Gauss Period, Sparse Polynomial, Redundant Basis, and Efficient Exponentiation for a Class of Finite Fields with Small Characteristic
We present an efficient exponentiation algorithm in a finite field GF(qn) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(qn) 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(qn). 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(qn), 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(qn)×.
KeywordsFinite field Gauss period exponentiation root of unity trinomial redundant basis
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