Abstract
In this paper we present a probabilistic polynomial-time algorithm for generating a large prime p such that Φ m (p 2) has a large prime factor, where Φ m (x) is the m − th cyclotomic polynomial and m = 3 or m = 6. An unconditionally polynomial time algorithm for generating primes of the above form is not yet known. Generating primes of such form is essential for the GH and the CEILIDH Public Key Systems, since they are key parameters in these cryptosystems.
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References
Agrawal, M., Kayal, K., Saxena, N.: Primes is P. Ann. of Math. 160, 781–793 (2004)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, New York (1993)
Davenport, H.: Multiplicative Number Theory. Springer, New York (1980)
Gong, G., Harn, L.: Public-Key Cryptosystems Based on Cubic Finite Field Extension. IEEE Transactions on Information Theory 45, 2601–2605 (1999)
Gong, G., Harn, L.: A New Approach on Public-key Distribution. In: Proceedings of China - Crypto, Chengdu, China, pp. 50–55 (1998)
Giuliani, K., Gong, G.: Generating Large Instances of the Gong-Harn Cryptosytem. In: Proceedings of Cryptography and Coding: 8th International Conference Cirencester. LNCS, vol. 2261, pp. 111–133. Springer, Heidelberg (2002)
Grześkowiak, M.: Analysis of Algorithms of Generating Key Parameters for the XTR Cryptosystem. In: Proceedings of Wartacrypt 2004, pp. 1–12. Tatra Mountains Mathematical Publications (2006)
Iwaniec, H.: Primes Represented by Quadratic Polynomials in Two Variables. Acta Arith. 24, 435–459 (1974)
Lenstra, A.K., Verhuel, E.R.: The XTR Public Key System. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 1–19. Springer, Heidelberg (2000)
Rubin, K., Silverberg, A.: Torus-based cryptography. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 349–365. Springer, Heidelberg (2003)
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Grześkowiak, M. (2008). Generating a Large Prime Factor of p 4±p 2 + 1 in Polynomial Time. In: Meersman, R., Tari, Z. (eds) On the Move to Meaningful Internet Systems: OTM 2008. OTM 2008. Lecture Notes in Computer Science, vol 5332. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88873-4_15
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DOI: https://doi.org/10.1007/978-3-540-88873-4_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88872-7
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