Abstract
Based on the FV scheme, we construct at first fully homomorphic encryption scheme \(\mathsf {FX}\) that can homomorphically compute addition and multiplication of encrypted fixed point numbers without knowing the secret key. Then, we show that in the \(\mathsf {FX}\) scheme one can efficiently and homomorphically compare magnitude of two encrypted numbers. That is, one can compute an encryption of the greater-than bit that indicates \(x > x'\) or not, given two ciphertexts c and \(c'\) of x and \(x'\), respectively, without knowing the secret key. Finally we show that these properties of the \(\mathsf {FX}\) scheme enables us to construct a fully homomorphic encryption scheme \(\mathsf {FL}\) that can homomorphically compute addition and multiplication of encrypted floating point numbers.
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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). doi:10.1007/978-3-642-45239-0_4
Brakerski, Z.: Fully homomorphic encryption without modulus switching from classical GapSVP. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 868–886. Springer, Heidelberg (2012). doi:10.1007/978-3-642-32009-5_50
Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S. (ed.) ITCS, pp. 309–325. ACM (2012)
Cheon, J.H., Coron, J.-S., Kim, J., Lee, M.S., Lepoint, T., Tibouchi, M., Yun, A.: Batch fully homomorphic encryption over the integers. In: Johansson, T., Nguyen, P.Q. (eds.) EUROCRYPT 2013. LNCS, vol. 7881, pp. 315–335. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38348-9_20
Chung, H.W., Kim, M.: Encoding rational numbers for FHE-based applications. IACR Cryptology ePrint Archive (2016/344)
Cheon, J.H., Kim, M., Lauter, K.: Homomorphic computation of edit distance. In: Brenner, M., Christin, N., Johnson, B., Rohloff, K. (eds.) FC 2015. LNCS, vol. 8976, pp. 194–212. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48051-9_15
Costache, A., Smart, N.P., Vivek, S., Waller, A.: Fixed-point arithmetic in SHE schemes. IACR Cryptology ePrint Archive (2016/250)
Fan, J., Vercauteren, F.: Somewhat practical fully homomorphic encryption. IACR Cryptology ePrint Archive (2012/144)
Gentry, C.: Fully homomorphic encryption using ideal lattices. In: Mitzenmacher, M. (ed.) STOC, pp. 169–178. ACM (2009)
Gentry, C., Halevi, S., Smart, N.P.: Better bootstrapping in fully homomorphic encryption. In: Fischlin, M., Buchmann, J., Manulis, M. (eds.) PKC 2012. LNCS, vol. 7293, pp. 1–16. Springer, Heidelberg (2012). doi:10.1007/978-3-642-30057-8_1
Golle, P.: A private stable matching algorithm. In: Crescenzo, G., Rubin, A. (eds.) FC 2006. LNCS, vol. 4107, pp. 65–80. Springer, Heidelberg (2006). doi:10.1007/11889663_5
Graepel, T., Lauter, K., Naehrig, M.: ML confidential: machine learning on encrypted data. In: Kwon, T., Lee, M.-K., Kwon, D. (eds.) ICISC 2012. LNCS, vol. 7839, pp. 1–21. Springer, Heidelberg (2013). doi:10.1007/978-3-642-37682-5_1
Ishimaki, Y., Shimizu, K., Nuida, K., Yamana, H.: Faster privacy-preserving search for genome sequences using fully homomorphic encryption. In: SCIS 2016, Japan (2016)
Lu, W-J., Kawasaki, S., Sakuma, J.: Cryptographically-secure outsourcing of statistical data analysis I: descriptive statistics. In: CSS 2015, Japan (2015)
Lauter, K., López-Alt, A., Naehrig, M.: Private computation on encrypted genomic data. In: Aranha, D.F., Menezes, A. (eds.) LATINCRYPT 2014. LNCS, vol. 8895, pp. 3–27. Springer, Cham (2015). doi:10.1007/978-3-319-16295-9_1
Liu, J., Li, J., Xu, S., Fung, B.C.M.: Secure outsourced frequent pattern mining by fully homomorphic encryption. In: Madria, S., Hara, T. (eds.) DaWaK 2015. LNCS, vol. 9263, pp. 70–81. Springer, Cham (2015). doi:10.1007/978-3-319-22729-0_6
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). doi:10.1007/978-3-319-06734-6_20
Lyubashevsky, V., Peikert, C., Regev, O.: On ideal lattices and learning with errors over rings. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 1–23. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13190-5_1
Alperin-Sheriff, J., Peikert, C.: Practical bootstrapping in quasilinear time. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 1–20. Springer, Heidelberg (2013). doi:10.1007/978-3-642-40041-4_1
Smart, N.P., Vercauteren, F.: Fully homomorphic SIMD operations. Des. Codes Crypt. 71(1), 57–81 (2014)
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This work was supported by CREST, JST.
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Arita, S., Nakasato, S. (2017). Fully Homomorphic Encryption for Point Numbers. In: Chen, K., Lin, D., Yung, M. (eds) Information Security and Cryptology. Inscrypt 2016. Lecture Notes in Computer Science(), vol 10143. Springer, Cham. https://doi.org/10.1007/978-3-319-54705-3_16
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