Article

Fully homomorphic encryption over the integers with shorter public keys

Authors:

Jean-Sébastien Coron,

Avradip Mandal,

David Naccache,

Mehdi TibouchiAuthors Info & Claims

CRYPTO'11: Proceedings of the 31st annual conference on Advances in cryptology

Pages 487 - 504

Published: 14 August 2011 Publication History

Abstract

At Eurocrypt 2010 van Dijk et al. described a fully homomorphic encryption scheme over the integers. The main appeal of this scheme (compared to Gentry's) is its conceptual simplicity. This simplicity comes at the expense of a public key size in Õ(λ¹⁰) which is too large for any practical system. In this paper we reduce the public key size to Õ(λ⁷) by encrypting with a quadratic form in the public key elements, instead of a linear form. We prove that the scheme remains semantically secure, based on a stronger variant of the approximate-GCD problem, already considered by van Dijk et al.

We alsodescribe the first implementation of the resulting fully homomorphic scheme. Borrowing some optimizations from the recent Gentry-Halevi implementation of Gentry's scheme, we obtain roughly the same level of efficiency. This shows that fully homomorphic encryption can be implemented using simple arithmetic operations.

References

[1]

Bach, E.: How to generate factored random numbers. SIAM J. Comput. 17, 179-193 (1988)

Digital Library

[2]

Boyar, J., Peralta, R., Pochuev, D.: On the multiplicative complexity of boolean functions over the basis (∧,⊕, 1). Theor. Comput. Sci. 235(1), 43-57 (2000)

Digital Library

[3]

Coron, J.S., Mandal, A., Naccache, D., Tibouchi, M.: Fully Homomorphic Encryption over the Integers with Shorter Public Keys, http://eprint.iacr.org

[4]

van Dijk, M., Gentry, C., Halevi, S., Vaikuntanathan, V.: Fully Homomorphic Encryption over the Integers. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 24-43. Springer, Heidelberg (2010)

Digital Library

[5]

Gentry, C.: A fully homomorphic encryption scheme. Ph.D. thesis, Stanford University (2009), http://crypto.stanford.edu/craig

Digital Library

[6]

Gentry, C., Halevi, S.: Implementing Gentry's Fully-Homomorphic Encryption Scheme. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 129- 148. Springer, Heidelberg (2011)

Digital Library

[7]

Goldreich, O., Goldwasser, S., Halevi, S.: Public-key cryptosystems from lattice reduction problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112-131. Springer, Heidelberg (1997)

Digital Library

[8]

Grandlung, T., et al.: The GNU Multiple Precision arithmetic library, Version 4.3.2 (2010), http://gmplib.org

[9]

Lidl, R., Niederreiter, H.: Finite Fields. In: Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley, Reading (1983)

[10]

Nguyêen, P.Q., Stern, J.: The Two Faces of Lattices in Cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146-180. Springer, Heidelberg (2001)

Digital Library

[11]

Nguyen, P.Q.: Personal Communication

[12]

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)

Digital Library

[13]

Stein, W.A., et al.: Sage Mathematics Software (Version 4.5.3), The Sage Development Team (2010), http://www.sagemath.org

[14]

Micciancio, D.: Improving Lattice Based Cryptosystems Using the Hermite Normal Form. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 126-145. Springer, Heidelberg (2001)

Digital Library

[15]

Pujol, X., Stehlé, D., et al.: Fplll lattice reduction library, http://perso.ens-lyon.fr/xavier.pujol/fplll/

[16]

Smart, N.P., Vercauteren, F.: Fully Homomorphic Encryption with Relatively Small Key and Ciphertext Sizes. In: Nguyen, P.Q., Pointcheval, D. (eds.) PKC 2010. LNCS, vol. 6056, pp. 420-443. Springer, Heidelberg (2010)

Digital Library

[17]

Stehlé, D., Steinfeld, R.: Faster Fully Homomorphic Encryption. In: Abe, M. (ed.) ASIACRYPT 2010. LNCS, vol. 6477, pp. 377-394. Springer, Heidelberg (2010)

[18]

Stehlé, D., Zimmermann, P.: A binary recursive gcd algorithm. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 411-425. Springer, Heidelberg (2004)

[19]

Wegman, M.N., Carter, J.L.: New hash functions and their use in authentication and set equality. Journal of Computer and System Sciences 22(3), 265-279 (1981)

Cited By

Wood ANajarian KKahrobaei D(2020)Homomorphic Encryption for Machine Learning in Medicine and BioinformaticsACM Computing Surveys10.1145/339465853:4(1-35)Online publication date: 25-Aug-2020
https://dl.acm.org/doi/10.1145/3394658
El-Yahyaoui AEl Kettani M(2019)About Fully Homomorphic Encryption Improvement TechniquesInternational Journal of Embedded and Real-Time Communication Systems10.4018/IJERTCS.201907010110:3(1-20)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.4018/IJERTCS.2019070101
Prema N(2019)Efficient Secure Aggregation in VANETs Using Fully Homomorphic Encryption (FHE)Mobile Networks and Applications10.1007/s11036-018-1095-y24:2(434-442)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s11036-018-1095-y
Show More Cited By

Fully homomorphic encryption over the integers with shorter public keys

Recommendations

Practical convertible authenticated encryption schemes using self-certified public keys

A convertible authenticated encryption scheme allows a designated receiver to recover and verify a message simultaneously, during which the recipient can prove the dishonesty of the sender to any third party if the sender repudiates her signature later. ...
Bootstrappable Identity-Based Fully Homomorphic Encryption
Proceedings of the 13th International Conference on Cryptology and Network Security - Volume 8813

It has been an open problem for a number of years to construct an identity-based fully homomorphic encryption IBFHE scheme first mentioned by Naccache at CHES/CRYPTO 2010. At CRYPTO 2013, Gentry, Sahai and Waters largely settled the problem by ...
CRT-based fully homomorphic encryption over the integers

In 1978, Rivest, Adleman and Dertouzos introduced the basic concept of privacy homomorphism that allows computation on encrypted data without decryption. It was an interesting work whose idea precedes the recent development of fully homomorphic ...

Comments

Information & Contributors

Information

Published In

cover image Guide Proceedings

CRYPTO'11: Proceedings of the 31st annual conference on Advances in cryptology

August 2011

779 pages

ISBN:9783642227912

Editor:
Phillip Rogaway
University of California, Department of Computer Science, Davis, CA

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 14 August 2011

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

44
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 06 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Wood ANajarian KKahrobaei D(2020)Homomorphic Encryption for Machine Learning in Medicine and BioinformaticsACM Computing Surveys10.1145/339465853:4(1-35)Online publication date: 25-Aug-2020
https://dl.acm.org/doi/10.1145/3394658
El-Yahyaoui AEl Kettani M(2019)About Fully Homomorphic Encryption Improvement TechniquesInternational Journal of Embedded and Real-Time Communication Systems10.4018/IJERTCS.201907010110:3(1-20)Online publication date: 1-Jul-2019
https://dl.acm.org/doi/10.4018/IJERTCS.2019070101
Prema N(2019)Efficient Secure Aggregation in VANETs Using Fully Homomorphic Encryption (FHE)Mobile Networks and Applications10.1007/s11036-018-1095-y24:2(434-442)Online publication date: 1-Apr-2019
https://dl.acm.org/doi/10.1007/s11036-018-1095-y
Bocu RCostache C(2018)A homomorphic encryption-based system for securely managing personal health metrics dataIBM Journal of Research and Development10.1147/JRD.2017.275552462:1(1:1-1:10)Online publication date: 1-Jan-2018
https://dl.acm.org/doi/10.1147/JRD.2017.2755524
Crockett EPeikert CSharp CLie DMannan MBackes MWang X(2018)ALCHEMYProceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security10.1145/3243734.3243828(1020-1037)Online publication date: 15-Oct-2018
https://dl.acm.org/doi/10.1145/3243734.3243828
Acar AAksu HUluagac AConti M(2018)A Survey on Homomorphic Encryption SchemesACM Computing Surveys10.1145/321430351:4(1-35)Online publication date: 25-Jul-2018
https://dl.acm.org/doi/10.1145/3214303
Kim JKoo DKim YYoon HShin JKim S(2018)Efficient Privacy-Preserving Matrix Factorization for Recommendation via Fully Homomorphic EncryptionACM Transactions on Privacy and Security10.1145/321250921:4(1-30)Online publication date: 27-Jun-2018
https://dl.acm.org/doi/10.1145/3212509
Shan ZRen KBlanton MWang C(2018)Practical Secure Computation OutsourcingACM Computing Surveys10.1145/315836351:2(1-40)Online publication date: 20-Feb-2018
https://dl.acm.org/doi/10.1145/3158363
Bogos SGaspoz JVaudenay S(2018)Cryptanalysis of a homomorphic encryption schemeCryptography and Communications10.1007/s12095-017-0243-810:1(27-39)Online publication date: 1-Jan-2018
https://dl.acm.org/doi/10.1007/s12095-017-0243-8
Youn TJho NChang K(2018)Design of additive homomorphic encryption with multiple message spaces for secure and practical storage services over encrypted dataThe Journal of Supercomputing10.1007/s11227-016-1796-674:8(3620-3638)Online publication date: 1-Aug-2018
https://dl.acm.org/doi/10.1007/s11227-016-1796-6
Show More Cited By

View Options

View options

Figures

Tables

Media

View Table of Conten