Abstract
We propose multi-bit versions of several single-bit cryptosystems based on lattice problems, the error-free version of the Ajtai-Dwork cryptosystem by Goldreich, Goldwasser, and Halevi [CRYPTO ’97], the Regev cryptosystems [JACM 2004 and STOC 2005], and the Ajtai cryptosystem [STOC 2005]. We develop a universal technique derived from a general structure behind them for constructing their multi-bit versions without increase in the size of ciphertexts. By evaluating the trade-off between the decryption errors and the hardness of underlying lattice problems, it is shown that our multi-bit versions encrypt O(logn)-bit plaintexts into ciphertexts of the same length as the original ones with reasonable sacrifices of the hardness of the underlying lattice problems. Our technique also reveals an algebraic property, named pseudohomomorphism, of the lattice-based cryptosystems.
Keywords
- Lattice Problem
- Security Proof
- Encryption Function
- Decryption Error
- Universal Technique
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Chapter PDF
References
Ajtai, M.: Generating hard instances of lattice problems. Electronic Colloquium on Computational Complexity (ECCC) 3(007) (1996)
Ajtai, M., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: STOC ’97, 284–293 (1997), See also ECCC TR96-065
Goldreich, O., Goldwasser, S., Halevi, S.: Eliminating decryption errors in the Ajtai-Dwork cryptosystem. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 105–111. Springer, Heidelberg (1997), See also ECCC TR097-018
Regev, O.: New lattice based cryptographic constructions. In: STOC 2003, pp. 407–416 (2003)
Ajtai, M.: Representing hard lattices with O(n logn) bits. In: STOC 2005, pp. 94–103 (2005)
Regev, O.: On lattices, learning with errors, random linear codes, and cryptography. In: STOC 2005, pp. 84–93 (2005)
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)
Hoffstein, J., Pipher, J., Silverman, J.H.: NTRU: A ring-based public key cryptosystem. In: Buhler, J.P. (ed.) Algorithmic Number Theory. LNCS, vol. 1423, pp. 267–288. Springer, Heidelberg (1998)
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)
Nguyen, P.Q.: Analysis and improvements of NTRU encryption paddings. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 210–225. Springer, Heidelberg (2002)
Howgrave-Graham, N., Nguyen, P.Q., Pointcheval, D., Proos, J., Silverman, J.H., Singer, A., Whyte, W.: The impact of decryption failures on the security of NTRU encryption. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 226–246. Springer, Heidelberg (2003)
Nguyen, P.Q.: Cryptanalysis of the Goldreich-Goldwasser-Halevi cryptosystem from Crypto ’97. In: Wiener, M.J. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 288–304. Springer, Heidelberg (1999)
Gentry, C.: Key recovery and message attacks on NTRU-composite. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 182–194. Springer, Heidelberg (2001)
Micciancio, D., Regev, O.: Worst-case to average-case reductions based on Gaussian measures. In: FOCS 2004, pp. 372–381 (2004)
Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. Electronic Colloquium on Computational Complexity (ECCC) 11(095) (2004)
Peikert, C., Rosen, A.: Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006)
Nguyen, P.Q., Stern, J.: Cryptanalysis of the Ajtai-Dwork cryptosystem. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 223–242. Springer, Heidelberg (1998)
Rappe, D.: Homomorphic Cryptosystems and Their Applications. PhD thesis, University of Dortmund (2004), Also available at, http://eprint.iacr.org/2006/001
Goldwasser, S., Kharchenko, D.: Proof of plaintext knowledge for the Ajtai-Dwork cryptosystem. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 529–555. Springer, Heidelberg (2005)
Micciancio, D., Goldwasser, S.: Complexity of Lattice Problems: a cryptographic perspective. Kluwer Academic Publishers, Boston (2002)
Cai, J.Y.: A new transference theorem in the geometry of numbers and new bounds for Ajtai’s connection factor. Discrete Applied Mathematics 126(1), 9–31 (2003)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58(301), 13–30 (1963)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Kawachi, A., Tanaka, K., Xagawa, K. (2007). Multi-bit Cryptosystems Based on Lattice Problems. In: Okamoto, T., Wang, X. (eds) Public Key Cryptography – PKC 2007. PKC 2007. Lecture Notes in Computer Science, vol 4450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71677-8_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-71677-8_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71676-1
Online ISBN: 978-3-540-71677-8
eBook Packages: Computer ScienceComputer Science (R0)