Abstract
In this chapter the idea of using optimal normal bases (o.n.b.) of second and third types in combination with polynomial basis of field F(q n) is detailed using a new modification of o.n.b. called reduced optimal normal basis –1, β 1, …, β n − 1 corresponding to a permutated o.n.b. β 1, …, β n − 1 Operations of multiplication, rising to power q i, rising to arbitrary power and inversion in reduced o.n.b. in combination with polynomial basis as well as converting operations between these bases in the fields of characteristic three has been described, estimated and expanded to the fields of characteristic two. This allows get efficient implementations of cryptographic protocols using operation of Tate pairing on supersingular elliptic curve.
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References
Lidl, R., Niderreiter, H.: Finite fields. Addison-Wesley publishing Company, London (1983)
Jungnickel, D.: Finite fields: Structure and arithmetics. Wissenschaftsverlag. Mannheim (1993)
Karatzuba, A.A., Offman, Y.P.: Umnozhenie mnogoznachnych chisel na avtomatah. DAN USSR 145(2), 293–294 (1962) (in Russian)
Schönhage, A.: Shnelle Multiplikation von Polynomen der Körpern der Charakteristik 2. Acta Informatica 7, 395–398 (1977)
Mullin, R.C., Onyszchuk, I.M., Vanstone, S.A., Wilson, R.M.: Optimal Normal Bases in GF(pn). Discrete Appl. Math. 22, 149–161 (1988/1989)
Bolotov, A.A., Gashkov, S.B.: On quick multiplication in normal bases of finite fields. Discrete Mathematics and Applications 11(4), 327–356 (2001)
Shokrollahi, J.: Efficient implementation of elliptic curve cryptography on FPGA. PhD thesis, universitet Bonn (2007)
von zur Gathen, J., Shokrollahi, A., Shokrollahi, J.: Efficient multiplication using type 2 optimal normal bases. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 55–68. Springer, Heidelberg (2007)
Bernstein, D.J., Lange, T.: Type-II Optimal Polynomial Bases. In: Hasan, M.A., Helleseth, T. (eds.) WAIFI 2010. LNCS, vol. 6087, pp. 41–61. Springer, Heidelberg (2010)
Kwon, S.: Efficient Tate Pairing for Supersingular Elliptic Curve over Binary Fields. Cryptology ePrint Archive. Report 2004/303 (2004)
Aho, A., Hopcroft, J., Ullman, J.: The design and analysis of computer algorithms. Addison-Wesley Publishing Company, Reading (1978)
Hankerson, D., Hernandez, J.L., Menezes, A.: Software implementation of elliptic curve cryptography over binary fields. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 1–23. Springer, Heidelberg (2000)
Gashkov, S.B., Frolov, A.B., Shilkin, S.O.: On some algorithms of inversion and division in finite rings and fields. MPEI Bulletin (6), 20–31 (2006) (in Russian)
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Gashkov, S., Frolov, A., Lukin, S., Sukhanova, O. (2015). Arithmetic in the Finite Fields Using Optimal Normal and Polynomial Bases in Combination. In: Zamojski, W., Mazurkiewicz, J., Sugier, J., Walkowiak, T., Kacprzyk, J. (eds) Theory and Engineering of Complex Systems and Dependability. DepCoS-RELCOMEX 2015. Advances in Intelligent Systems and Computing, vol 365. Springer, Cham. https://doi.org/10.1007/978-3-319-19216-1_15
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DOI: https://doi.org/10.1007/978-3-319-19216-1_15
Publisher Name: Springer, Cham
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