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- W2055762904 abstract "Algorithms for factoring polynomials over finite fields are discussed. A construction is shown which reduces the final step of Berlekamp’s algorithm to the problem of finding the roots of a polynomial in a finite field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Z Subscript p> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>{Z_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that if the characteristic of the field is of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p equals upper L dot 2 Superscript l Baseline plus 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>L</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>l</mml:mi> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p = L cdot {2^l} + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=l asymptotically-equals upper L> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>≃<!-- ≃ --></mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>l simeq L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the roots of a polynomial of degree <italic>n</italic> may be found in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis n squared log p plus n log squared p right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O({n^2}log p + n{log ^2}p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> steps. As a result, a modification of Berlekamp’s method can be performed in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis n cubed plus n squared log p plus n log squared p right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O({n^3} + {n^2}log p + n{log ^2}p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> steps. If <italic>n</italic> is very large then an alternative method finds the factors of the polynomial in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper O left-parenthesis n squared log squared n plus n squared log n log p right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>log</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>n</mml:mi> <mml:mi>log</mml:mi> <mml:mo><!-- --></mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>O({n^2}{log ^2}n + {n^2}log nlog p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Some consequences and empirical evidence are discussed." @default.
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- W2055762904 date "1977-01-01" @default.
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- W2055762904 title "On the efficiency of algorithms for polynomial factoring" @default.
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