ApCoCoA-1:GLPK.L01PSolve

GLPK.L01PSolve

Solve a system of polynomial equations over F_2 for one solution in F_2^n.

Syntax

GLPK.L01PSolve(F:LIST, CuttingNumber:INT, QStrategy:INT, CStrategy:INT, MinMax:STRING):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 finds one solution in F_2^n of a system of polynomial equations over the field F_2. It operates in two stages. Firstly, it models the problem of finding one solution of given polynomial system into a mixed integer linear programming problem. For this the system is first converted into an equivalent CNF form and then the CNF form is converted into a system of equalities and inequalities. Secondly, the mixed integer linear programming model is solved using glpk.

• @param F: A List containing the polynomials of the given system.

• @param CuttingNumber: Maximal support-length of the linear polynomials for conversion to CNF. The possible value could be from 3 to 6.

• @param QStrategy: Strategy for quadratic substitution. 0 - Standard; 1 - Linear Partner; 2 - Double Linear Partner; 3 - Quadratic Partner;

• @param CStrategy: Strategy for cubic substitution. 0 - Standard; and 1 - Quadratic Partner;

• @param MinMax: Optimization direction i.e. minimization ("Min") or maximization ("Max").

• @return A list containing a zero of the system F.

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
    ];

CuttingNumber:=6;
QStrategy:=0;
CStrategy:=0;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: Standard, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 2
[0, 1, 0, 1]
-------------------------------

Example

Use S::=Z/(2)[x[1..5]];
F:=[
 x[1]x[5] + x[3]x[5] + x[4]x[5] + x[1] + x[4],
 x[1]x[2] + x[1]x[4] + x[3]x[4] + x[1]x[5] + x[2]x[5] + x[3]x[5] + x[1] + x[4] + x[5] + 1,
 x[1]x[2] + x[4]x[5] + x[1] + x[2] + x[4],
 x[1]x[4] + x[3]x[4] + x[2]x[5] + x[1] + x[2] + x[4] + x[5] + 1,
 x[1]x[4] + x[2]x[4] + x[3]x[4] + x[2]x[5] + x[4]x[5] + x[1] + x[2] + x[4] + x[5]
];

CuttingNumber:=6;
QStrategy:=1;
CStrategy:=0;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 6, QStrategy: LinearPartner, CStrategy: Standard.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 4
[1, 1, 1, 1, 0]
-------------------------------

Example

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

CuttingNumber:=5;
QStrategy:=0;
CStrategy:=1;
MinMax:=<quotes>Max</quotes>;

-- Then we compute the solution with

GLPK.L01PSolve(F, CuttingNumber, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Converting to CNF with CuttingLength: 5, QStrategy: Standard, CStrategy: CubicParnterDegree2.
Converting CNF to system of equalities and inequalities...
Model is ready to solve with GLPK...
Solution Status: INTEGER OPTIMAL
Value of objective function: 1
[0, 0, 1]
-------------------------------