Abstract
One way of improving efficiency of Gentry’s fully homomorphic encryption is controlling the number of operations, but our recollection is that any scheme which controls the bound has not proposed.
In this paper, we propose a key generation algorithm for Gentry’s homomorphic encryption scheme that controls the bound of the circuit depth by using the relation between the circuit depth and the eigenvalues of a basis of a lattice. We present experimental results that show that the proposed algorithm is practical. We discuss security of the basis of the lattices generated by the algorithm for practical use.
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References
Ajtai, M., Kumar, R., Sivakumar, D.: A Sieve Algorithm for the Shortest Lattice Vector Problem. In: STOC 2001, pp. 266–275 (2001)
Cohen, H.: A Course in Computational Algebraic Number Theory. In: GTM138. Springer, Heidelberg (1996)
ElGamal, T.: A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. IEEE Transactions on Information Theory IT-31, 469–472 (1985)
Gama, N., Nguyen, P.Q.: Predicting Lattice Reduction. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 31–51. Springer, Heidelberg (2008), http://www.di.ens.fr/~pnguyen/pub_GaNg08.htm
Gentry, C.: Fully Homomorphic Encryption Using Ideal Lattices. In: STOC 2009, pp. 169–178 (2009)
Gentry, C.: A Fully Homomorphic Encryption Scheme. PhD thesis, Stanford University (2009), http://crypto.stanford.edu/craig
Gentry, C., Halevi, S.: A Working Implementation of Fully Homomorphic Encryption. In: EUROCRYPT 2010 rump session (2010), http://eurocrypt2010rump.cr.yp.to/9854ad3cab48983f7c2c5a2258e27717.pdf
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)
Gray, R.M.: Toeplitz and Circulant Matrices: A Review. In: Foundation and Trends in Communications and Information Theory, vol. 2(3), Now Publishers Inc., USA (2006)
Hoffstein, J., Pipher, J., Silverman, J.: NTRU: A Ring Based Public Key Cryptosystem. In: Buhler, J.P. (ed.) ANTS 1998. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)
Kitaev, A.Y., Shen, A.H., Vyalyi, M.N.: Classical and Quantum Computation. Graduate Studies in Mathematics, vol. 47. AMS, Providence (2002)
Lenstra, A.K., Lenstra Jr., H.W., Lov’asz, L.: Factoring Polynomials with Rational Coefficients. Mathematische Annalen 261, 513–534 (1982)
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)
Okamoto, T., Uchiyama, S.: A New Public-Key Cryptosystem as Secure as Factoring. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 308–318. Springer, Heidelberg (1998)
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)
Rivest, R.L., Shamir, A., Adleman, L.: A Method for Obtaining Digital Signatures and Public-Key Cryptosystems. Communications of ACM 21(2), 120–126 (1978)
Schnorr, C.P.: A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms. Theoretical Computer Science 53(2-3), 201–224 (1987)
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), http://eprint.iacr.org/2009/571
Stehl’e, D., Steinfeld, R.: Faster Fully Homomorphic Encryption. In: Cryptology ePrint archive (2010), http://eprint.iacr.org/2010/299
Turing Machines, http://www.math.ku.dk/~wester/turing.html
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Ogura, N., Yamamoto, G., Kobayashi, T., Uchiyama, S. (2010). An Improvement of Key Generation Algorithm for Gentry’s Homomorphic Encryption Scheme. In: Echizen, I., Kunihiro, N., Sasaki, R. (eds) Advances in Information and Computer Security. IWSEC 2010. Lecture Notes in Computer Science, vol 6434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16825-3_6
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DOI: https://doi.org/10.1007/978-3-642-16825-3_6
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