Abstract
In 1999 Gong and Harn proposed a new cryptosystem based on third-order characteristic sequences with period p 2k + p k + 1 for a large prime p and fixed k. In order to find key parameters and therefore to construct a polynomial whose characteristic sequence is equal to p 2k + p k + 1 one should generate a prime p such that the prime factorization of the number p 2k + p k + 1 is known. In this paper we propose new, efficient methods for finding the prime p and the factorization of the aforementioned number. Our algorithms work faster in practice than those proposed before. Moreover, when used for generating of XTR key parameters, they are a significant improvement over the Lenstra-Verheul Algorithm. Our methods have been implemented in C++ using LiDIA and numerical test are presented.
Partially supported by Ministry of Science and Higher Education, grant N N206 2701 33, 2007–2010.
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Grześkowiak, M. (2008). On Generating Elements of Orders Dividing p 2k±p k + 1. In: Matsuura, K., Fujisaki, E. (eds) Advances in Information and Computer Security. IWSEC 2008. Lecture Notes in Computer Science, vol 5312. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89598-5_1
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DOI: https://doi.org/10.1007/978-3-540-89598-5_1
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