Abelian Groups, Gauss Periods, and Normal Bases

Abstract

DEDICATED TO PROFESSOR CHAO KO ON HIS 90TH BIRTHDAYA result on "nite abelian groups is "rst proved and then used to solve problems in "nite "elds. Particularly, all "nite "elds that have normal bases generated by general Gauss periods are characterized and it is shown how to "nd normal bases of low complexity. Academic Press

On the complexity of the normal bases via prime Gauss period over finite fields

“…If a normal basis B of F q n /F q is not optimal, it is proven that C(B) ≥ 3n−3. Several normal bases via Gauss period with lower complexity have been searched in [8]. An interesting way to construct normal bases with low complexity via Gauss period is suggested and researched intensively (see [7][8] and the references therein).…”

“…is called prime Gauss period of type (n, k)(more general Gauss period can be defined for arbitrary integer l ≥ 2, see [8]). …”

“…For a proof of Lemma 1.2, see [8,Theorem 1.5], where the result is stated and proven in general Gauss period case. The upper bound C(B) ≤ nk − 1 is meanless for k > n since we have trivial bound C(B) ≤ n 2 .…”

A new criterion on normal bases of finite field extensions

“…A series of criterions on normal bases has been given [11,17], many series of normal bases with lower complexity have been found [1,3,5,6,9,10,12,16,19], and explicit description to construct normal bases for specific cases of finite field have been presented [2,4,8,[13][14][15]18].…”

Normal Bases on Galois Ring Extensions

“…Low complexity operation, particularly the multiplicative operation, squaring, and exponentiation operations, are preferred in various applications, including coding, cryptography, and communication. The performance of these operations is closely related to the representation of the finite elements; they are desired for efficient hardware implementation, and in this respect, many useful bases for F q n /F q with low complexity have been found [1][2][3][4][5][6][7][8][9][10][11]. An efficient algorithm for field multiplication using a normal basis was proposed by Massey and Omura in 1985 [12].…”