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| <command>
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| <title>CharP.LAAlgorithm</title>
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| <short_description>Computes the unique <tt>F_2-</tt>rational zero of a given polynomial system over <tt>F_2</tt>.</short_description>
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| <syntax>
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| CharP.LAAlgorithm(F:LIST):LIST
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| </syntax>
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| <description>
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| <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
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| <par/>
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| This function computes the unique zero in <tt>F_2^n</tt> of a polynomial system over <tt>F_2 </tt>. It uses LA-Algorithm to find the unique zero. The LA-Algorithm generates a sequence of linear systems to solve the given system. The LA-Algorithm can find the unique zero only. If the given polynomial system has more than one zero's in <tt>F_2^n </tt> then this function does not find any zero. In this case a massage for non-uniqueness will be displayed to the screen after reaching the maximum degree bound. To solve linear system naive Gaußian elimination is used.
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| <itemize>
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| <item>@param <em>F:</em> List of polynomials of given system.</item>
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| <item>@return The unique solution of the given system in <tt>F_2^n</tt>. </item>
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| </itemize>
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|
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| <example>
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| Use Z/(2)[x[1..4]];
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| F:=[
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| x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1,
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| x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1,
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| x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1,
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| x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
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| ];
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|
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| -- Then we compute the solution with
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| CharP.LAAlgorithm(F);
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| [0, 1, 0, 1]
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|
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| </example>
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| <example>
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| Use Z/(2)[x[1..4]];
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| F:=[
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| x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4],
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| x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4],
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| x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2],
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| x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]
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| ];
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|
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| -- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions
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|
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| -- Then we compute the solution with
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| CharP.LAAlgorithm(F);
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|
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| x[4] = NA
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| Please Check the uniqueness of solution.
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| The Given system of polynomials does not
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| seem to have a unique solution or it has
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| no solution over the finite field F2.
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|
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|
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| </example>
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| </description>
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| <seealso>
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| <see>CharP.MXLSolve</see>
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| <see>Introduction to CoCoAServer</see>
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| <see>Introduction to Groebner Basis in CoCoA</see>
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| <see>CharP.GBasisF2</see>
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| <see>CharP.XLSolve</see>
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| <see>CharP.IMXLSolve</see>
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| <see>CharP.IMNLASolve</see>
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| <see>CharP.MNLASolve</see>
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| </seealso>
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|
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| <types>
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| <type>apcocoaserver</type>
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| <type>poly_system</type>
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| </types>
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|
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| <key>charP.nlasolve</key>
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| <key>nlasolve</key>
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| <key>finite field</key>
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| <wiki-category>Package_charP</wiki-category>
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| </command>
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