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A new hardware scheme for computing the transition and control matrix of a parallel cyclic redundancy checksum is proposed. This opens possibilities for parallel high-speed cyclic redundancy checksum circuits that reconfigure very rapidly to new polynomials. The area requirements are lower than those for a realization storing a precomputed matrix. An additional simplification arises as only the polynomial needs to be supplied. The derived equations allow the width of the data to be processed in parallel to be selected independently of the degree of the polynomial. The new design has been simulated and outperforms a recently proposed architecture significantly in speed, area, and energy efficiency.