Abstract
Bit-parallel finite field multiplication in F 2m using polynomial basis can be realized in two steps: polynomial multiplication and reduction modulo the irreducible polynomial. In this article, we prove that the modular polynomial reduction can be done with (r − 1)(m − 1) bit additions, where r is the Hamming weight of the irreducible polynomial. We also show that a bit-parallel squaring operation using polynomial basis costs not more than ⌊m+k-1⌋ bit operations if an irreducible trinomial of form x m + x k +1 over F 2 is used. Consequently, it is argued that to solve multiplicative inverse in F m using polynomial basis can be as good as using normal basis.
Acknowledgements
This work was done when the author worked for his Ph.D degree with the Dept of ECE, University of Waterloo. The author thanks Professor Hasan and Professor Blake for their encouragement and valuable comments.
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Wu, H. (1999). Low Complexity Bit-Parallel Finite Field Arithmetic Using Polynomial Basis. In: Koç, Ç.K., Paar, C. (eds) Cryptographic Hardware and Embedded Systems. CHES 1999. Lecture Notes in Computer Science, vol 1717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48059-5_24
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DOI: https://doi.org/10.1007/3-540-48059-5_24
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