Abstract: First, we design an approach to determining the corresponding trace function of a linear transformation over finite fields, on the basis of which, we have proposed an effective methodology to compute the algebraic representations of coordinates of finite field elements in the form of trace functions. Second, by calculating the dual basis of the standard basis, we develope another method to find the algebraic representations of coordinates of finite field elements, also in the form of trace functions. The necessary data for both methods are only n elements over the corresponding fields with time complexity each of O(n³). This is a great improvement in contrast to the existing data and time complexity of O(p^n-1) and O(np^2n-2) respectively. Finally, based on the algebraic representations of coordinates which could be computed easily with our new approaches, we have given a direct proof of the equivalence between any two coordinate functions of Rijndael S-box and we have depicted this equivalence with only one matrix of order eight over GF(2⁸). Compared to the existing ones, in which the equivalence was found by the search of the affine parameters and was illustrated with no less than 56 matrices of order eight over GF(2), our proof seems more straightforward and simpler.

Key words: finite field, linear transformation, trace, dualbasis, Rijndael