Speeding up the Arithmetic Operations over Optimal Extension Fields in the Lagrange Representation Using DFT

In this paper, efficient algorithms of the arithmetic operations over Optimal Extension Field have been considered. An arbitrary field element is represented by the Lagrange Representation, and transformation forward and backward between the Lagrange Representation and the vector of polynomial coefficients of the field element has been speeded up by applying DFT (discrete Fourier transform) over a ring. Our contribution is of establishing modular multiplication and inversion algorithms over Optimal Extension Field in the Lagrange Representation using the discrete Fourier transform.