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Multiplication and squaring in cubic and quartic extensions for pairing based cryptography. (English) Zbl 07358950

Davis, James A. (ed.), Finite fields and their applications. Proceedings of the 14th international conference on finite fields and their applications (Fq14), Vancouver, Canada, June 3–7, 2019. Berlin: De Gruyter. De Gruyter Proc. Math., 71-86 (2020).
In this paper the authors study the cyclic vector multiplication algorithm (CVMA) for multiplication and squaring in cubic and quartic extensions of finite fields. This is done with a careful analysis of the CVMA based on Gauss periods of type \((h,m)\). Such Gauss periods of type \((h,m)\) gives a way for constructing normal bases of the field \(\mathbb F_{p^m}\) with \(p^m\) elements using special arithmetic properties of \(h\) and \(m\). These normal bases are then used for the CVMA. The paper focuses on optimizing the vector arithmetic in the CVMA in dependence of \(h\) and \(m\). Specifically, the authors restrict themselves to the study of the cases \(m=3\) and \(m=4\). Finally, the paper contains detailed considerations on the CVMA and comparisons with classical algorithms. In particular, it is shown that the proposed algorithm’s costs are lower than those of the conventional algorithms, such as Karatsuba-based methods.
For the entire collection see [Zbl 1455.11009].

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
94A60 Cryptography
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