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Demonstrates improved logarithm calculations in fields of characteristic two, suggesting enhanced computational efficiency.
A method for determining logarithms in GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2ⁿ)</tex> is presented. Its asymptotic running time is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(exp (cn1/3 log2/3 n))</tex> for a small constant <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> , while, by comparison, Adleman's scheme runs in time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(exp (c'n1/2 log1/2 n ))</tex> . The ideas give a dramatic improvement even for moderate-sized fields such as GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2¹²⁷)</tex> , and make (barely) possible computations in fields of size around <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2⁴⁰⁰</tex> . The method is not applicable to GF <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(q)</tex> for a large prime <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> .
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Don Coppersmith (1984) studied this question.
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