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This paper presents the generalized theory and an efficient graph-based technique for the calculation and representation of coefficients of multivariate canonic polynomials over arbitrary finite fields in any polarity. The technique presented for computing coefficients is unlike polynomial interpolation or matrix-based techniques and takes into consideration efficient graph-based forms which can be available as an existing resource during synthesis, verification, or simulation of digital systems. Techniques for optimization of the graph-based forms for representing the coefficients are also presented. The efficiency of the algorithm increases for larger fields. As a test case, the proposed technique has been applied to benchmark circuits over GF(2m). The experimental results show that the proposed technique can significantly speed up execution time.