Efficient Multiplication over Extension Fields

  • Nadia El Mrabet
  • Nicolas Gama
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7369)


The efficiency of cryptographic protocols rely on the speed of the underlying arithmetic and finite field computation. In the literature, several methods on how to improve the multiplication over extensions fields \(\mathbb{F}_{q^{m}}\), for prime q were developped. These optimisations are often related to the Karatsuba and Toom Cook methods. However, the speeding-up is only interesting when m is a product of powers of 2 and 3. In general cases, a fast multiplication over \(\mathbb{F}_{q^{m}}\) is implemented through the use of the naive school-book method. In this paper, we propose a new efficient multiplication over \(\mathbb{F}_{q^{m}}\) for any power m. The multiplication relies on the notion of Adapted Modular Number System (AMNS), introduced in 2004 by [3]. We improve the construction of an AMNS basis and we provide a fast implementation of the multiplication over \(\mathbb{F}_{q^{m}}\), which is faster than GMP and NTL.


Discrete Fourier Transform Elliptic Curve Cryptography Short Vector Fast Fourier Transformation Method Homomorphic Encryption Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Nadia El Mrabet
    • 1
  • Nicolas Gama
    • 2
  1. 1.LIASD - Université Paris 8France
  2. 2.Université de Versailles - PRISM - CNRSFrance

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