Abstract
Finite field multiplication is a crucial building block for cryptography, especially the elliptic curve public key cryptosystem. Recently, various algorithms for efficient finite field multiplication over devices whose resources are extremely constrained have been proposed. However, most of these proposals only take speed optimization into account, but they do not pay much attention to optimization of memory usage. In this paper, we propose a multiplication algorithm on \(F_{2^{m}}\), which minimizes the RAM requirement by rescheduling operation sequences. According to our experimental results on the ATmega128L microprocessor, the proposed algorithm reduces the amount of required RAM by up to 50 % while maintaining the speed at the same level. We also verify the feasibility of our algorithm by applying it to the elliptic curve cryptosystem.
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Notes
Note that a polynomial addition and a polynomial subtraction are the same operations over \({\mathbb{F}}_{2^{m}}\), because a+bmod2=a−bmod2 for a,b∈{0,1}.
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Acknowledgements
This research was supported by Next Generation Information Computing Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (No. 2011-0029925) and Inha University.
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Appendix: Proposed algorithm
Appendix: Proposed algorithm
Proposed method with four temporary memory slots
Proposed method with three temporary memory slots
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Han, T.Y., Lee, MK. Reordering computation sequences for memory-efficient binary field multiplication. J Supercomput 66, 936–949 (2013). https://doi.org/10.1007/s11227-013-0930-y
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DOI: https://doi.org/10.1007/s11227-013-0930-y