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Fast evaluation of logarithms in fields of characteristic two

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A method for determining logarithms in GF (2^{n}) is presented. Its asymptotic running time is O(\exp (cn^{1/3} \log ^{2/3} n)) for a small constant c , while, by comparison, Adleman's scheme runs in time O(\exp (c^{'}n^{1/2} \log ^{1/2} n )) . The ideas give a dramatic improvement even for moderate-sized fields such as GF (2^{127}) , and make (barely) possible computations in fields of size around 2^{400} . The method is not applicable to GF (q) for a large prime q .

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Information Theory, IEEE Transactions on  (Volume:30 ,  Issue: 4 )