Abstract
In this paper, we generalize an approach of switching between different bases of a finite field to efficiently implement distinct stages of algebraic algorithms. We consider seven bases of finite fields supporting optimal normal bases of types 2 and 3: polynomial, optimal normal, permuted, redundant, reduced, doubled polynomial, and doubled reduced bases. With respect to fields of characteristic q = 7 we provide complexity estimates for conversion between the bases, multiplication, and exponentiation to a power \( q^{k} \), q-th root extraction. These operations are basic for inversion and exponentiation in \( GF\left( {7^{n} } \right) \). One needs a fast arithmetic in \( GF\left( {7^{n} } \right) \) for efficient computations in field extensions \( \left( {7^{2n} } \right) \), \( GF\left( {7^{3n} } \right),GF\left( {7^{6n} } \right) GF\left( {7^{14n} } \right),GF(7^{3 \times 14n} ) \) which are the core of the Tate pairing on a supersingular hyperelliptic curve of genus three. The latter serves for an efficient implementation of cryptographic protocols.
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This research was supported by the Russian Foundation for Basic Research, project 14-01-00671a.
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Gashkov, S., Frolov, A., Sergeev, I. (2016). Arithmetic in Finite Fields Supporting Type-2 or Type-3 Optimal Normal Bases. In: Zamojski, W., Mazurkiewicz, J., Sugier, J., Walkowiak, T., Kacprzyk, J. (eds) Dependability Engineering and Complex Systems. DepCoS-RELCOMEX 2016. Advances in Intelligent Systems and Computing, vol 470. Springer, Cham. https://doi.org/10.1007/978-3-319-39639-2_14
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DOI: https://doi.org/10.1007/978-3-319-39639-2_14
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