Ring-LWE in Polynomial Rings
The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal R ∨ , the dual of the ring R, which is the source of many theoretical and implementation technicalities. Until now, getting rid of R ∨ , required some relatively complex transformation that substantially increase the magnitude of the error polynomial and the practical complexity to sample it. It is only for rings R = ℤ[X]/(Xn + 1) where n a power of 2, that this transformation is simple and benign.
In this work we show that by applying a different, and much simpler transformation, one can transfer the results from [LPR10] into an “easy-to-use” Ring-LWE setting (i.e. without the dual ring R ∨ ), with only a very slight increase in the magnitude of the noise coefficients. Additionally, we show that creating the correct noise distribution can also be simplified by generating a Gaussian distribution over a particular extension ring of R, and then performing a reduction modulo f(X). In essence, our results show that one does not need to resort to using any algebraic structure that is more complicated than polynomial rings in order to fully utilize the hardness of the Ring-LWE problem as a building block for cryptographic applications.
- [BGV11]Brakerski, Z., Gentry, C., Vaikuntanathan, V.: Fully homomorphic encryption without bootstrapping, Cryptology ePrint Archive, Report 2011/277 (2011); To appear at ITCS 2012Google Scholar
- [BV11a]Brakerski, Z., Vaikuntanathan, V.: Efficient fully homomorphic encryption from (standard) LWE. In: FOCS (2011)Google Scholar
- [BV11b]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)Google Scholar
- [Con09]Conrad, K.: The different ideal (2009), http://www.math.uconn.edu/~kconrad/blurbs/
- [DPSZ11]Damgard, I., Pastro, V., Smart, N.P., Zakarias, S.: Multiparty computation from somewhat homomorphic encryption. Cryptology ePrint Archive, Report 2011/535 (2011)Google Scholar
- [Gen10]Gentry, C.: Toward Basing Fully Homomorphic Encryption on Worst-Case Hardness. In: Rabin, T. (ed.) CRYPTO 2010. LNCS, vol. 6223, pp. 116–137. Springer, Heidelberg (2010)Google Scholar
- [LATV11]Lopez-Alt, A., Tromer, E., Vaikuntanathan, V.: Cloud-assisted multiparty computation from fully homomorphic encryption. Cryptology ePrint Archive, Report 2011/663 (2011)Google Scholar
- [Ste05]Stein, W.: Introduction to algebraic number theory (2005), http://wstein.org/courses/