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

Abstract

A formula on the complexity of the normal bases generated by prime Gauss period over finite fields is presented in terms of cyclotomic numbers. Then, the authors determine explicitly the complexity of such normal bases and their dual bases in several cases where the related cyclotomic numbers have been calculated. Particularly, the authors find several series of such normal bases with low complexity.

Construction of self-dual normal bases and their complexity

“…In Section 3 we present our algorithms, experimental results and conclusions. For semi-simple extensions in odd characteristic, the lowest complexity we find is close to that obtained for normal bases from exhaustive computer search [3] or from theoretical constructions [14], as this was already the case in even characteristic. We also observe an interesting behaviour under base field extension.…”

“…The search for normal bases with low complexity has taken two complementary directions. On the theoretical side, several authors have attempted to build them either from roots of unity in larger extensions, using Gauss periods [1,8,14] or traces of optimal normal bases [5], again with some limitations on the degree; or from the extension itself, using division points of a torus [3,7] or of an elliptic curve [6]. In the latter case the authors show that fast arithmetic can be implemented using their bases, as was also shown to be the case for normal bases generated by Gauss periods in [9].…”

Normal Bases on Galois Ring Extensions

“…For α ∈ R(= R (r) ), α is an NBG for R/Z p r if and only ifᾱ is an NBG for F q /F p by Theorem 1, then this is also equivalent to α (λ) being an NBG for R (λ) /Z p r for any λ ≥ 1. Moreover, by the diagram (6), we get that for any λ, the equality (10) implies that:…”

“…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].…”

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].…”