Tutorial for prime finite fields with modulo > 2^32

There are three main ways to use the openf4 library (from the simplest to the most involved):

1. In Sage (not yet available):
   1. R.<x0,x1,x2,x3,x4,x5> = Zmod(115792089237316195423570985008687907853269984665640564039457584007913129640233)[]
   2. I = sage.rings.ideal.Cyclic(R,6)
   3. I.groebner_basis('openf4')
2. As a C++ library:
   1. Include the library header: libopenf4.h.
   2. Link with the library during the compilation: -lopenf4
   3. Use the function groebnerBasisGivaroIntegerF4(std::string modulo, int nbVariable, std::vector<std::string> variableName, std::vector<std::string> polynomialList, int nbThread, int verbose);
   4. Look at the example tutorial-big-modulo-method2.cpp into the section example.
3. As a template library:
   1. Include the header openf4.h.

   2. Create and configure the element type with your modulo:

      • Define the Givaro field type: typedef Givaro::Modular<Givaro::Integer> Field;
      • Create a field with your modulo: Field F(Givaro::Integer("115792089237316195423570985008687907853269984665640564039457584007913129640233"));
      • Use the field type to define your element type: typedef ElementGivaro<Field> eltType;
      • Initialise your element type with the field F: eltType::setField(F);

   3. Configure the class monomial with the number of variables and an initial degree for the monomial precomputation.
   4. Create an ideal with a vector of polynomials and others informations: F4::Ideal::Ideal(std::vector<Polynomial<Element>> & polynomialArray, int nbVariable, int capacity, int degree);
   5. Call the method f4: F4::Ideal::f4();
   6. Compile with:
      • -std=c++11 to enable c++11 support
      • -DNDEBUG to disable assertion
      • -fopenmp to use parallelisation
      • -march=native to use vectorisation (if your system support SSE instructions).
      • -03 -funroll-loops -ftree-vectorize (for fast)
      • -Wall -Wno-strict-overflow to enable useful warning.
   7. Link with:
      • -lgmp for big number.
      • -lgmpxx for big number.
      • -lgivaro for prime field based on GMP.
      • -fopenmp to use parallelisation
   8. Look at the example tutorial-big-modulo-method3.cpp into the section example.