ApCoCoA-1:CharP.LAAlgorithm

CharP.LAAlgorithm

Computes the unique F_2-rational zero of a given polynomial system over F_2.

Syntax

CharP.LAAlgorithm(F:LIST):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function computes the unique zero in F_2^n of a polynomial system over F_2 . 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 F_2^n then this function does not find any zero. In this case the trivial solution is given. To solve linear systems naive Gaußian elimination is used.

• @param F: List of polynomials of given system.

• @return The unique solution of the given system in F_2^n.

Example

Use Z/(2)[x[1..4]];
F:=[
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[1]x[4] + x[3]x[4] + x[2] + x[3] + 1, 
    x[1]x[2] + x[1]x[3] + x[2]x[3] + x[3]x[4] + x[1] + x[4] + 1, 
    x[1]x[3] + x[2]x[3] + x[1]x[4] + x[2]x[4] + 1
    ];

-- Then we compute the solution with
CharP.LAAlgorithm(F);

[0, 1, 0, 1]

Example

Use Z/(2)[x[1..4]];
F:=[ 
    x[2]x[3] + x[1]x[4] + x[2]x[4] + x[3]x[4] + x[1] + x[2] + x[3] + x[4], 
    x[2]x[3] + x[2]x[4] + x[3]x[4] + x[2] + x[3] + x[4],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2],  
    x[1]x[2] + x[2]x[3] + x[2]x[4] + x[3]x[4] + x[1] + x[2]
   ];

-- Solution is not unique i.e. [0, 1, 1, 1], [0, 0, 0, 0], and [1, 1, 1, 1] are solutions 

-- Then we compute the solution with
CharP.LAAlgorithm(F);

[0, 0, 0, 0]

See also

CharP.MXLSolve

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

CharP.GBasisF2

CharP.XLSolve

CharP.IMXLSolve

CharP.IMNLASolve

CharP.MNLASolve