Document Zbl 1081.94021 - zbMATH Open
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More generalized Mersenne numbers. (English) Zbl 1081.94021
Matsui, Mitsuru (ed.) et al., Selected areas in cryptography. 10th annual international workshop, SAC 2003, Ottawa, Canada, August 14–15, 2003. Revised papers. Berlin: Springer (ISBN 3-540-21370-8/pbk). Lecture Notes in Computer Science 3006, 335-347 (2004).
Summary: In 1999, J. Solinas [Generalized Mersenne numbers. Technical Report CORR 99-39, Centre for Applied Cryptographic Research, University of Waterloo (1999), http://www.cacr.math.uwaterloo.ca] 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.
For the entire collection see [Zbl 1051.94004].