An efficient basis conversion algorithm for composite fields with given representations

We describe an efficient method for constructing the basis conversion matrix between two given finite field representations where one is composite. We are motivated by the fact that using certain representations, e.g., low-Hamming weight polynomial or composite field representations, permits arithmetic operations such as multiplication and inversion to be computed more efficiently. An earlier work by Paar defines the conversion problem and outlines an exponential time algorithm that requires an exhaustive search in the field. Another algorithm by Sunar et al. provides a polynomial time algorithm for the limited case where the second representation is constructed (rather than initially given). The algorithm we present facilitates existing factorization algorithms and provides a randomized polynomial time algorithm to solve the basis conversion problem where the two representations are initially given. We also adapt a fast trace-based factorization algorithm to work in the composite field setting which yields a subcubic complexity algorithm for the construction of the basis conversion matrix.