Abstract
The original algorithm of D.V. Chudnovsky and G.V. Chudnovsky for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Randriambololona generalized the method, allowing asymmetry in the interpolation procedure. The aim of this article is to make effective this method. We first make explicit this generalization in order to construct the underlying asymmetric algorithms. Then, we propose a generic strategy to construct these algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in \(\mathbb {F}_{16^{13}}\) using only rational places, in \(\mathbb {F}_{4^{5}}\) using also places of degree two, and in \(\mathbb {F}_{2^{5}}\) using also places of degree four.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”.
Magma implementation of the multiplication algorithms in the finite fields
Magma implementation of the multiplication algorithms in the finite fields
Appendix A gives a Magma implementation for each of the examples of Sects. 4 to 6.
1.1 \(\mathbb {F}_{16^{13}}\) over \(\mathbb {F}_{16}\)
1.2 \(\mathbb {F}_{4^{5}}\) over \(\mathbb {F}_{4}\)
1.3 \(\mathbb {F}_{2^{5}}\) over \(\mathbb {F}_{2}\)
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Ballet, S., Baudru, N., Bonnecaze, A. et al. Construction of asymmetric Chudnovsky-type algorithms for multiplication in finite fields. Des. Codes Cryptogr. 90, 2783–2811 (2022). https://doi.org/10.1007/s10623-021-00986-1
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DOI: https://doi.org/10.1007/s10623-021-00986-1