On bit-serial multiplication and dual bases in GF(2/sup m/)

The existence of certain types of dual bases in finite fields GF(2/sup m/) is discussed. These special types of dual bases are needed for efficient implementation of (generalized) bit-serial multiplication in GF(2/sup m/). In particular, the question of choosing a polynomial basis of GF(2/sup m/), for example (1, alpha , alpha /sup 2/, alpha /sup 3/, . . ., alpha /sup m-1/), such that the change of basis matrix from the dual basis to a scalar multiple of the original basis has as few '1' entries as possible, is studied. It was previously shown by M. Wang and I. F. Blake (1990) that the optimal situation occurs when the minimal polynomial of alpha is an irreducible trinomial of degree m; then, an appropriate scalar multiple, beta , yields a change of basis matrix that is a permutation matrix. A construction is presented that often yields bases where the change of basis matrix has low weight, in the case where no irreducible trinomial of degree m exists. A simple formula can be used to compute beta and the weight of the change of basis matrix, given the minimal polynomial of alpha .<
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