Analysis of Constructing Fully Homomorphic Encryption Based on the Abstract Decryption Structure
SONG Xinxia① CHEN Zhigang②③
①(College of Junior, Zhejiang Wanli University, Ningbo 315100, China) ②(College of Electronics and Computer, Zhejiang Wanli University, Ningbo 315100, China) ③(State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China)
Abstract：Why can fully homomorphic encryption be constructed based on lattice What is the essence and construction of the matrix An important concept is proposed: Abstract decryption structure. Based on the abstract decryption structure, the main factors related to the homomorphic encryption are analyzed and relationship between abstract decryption structure, homomorphism and noise control is studied. The construction of the homomorphic encryption is attributed to the problem of how to obtain the final decryption structure. So the formal method of homomorphic encryption can be established. Thus the essential law of the construction method of the homomorphic encryption construction is expounded, which provides the clue and clue for the construction of the new full homomorphic encryption. The general reason of the full homomorphic encryption of the ciphertext matrix from the point of view of the ciphertexts stack method is studied. The relation between the full homomorphic encryption and the other homomorphic encryption is obtained. Finally, this paper gives a general method of constructing fully homomorphic encryption.
GENTRY C. Fully homomorphic encryption using ideal lattices[C]. Proceedings of the 41st Annual ACM Symposium on Theory of Computing, Bethesda, USA, 2009: 169-178. doi: 10.1145/1536414.1536440.
[2]
SMART N P and VERCAUTEREN F. Fully homomorphic encryption with relatively small key and ciphertext sizes[C]. International Conference on Practice and Theory in Public- Key Cryptography, Berlin, Heidelberg, 2010: 420-443. doi: 10.1007/978-3-642-13013-7_25.
[3]
DIJK M, GENTRY C, HALEVI S, et al. Fully homomorphic encryption over the integers[C]. Advances in Cryptology- EUROCRYPT 2010, Berlin, Heidelberg, 2010: 24-43.
[4]
CORON J S, NACCACHE D, and TIBOUCHI M. Public key compression and modulus switching for fully homomorphic encryption over the integers[C]. Advances in Cryptology-EUROCRYPT 2012, Berlin, Heidelberg, 2012: 446-464. doi: 10.1007/978-3-642-29011-4_27.
[5]
CORON J S, MANDAL A, NACCACHE D, et al. Fully homomorphic encryption over the integers with shorter public keys[C]. Advances in Cryptology-CRYPTO 2011, Berlin, Heidelberg, 2011: 487-504. doi: 10.1007/978-3-642- 22792-9_28.
[6]
CHEON J H and STEHL D. Fully homomophic encryption over the integers revisited[C]. Advances in Cryptology- EUROCRYPT 2015, Sofia, Bulgaria, 2015: 513-536. doi: 10.1007/978-3-662-46800-5_20.
[7]
BRAKERSKI Z and VAIKUNTANATHAN V. Efficient fully homomorphic encryption from (standard) LWE[C]. IEEE 52nd Annual Symposium on Foundations of Computer Science, Los Alamitos, 2011: 97-106. doi: 10.1109/FOCS. 2011.12.
[8]
BRAKERSKI Z. Fully homomorphic encryption without modulus switching from classical gapsvp[C]. Advances in Cryptology-CRYPTO 2012, Berlin, Heidelberg, 2012: 868-886. doi: 10.1007/978-3-642-32009-5_50.
[9]
BRAKERSKI Z, GENTRY C, and VAIKUNTANATHAN V. (Leveled) Fully homomorphic encryption without bootstrapping[C]. The 3rd Innovations in Theoretical Computer Science Conference, Cambridge, Massachusetts, 2012: 1-36. doi: 10.1145/2090236.2090262.
[10]
GENTRY C, SAHAI A, and WATERS B. Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-Based[C]. Advances in Cryptology - CRYPTO 2013, Berlin, Heidelberg, 2013: 75-92. doi: 10.1007/978-3-642-40041-4_5.
[11]
REGEV O. On lattices, learning with errors, random linear codes, and cryptography[C]. The 37th Annual ACM Symposium on Theory of Computing, Baltimore, 2005: 84-93. doi: 10.1145/1060590.1060603.
[12]
COSTACHE A and SMART N P. Which ring based somewhat homomorphic encryption scheme is best?[C]. CT-RSA 2016, San Francisco, CA, 2016: 325-340. doi: 10.1007/978-3-319-29485-8_19.
[13]
GENTRY C, HALEVI S, and SMART N. Fully homomorphic encryption with polylog overhead[C]. Advances in Cryptology-EUROCRYPT 2012, Berlin, Heidelberg, 2012: 465-482. doi: 10.1007/978-3-642-29011-4_28.
[14]
OZTURK E, DOROZ Y, SAVAS E, et al. A custom accelerator for homomorphic encryption applications[J]. IEEE Transactions on Computers, 2017, 66(1): 3-16. doi: 10.1109/TC.2016.2574340.
[15]
CANETTI R, RAGHURAMAN S, RICHELSON S, et al. Chosen-ciphertext secure fully homomorphic encryption[C]. International Conference on Practice and Theory in Public- Key Cryptography, Amsterdam, 2017: 213-240. doi: 10.1007/ 978-3-662-54388-7_8.
[16]
GAVIN G. An efficient somewhat homomorphic encryption scheme based on factorization[C]. The 15th International Conference Cryptology and Network Security, Milan, 2016: 451-464. doi: 10.1007/978-3-319-48965-0_27.
[17]
BENARROCH D, BRAKERSKI Z, and LEPOINT T. FHE over the integers: decomposed and batched in the post-quantum regime[C]. International Conference on Practice and Theory in Public-Key Cryptography, Amsterdam, Netherlands, 2017: 271-301. doi: 10.1007/978- 3-662-54388-7_10.
[18]
CHILLOTTI I, GAMA N, GEORGIEVA M, et al. Faster fully homomorphic encryption: Bootstrapping in less than 0.1 seconds[C]. International Conference on the Theory and Application of Cryptology and Information Security, Hanoi, Vietnam, 2016: 3-33. doi: 10.1007/978-3-662-53887-6_1.
[19]
HALEVI S and SHOUP V. Algorithms in HElib[C]. Advances in Cryptology-CRYPTO 2014, Santa Barbara, CA, 2014: 554-571. doi: 10.1007/978-3-662-44371-2_31.
[20]
CHEN H, LAINE K, PLAYER R, et al. Simple encrypted arithmetic library-SEAL v2.1[C]. Proceedings of the Financial Cryptography and Data Security, Sliema, Malta, 2017: 3-18. doi: 10.1007/978-3-319-70278-0_1.
[21]
CROCKETT E and PEIKERT C. : Functional lattice cryptography[C]. Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, Vienna, Austria, 2016: 993-1005. doi: 10.1145/2976749. 2978402.
[22]
L PEZ-ALT A, TROMER E, and VAIKUNTANATHAN V. On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption[C]. Proceedings of the 44th Symposium on Theory of Computing, New York, USA, 2012: 1219-1234. doi: 10.1145/2213977.2214086.
