Abstract
Homomorphic encryption allows to perform various calculations on encrypted data without decryption. In this paper, we propose an efficient method for secure multiple matrix multiplications over the somewhat homomorphic encryption scheme proposed by Brakerski and Vaikuntanathan. Our method is a generalization of Duong et al.’s method, which computes only one multiplication between two matrices. In order to minimize both the ciphertext size and the computation cost, our method packs every matrix into a single ciphertext so that it enables efficient matrix multiplications over the packed ciphertexts. We also propose several modifications to obtain practical performance of secure multiplications among matrices with larger size and entries. We show implementation results of our packing method with modifications for secure multiplications among two and three matrices with \(32 \times 32\) and \(64 \times 64\) sizes and entries from 16-bit to 64-bit.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Boneh, D., Goh, E.-J., Nissim, K.: Evaluating 2-DNF formulas on ciphertexts. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 325–341. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30576-7_18
Bos, J.W., Lauter, K., Loftus, J., Naehrig, M.: Improved security for a ring-based fully homomorphic encryption scheme. In: Stam, M. (ed.) IMACC 2013. LNCS, vol. 8308, pp. 45–64. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45239-0_4
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. ACM Trans. Comput. Theory (TOCT) 6(3) (2014). Article No. 13, Special issue on innovations in theoretical computer science 2012-Part II
Brakerski, Z., Vaikuntanathan, V.: Fully homomorphic encryption from ring-LWE and security for key dependent messages. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 505–524. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22792-9_29
Chillotti, I., Gama, N., Georgieva, M., Izabachène, M.: Faster fully homomorphic encryption: bootstrapping in less than 0.1 seconds. In: Cheon, J.H., Takagi, T. (eds.) ASIACRYPT 2016. LNCS, vol. 10031, pp. 3–33. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53887-6_1
Costache, A., Smart, N.P.: Which ring based somewhat homomorphic encryption scheme is best? In: Sako, K. (ed.) CT-RSA 2016. LNCS, vol. 9610, pp. 325–340. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29485-8_19
Duong, D.H., Mishra, P.K., Yasuda, M.: Efficient secure matrix multiplication over LWE-based homomorphic encryption. Tatra Mountains Math. Publ. 67(1), 69–83 (2016)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Symposium on Theory of Computing-STOC 2009, pp. 169–178. ACM (2009)
Gentry, C., Halevi, S., Smart, N.P.: Homomorphic evaluation of the AES circuit. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 850–867. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_49
Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. IACR Cryptology ePrint 2012/144 (2014). https://eprint.iacr.org/2012/144
Naehrig, M., Lauter, K.E., Vaikuntanathan, V.: Can homomorphic encryption be practical? In: ACM Workshop on Cloud Computing Security Workshop-CCSW 2011, pp. 113–124. ACM (2011)
Lepoint, T., Naehrig, M.: A comparison of the homomorphic encryption schemes FV and YASHE. In: Pointcheval, D., Vergnaud, D. (eds.) AFRICACRYPT 2014. LNCS, vol. 8469, pp. 318–335. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06734-6_20
Lòpez-Alt, A., Tromer, E., Vaikuntanathan, V.: On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Symposium on Theory of Computing-STOC 2012, pp. 1219–1234. ACM (2012)
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). https://doi.org/10.1007/3-540-48910-X_16
Pedrouzo-Ulloa, A., Troncoso-Pastoriza, J.R., Pérez-González, F.: Number theoretic transforms for secure signal processing. IEEE Trans. Inf. Forensics Secur. 12(5), 1125–1140 (2017)
The PARI Group, Bordeaux, PARI/GP. http://pari.math.u-bordeaux.fr/doc.html
Rivest, R.L., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21(2), 120–126 (1978)
Yasuda, M., Shimoyama, T., Kogure, J., Yokoyama, K., Koshiba, T.: New packing method in somewhat homomorphic encryption and its applications. Secur. Commun. Netw. (SCN) 8(13), 2194–2213 (2015)
Yasuda, M., Shimoyama, T., Kogure, J., Yokoyama, K., Koshiba, T.: Secure statistical analysis using RLWE-based homomorphic encryption. In: Foo, E., Stebila, D. (eds.) ACISP 2015. LNCS, vol. 9144, pp. 471–487. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19962-7_27
Acknowledgments
This work was supported by JST CREST Grant Number JPMJCR14D6, Japan. This work was also supported by JSPS KAKENHI Grant Numbers 16K17644 and 16H02830.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Mishra, P.K., Duong, D.H., Yasuda, M. (2017). Enhancement for Secure Multiple Matrix Multiplications over Ring-LWE Homomorphic Encryption. In: Liu, J., Samarati, P. (eds) Information Security Practice and Experience. ISPEC 2017. Lecture Notes in Computer Science(), vol 10701. Springer, Cham. https://doi.org/10.1007/978-3-319-72359-4_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-72359-4_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72358-7
Online ISBN: 978-3-319-72359-4
eBook Packages: Computer ScienceComputer Science (R0)