. Use of elliptic curves in cryptography. In: Proceedings of Annual International Cryptology Conference, Santa Barbara, 1985. 417–426
Hoffstein J, Pipher J, Silverman J H. NTRU: a ring-based public key cryptosystem. In: Proceedings of the 3rd International Symposium on Algorithmic Number Theory, Portland, 1998. 267–288
Goldreich O, Goldwasser S, Halevi S. Public-key cryptosystems from lattice reduction problems. In: Proceedings of Annual International Cryptology Conference, Santa Barbara, 1997. 112–131
Hu Y P, Jia H W. Cryptanalysis of GGH map. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, 2016. 537–565
Benaloh J. Dense probabilistic encryption. In: Proceedings of the Workshop on Selected Areas of Cryptography, Kingston, 1994. 120–128
Naccache D, Stern J. A new public key cryptosystem based on higher residues. In: Proceedings of the 5th ACM conference on Computer and Communications Security, San Francisco, 1998. 59–66
Damgard I, Jurik M. A generalisation, a simplification and some applications of Paillier’s probabilistic public-key system. In: Proceedings of Public Key Cryptosystem, 2001. 119–136
Ishai Y, Paskin A. Evaluating branching programs on encrypted data. In: Proceedings of International Theory of Cryptography Conference, Amsterdam, 2007. 575–594
Goldwasser S, Micali S. Probabilistic encryption. J Comput Syst Sci, 1984, 28: 270–299
Boneh D, Goh E J, Nissim K. Evaluating 2-DNF formulas on ciphertexts. In: Proceedings of International Theory of Cryptography Conference, Cambridge, 2005. 325–341
Gentry C. Fully homomorphic encryption using ideal lattices. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, Bethesda, 2009. 169–178
Brakerski Z, Gentry C, Vaikuntanathan V. (Leveled) fully homomorphic encryption without bootstrapping. ACM Trans Comput Theor, 2014, 6: 1–36
Brakerski Z, Vaikuntanathan V. Efficient fully homomorphic encryption from (standard) LWE. SIAM J Comput, 2014, 43: 831–871
Brakerski Z, Gentry C, Vaikuntanathan V. (Leveled) fully homomorphic encryption without bootstrapping. In: Proceedings of Innovations in Theoretical Computer Science, Cambridge, 2012. 309–325
Fan J F, Vercauteren F. Somewhat practical fully homomorphic encryption. Cryptology ePrint Archive, Report 2012/144. http://eprint.iacr.org/
Cheon J H, Kim A, Kim M, et al. Homomorphic encryption for arithmetic of approximate numbers. In: Proceedings of International Conference on the Theory and Application of Cryptology and Information Security, 2017. 409–437
Chillotti I, Gama N, Georgieva M, et al. Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Proceedings of International Conference on the Theory and Application of Cryptology and Information Security, Hanoi, 2016. 3–33
Chillotti I, Gama N, Georgieva M, et al. Faster packed homomorphic operations and efficient circuit bootstrapping for TFHE. In: Proceedings of International Conference on the Theory and Application of Cryptology and Information Security, 2017. 377–408
Chillotti I, Gama N, Georgieva M, et al. TFHE: fast fully homomorphic encryption over the torus. IACR Cryptol ePrint Arch, 2018, 2018: 421
Chillotti I, Gama N, Georgieva M, et al. Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Proceedings of International Conference on the Theory and Application of Cryptology and Information Security, Hanoi, 2016. 3–33
Chillotti I, Gama N, Georgieva M, et al. TFHE: fast fully homomorphic encryption over the torus. https://eprint.iacr.org/2018/421
Goldreich O. Foundations of Cryptography: Basic Applications. Cambridge: Cambridge University Press, 2004
Nielsen J B, Nordholt P S, Orlandi C, et al. A new approach to practical active-secure two-party computation. In: Proceedings of Annual International Cryptology Conference, Santa Barbara, 2012. 681–700
Naehrig M, Lauter K, Vaikuntanathan V. Can homomorphic encryption be practical? In: Proceedings of the 3rd ACM Cloud Computing Security Workshop, Chicago, 2011. 113–124
Sander T, Young A, Yung M. Non-interactive crypto-computing for NC1. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999. 554–566
Fischlin M. A cost-effective pay-per-multiplication comparison method for millionaires. In: Proceedings of the Cryptographer’s Track at the RSA Conference, San Jose, 2001. 457–471
Barbulescu R, Gaudry P, Joux A, et al. A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic: improvements over FFS in small to medium characteristic. In: Proceedings of Annual International Conference on the Theory and Applications of Cryptographic Techniques, Copenhagen, 2014. 1–16
Menezes A, Sarkar P, Singh S. Challenges with assessing the impact of NFS advances on the security of pairing-based cryptography. In: Proceedings of International Conference on Cryptology in Malaysia, Kuala Lumpur, 2016. 83–108
Thomas W J, Stephen F. Abstract Algebra Theory and Applications. Nacogdoches: Austin State University Press, 2014
Meadows C. A more efficient cryptographic matchmaking protocol for use in the absence of a continuously available third party. In: Proceedings of 1986 IEEE Symposium on Security and Privacy, Oakland, 1986. 134
Freedman M J, Hazay C, Nissim K, et al. Efficient set intersection with simulation-based security. J Cryptol, 2016, 29: 115–155
Kissner L, Song D. Privacy-preserving set operations. In: Proceedings of Annual International Cryptology Conference, Santa Barbara, 2005. 241–257
Pinkas B, Schneider T, Segev G, et al. Phasing: private set intersection using permutation-based hashing. In: Proceedings of the 24th USENIX Security Symposium, Washington, 2015. 515–530
Pinkas B, Schneider T, Zohner M. Scalable private set intersection based on OT extension. ACM Trans Priv Secur, 2018, 21: 1–35
Orrú M, Orsini E, Scholl P. Actively secure 1-out-of-n OT extension with application to private set intersection. In: Proceedings of Cryptographers’ Track at the RSA Conference, San Francisco, 2017. 381–396
Chen H, Laine K, Rindal P. Fast private set intersection from homomorphic encryption. In: Proceedings of the ACM SIGSAC Conference on Computer and Communications Security, Dallas, 2017. 1243–1255
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 62172435). We are deeply grateful to all reviewers.
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Li, S., Zhou, S., Dou, J. et al. Polynomial AND homomorphic cryptosystem and applications. Sci. China Inf. Sci. 63, 112105 (2020). https://doi.org/10.1007/s11432-018-9789-y
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DOI: https://doi.org/10.1007/s11432-018-9789-y