More Generalized Mersenne Numbers

  • Jaewook Chung
  • Anwar Hasan
Conference paper

DOI: 10.1007/978-3-540-24654-1_24

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3006)
Cite this paper as:
Chung J., Hasan A. (2004) More Generalized Mersenne Numbers. In: Matsui M., Zuccherato R.J. (eds) Selected Areas in Cryptography. SAC 2003. Lecture Notes in Computer Science, vol 3006. Springer, Berlin, Heidelberg

Abstract

In 1999, Jerome Solinas introduced families of moduli called the generalized Mersenne numbers. The generalized Mersenne numbers are expressed in a polynomial form, p = f(t), where t is a power of 2. It is shown that such p’s lead to fast modular reduction methods which use only a few integer additions and subtractions. We further generalize this idea by allowing any integer for t. We show that more generalized Mersenne numbers still lead to a significant improvement over well-known modular multiplication techniques. While each generalized Mersenne number requires a dedicated implementation, more generalized Mersenne numbers allow flexible implementations that work for more than one modulus. We also show that it is possible to perform long integer modular arithmetic without using multiple precision operations when t is chosen properly. Moreover, based on our results, we propose efficient arithmetic methods for XTR cryptosystem.

Keywords

Generalized Mersenne Numbers RSA XTR Montgomery Karatsuba-Ofman Modular Reduction 

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jaewook Chung
    • 1
  • Anwar Hasan
    • 1
  1. 1.Centre for Applied Cryptographic ResearchUniversity of WaterlooCanada

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