Soft Computing

, Volume 23, Issue 5, pp 1735–1744 | Cite as

A multi-key SMC protocol and multi-key FHE based on some-are-errorless LWE

  • Huiyong Wang
  • Yong Feng
  • Yong DingEmail author
  • Shijie Tang
Methodologies and Application


We study the hardness of learning with errors (LWE) problem with some equation constraints (\(\hbox {LWE}_{n,l,q,\chi }\), some-are-errorless LWE). Previously, it was proved that LWE with one equation (first-is-errorless LWE) can be made as hard as the standard \(\hbox {LWE}_{n,q,\chi }\), given a large lattice dimension n. We show that the some-are-errorless LWE problem can also be made equivalently hard as long as n is big enough and \(n \gg l\) (A similar conclusion was given using fuzzy extrators by Fuller). A second work in this paper is to construct a multi-key secure multi-party computation (SMC) protocol, whose security relies on LWE and the some-are-errorless LWE problem in semi-honest and semi-malicious environments assuming the common random string model. We study the Gentry–Sahai–Waters (GSW13) fully homomorphic encryption (FHE) scheme and its key homomorphism, which is essential for the construction of our multi-key SMC protocol. The proposed protocol naturally constitutes a multi-key FHE scheme in the same settings. Finally, we show the excellence of the proposed SMC protocol in time and space complexity by comparisons with existing relative schemes.


LWE FHE Key homomorphism SMC 



This work is partially supported by National Natural Science Foundation of China (Grant Nos. 61772150, 61262008) and the open project of Guangxi Key Lab. of Crypto. and Info. Security (Grant Nos. GCIS201621, GCIS201622). We thank Xiaolin Qin, Jikui Wang, Jianguang Lu, Juyi Fan and Zhi Sun for helpful comments and discussions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest with any individual or organization.

Human and animals rights

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent

Informed consent was obtained from all individual participants included in the study.


  1. Agrawal S, Boneh D, Boyen X (2010) Efficient lattice (H) IBE in the standard model. In: Advances in cryptology—EUROCRYPT 2010, Springer, pp 553–572Google Scholar
  2. Ajtai M (1996) Generating hard instances of lattice problems. In: Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, ACM, pp 99–108Google Scholar
  3. Alperin-Sheriff J, Peikert C (2014) Faster bootstrapping with polynomial error. In: Advances in cryptology—CRYPTO 2014, Springer, pp 297–314Google Scholar
  4. Asharov G, Jain A, López-Alt A, Tromer E, Vaikuntanathan V, Wichs D (2012) Multiparty computation with low communication, computation and interaction via threshold FHE. In: Advances in cryptology—EUROCRYPT 2012, Springer, pp 483–501Google Scholar
  5. Boneh D, Lewi K, Montgomery H, Raghunathan A (2013) Key homomorphic PRFs and their applications. In: Advances in cryptology—CRYPTO 2013, Springer, pp 410–428Google Scholar
  6. Brakerski Z, Vaikuntanathan V (2014) Efficient fully homomorphic encryption from (standard) LWE. SIAM J Comput 43(2):831–871MathSciNetCrossRefzbMATHGoogle Scholar
  7. Brakerski Z, Gentry C, Vaikuntanathan V (2012) (Leveled) fully homomorphic encryption without bootstrapping. In: Proceedings of the 3rd innovations in theoretical computer science conference, ACM, pp 309–325Google Scholar
  8. Brakerski Z, Langlois A, Peikert C, Regev O, Stehlé D (2013) Classical hardness of learning with errors. In: Proceedings of the forty-fifth annual ACM symposium on theory of computing, ACM, pp 575–584Google Scholar
  9. Cash D, Hofheinz D, Kiltz E, Peikert C (2012) Bonsai trees, or how to delegate a lattice basis. J Cryptol 25(4):601–639MathSciNetCrossRefzbMATHGoogle Scholar
  10. Clear M, McGoldrick C (2015) Multi-identity and multi-key leveled FHE from learning with errors. In: Advances in cryptology—CRYPTO 2015, Springer, pp 630–656Google Scholar
  11. Gentry C et al (2009) Fully homomorphic encryption using ideal lattices. STOC 9:169–178MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the fortieth annual ACM symposium on theory of computing, ACM, pp 197–206Google Scholar
  13. Gentry C, Sahai A, Waters B (2013) Homomorphic encryption from learning with errors: conceptually-simpler, asymptotically-faster, attribute-based. In: Advances in cryptology—CRYPTO 2013, Springer, pp 75–92Google Scholar
  14. Jiang L, Xu C, Wang X, Lin C (2016) Statistical learning based fully homomorphic encryption on encrypted data. Soft Comput.
  15. Liu Z, Weng J, Li J, Yang J, Fu C, Jia C (2016) Cloud-based electronic health record system supporting fuzzy keyword search. Soft Comput 20(8):3243–3255Google Scholar
  16. Lopez-Alt A, Tromer E, Vaikuntanathan V (2012) On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Proceedings of the forty-fourth annual ACM symposium on theory of computing, ACM, pp 1219–1234Google Scholar
  17. Lyubashevsky V, Micciancio D (2009) On bounded distance decoding, unique shortest vectors, and the minimum distance problem. In: Advances in cryptology—CRYPTO 2009, Springer, pp 577–594Google Scholar
  18. Micciancio D, Goldwasser S (2012) Complexity of lattice problems: a cryptographic perspective, vol 671. Springer, New YorkzbMATHGoogle Scholar
  19. Micciancio D, Peikert C (2012) Trapdoors for lattices: simpler, tighter, faster, smaller. In: Advances in cryptology—EUROCRYPT 2012, Springer, pp 700–718Google Scholar
  20. Mukherjee P, Wichs D (2016) Two round multiparty computation via multi-key FHE. In: Annual international conference on the theory and applications of cryptographic techniques. Springer, Berlin, Heidelberg, pp 735–763Google Scholar
  21. Peikert C (2009) Public-key cryptosystems from the worst-case shortest vector problem. In: Proceedings of the forty-first annual ACM symposium on Theory of computing, ACM, pp 333–342Google Scholar
  22. Peikert C, Shiehian S (2016) Multi-key FHE from lwe, revisited. In: Theory of cryptography conference. Springer, Berlin, Heidelberg. pp 217–238Google Scholar
  23. Peikert C, Waters B (2011) Lossy trapdoor functions and their applications. SIAM J Comput 40(6):1803–1844MathSciNetCrossRefzbMATHGoogle Scholar
  24. Peikert C, Vaikuntanathan V, Waters B (2008) A framework for efficient and composable oblivious transfer. In: Advances in cryptology—CRYPTO 2008, Springer, pp 554–571Google Scholar
  25. Regev O (2009) On lattices, learning with errors, random linear codes, and cryptography. J ACM (JACM) 56(6):34MathSciNetCrossRefzbMATHGoogle Scholar
  26. Regev O (2010) The learning with errors problem (invited survey). In: IEEE conference on computational complexity, IEEE Computer Society, pp 191–204Google Scholar
  27. Reyzin L, Fuller B, Meng X (2013) Computational fuzzy extractors. In: International conference on the theory and application of cryptology and information security, Springer, Berlin, pp 174–193Google Scholar
  28. Ross SM (2014) Introduction to probability and statistics for engineers and scientists. Academic Press, OxfordzbMATHGoogle Scholar
  29. Van Dijk M, Gentry C, Halevi S, Vaikuntanathan V (2010) Fully homomorphic encryption over the integers. In: Advances in cryptology—EUROCRYPT 2010, Springer, pp 24–43Google Scholar
  30. Xiang C, Tang C, Cai Y, Xu Q (2016) Privacy-preserving face recognition with outsourced computation. Soft Comput 20(9):3735–3744Google Scholar
  31. Yao AC (1982) Protocols for secure computations. In: Foundations of computer science, 1982. SFCS’08. 23rd annual symposium on, IEEE, pp 160–164Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of Mathematics and Computing ScienceGuilin University of Electronic TechnologyGuilinPeople’s Republic of China
  2. 2.Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent TechnologyChinese Academy of ScienceChongqingPeople’s Republic of China
  3. 3.Guangxi Key Laboratory of Cryptography and Information Security, School of Computer Science and Information SecurityGuilin University of Electronic TechnologyGuilinPeople’s Republic of China
  4. 4.Guangxi Key Laboratory of Intelligent Integrated Automation, School of Electronic Engineering and AutomationGuilin University of Electronic TechnologyGuilinPeople’s Republic of China

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