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The Complexity and Depth of Boolean Circuits for Multiplication and Inversion in some Fields GF(2ⁿ) / S. B. Gashkov, I. S. Sergeev // Vestn. Mosk. Univ., Matem. Mekhan. 2009. № 4. P. 3-7 [Moscow Univ. Math. Bulletin. Vol. 64, No 4, 2009. P. 139-143.]

Let n=(p-1)·p^k, where p is a prime number such that 2 is a primitive root modulo p, and 2^p-1-1 is not a multiple of p². For a standard basis of the field GF(2ⁿ), a multiplier of complexity O(log log p)nlog n log log_pn and an inverter of complexity O(log p log log p)n log n log log_pn are constructed. In particular, in the case p=3 the upper bound 5·5/8·n log₃n log₂log₃n+O(nlog n) for the multiplication complexity and an upper bound for the inversion complexity which is asymptotically 2.5-times greater are obtained (hereafter, if not indicated explicitly, all logarithms are base two).

Key words: Boolean schemes, finite fields, multiplier, invertor.