Abstract
In 2000, Paulus and Takagi introduced a public key cryptosystem called NICE that exploits the relationship between maximal and non-maximal orders in imaginary quadratic number fields. Relying on the intractability of integer factorization, NICE provides a similar level of security as RSA, but has faster decryption. This paper presents REAL-NICE, an adaptation of NICE to orders in real quadratic fields. REAL-NICE supports smaller public keys than NICE, and while preliminary computations suggest that it is somewhat slower than NICE, it still significantly outperforms RSA in decryption.
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References
Buchmann, J., Sakurai, K., Takagi, T.: An IND-CCA2 public-key cryptosystem with fast decryption. In: Kim, K.-c. (ed.) ICISC 2001. LNCS, vol. 2288, pp. 51–71. Springer, Heidelberg (2002)
Buchmann, J., Thiel, C., Williams, H.: Short representation of quadratic integers. In: Computational Algebra and Number Theory (Sydney, 1992). Math. Appl., vol. 325, pp. 159–185. Kluwer, Dordrecht (1995)
Cheng, K.H.F., Williams, H.C.: Some results concerning certain periodic continued fractions. Acta Arith. 117, 247–264 (2005)
Cox, D.A.: Primes of the Form x 2 + ny 2. John Wiley & Sons, Inc, New York (1989)
Hardy, G.H., Littlewood, J.E.: Partitio numerorum III: On the expression of a number as a sum of primes. Acta Math. 44, 1–70 (1923)
Hühnlein, D., Jacobson Jr., M.J., Paulus, S., Takagi, T.: A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 294–307. Springer, Heidelberg (1998)
Jaulmes, E., Joux, A.: A NICE Cryptanalysis. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 382–391. Springer, Heidelberg (2000)
Jacobson Jr., M.J., Scheidler, R., Stein, A.: Cryptographic protocols on real hyperelliptic curves. Adv. Math. Commun. 1, 197–221 (2007)
Jacobson Jr., M.J., Sawilla, R.E., Williams, H.C.: Efficient Ideal Reduction in Quadratic Fields. Internat. J. Math. Comput. Sci. 1, 83–116 (2006)
Lenstra, A.K.: Unbelievable Security. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 67–86. Springer, Heidelberg (2001)
Mollin, R.A., Williams, H.C.: Computation of the class number of a real quadratic field. Util. Math. 41, 259–308 (1992)
National Institute of Standards and Technology (NIST), Recommendation for key management - part 1: General (revised). NIST Special Publication 800-57 (March 2007), http://csrc.nist.gov/groups/ST/toolkit/documents/SP800-57Part1_3-8-07.pdf
Paulus, S., Takagi, T.: A new public key cryptosystem over quadratic orders with quadratic decryption time. J. Cryptology 13, 263–272 (2000)
van der Poorten, A.J., te Riele, H.J.J., Williams, H.C.: Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000. Math. Comp. 70, 1311–1328 (2001)
Schinzel, A.: On some problems of the arithmetical theory of continued fractions. Acta Arith. 6, 393–413 (1961)
Shoup, V.: NTL: A Library for Doing Number Theory. Software (2001), Available at http://www.shoup.net/ntl
Takagi, T.: Fast RSA-type cryptosystem modulo p k q. In: Krawczyk, H. (ed.) CRYPTO 1998. LNCS, vol. 1462, pp. 318–326. Springer, Heidelberg (1998)
Takagi, T.: A New Public-Key Cryptosystems with Fast Decryption. PhD Thesis, Technische Universität Darmstadt (Germany) (2001)
Weimer, D.: An Adaptation of the NICE Cryptosystem to Real Quadratic Orders. Master’s Thesis, Technische Universität Darmstadt (Germany) (2004), http://www.cdc.informatik.tu-darmstadt.de/reports/reports/DanielWeimer.diplom.pdf
Williams, H.C.: Édouard Lucas and Primality Testing. John Wiley & Sons, New York (1998)
Williams, H.C., Wunderlich, M.C.: On the parallel generation of the residues for the continued fraction factoring algorithm. Math. Comp. 48, 405–423 (1987)
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Jacobson, M.J., Scheidler, R., Weimer, D. (2008). An Adaptation of the NICE Cryptosystem to Real Quadratic Orders. In: Vaudenay, S. (eds) Progress in Cryptology – AFRICACRYPT 2008. AFRICACRYPT 2008. Lecture Notes in Computer Science, vol 5023. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68164-9_13
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DOI: https://doi.org/10.1007/978-3-540-68164-9_13
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