Abstract
We present an efficient exponentiation algorithm in a finite field GF(q n) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n)×.
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Kwon, S., Kim, C.H., Hong, C.P. (2003). Gauss Period, Sparse Polynomial, Redundant Basis, and Efficient Exponentiation for a Class of Finite Fields with Small Characteristic. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_75
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DOI: https://doi.org/10.1007/978-3-540-24587-2_75
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