ApCoCoA-1:GLPK.RPCSolve

This article is about a function from ApCoCoA-1.

GLPK.RPCSolve

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

Syntax

GLPK.RPCSolve(F:LIST, 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 uses Real Polynomial Conversion (RPC) along with some strategies from propositional logic to model a mixed integer linear programming problem. Then the modelled mixed integer linear programming problem is solved using glpk.

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

• @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
    ];

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

-- Then we compute the solution with

GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);

-- The result will be the following:
Modelling the system as a mixed integer programming problem. 
QStrategy: Standard, CStrategy: Standard.
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]
];


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

-- Then we compute the solution with

GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Modelling the system as a mixed integer programming problem. 
QStrategy: LinearPartner, CStrategy: Standard.
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]
     ];


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

-- Then we compute the solution with

GLPK.RPCSolve(F, QStrategy, CStrategy, MinMax);

-- The result will be the following:

Modelling the system as a mixed integer programming problem. 
QStrategy: Standard, CStrategy: CubicParnterDegree2.
Model is ready to solve with GLPK...

Solution Status: INTEGER OPTIMAL
Value of objective function: 1

[0, 0, 1]
-------------------------------