Difference between revisions of "ApCoCoA-1:Representation of finite fields"
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| − | To represent a finite field of | + | {{Version|1}} |
| − | + | <command> | |
| − | For example, | + | <title>Representation of finite fields</title> |
| − | So we have a field with | + | <description> |
| − | 0, 1, x, and x+1 (choosing their | + | To represent a finite field of characteristic 2 in ApCoCoA an integer is used. |
| + | <par/> | ||
| + | For example, let K = F_2[x]/(x^2+x+1). | ||
| + | So we have a field with 2^2 = 4 elements, namely one representation of F_4. The elements of the field are 0, 1, x, and x+1 (choosing their canonical representatives). | ||
| + | <par/> | ||
In the ring of integers, the elements 0,1,2 and 3 are chosen to represent the field. To map between both sets the substitution map, substituting x with 2 is chosen, so we have | In the ring of integers, the elements 0,1,2 and 3 are chosen to represent the field. To map between both sets the substitution map, substituting x with 2 is chosen, so we have | ||
| − | + | <example> | |
| − | + | 0 maps to 0 | |
| − | + | 1 maps to 1 | |
| − | + | 2 maps to x | |
| + | 3 maps to x+1. | ||
| + | </example> | ||
| − | A polynomial Ring Z[y[1..3]] | + | A polynomial Ring Z[y[1..3]] can be used to represent F_4[y_1, y_2, y_3]. Therefore, we map Z[y[1..3]] to Z/(4)[y[1..3]] and interpret each integer via the above described map. So the polynomial 3y[1] + 1y[2] + 5y[3] - 1 y[1]y[2] is first mapped to 3y[1] + 1y[2] + 1y[3] +3 y[1]y[2] which means the polynomial (x+1)y_1 + y_2 + y_3 + (x_1)y_1 y_2. |
| − | + | <par/> | |
| − | Z[y[1..3]] | + | To compute a Groebner basis in one of the implemented finite fields, polynomials with integer coefficients are used and sent to the ApCoCoAServer. These polynomials are then interpreted as described above. Then a Groebner basis is computed and the results were sent back, again as polynomials with integer coefficients. |
| − | + | <par/> | |
| − | To compute a Groebner basis in one of the implemented finite fields, polynomials with integer coefficients are used and sent to the ApCoCoAServer. These polynomials are then interpreted as described above. Then a | + | In case you have represented your polynomials in the graphical user interface, using an additional indeterminate (e.g. x like above), you can use the <ref>ApCoCoA-1:Subst|Subst</ref> command of CoCoA and substitute x with 2 to compute a integer representation and send this to the server. |
| − | |||
| − | In case you have represented your polynomials in the graphical user interface, using an additional indeterminate (e.g. x like above), you can use CoCoA | ||
The corresponding CoCoAL code could be as follows: | The corresponding CoCoAL code could be as follows: | ||
| + | <example> | ||
Use Z/(2)[x,y[1..3]]; | Use Z/(2)[x,y[1..3]]; | ||
Modulus := x^2 + x + 1; -- the polynomial to create the field | Modulus := x^2 + x + 1; -- the polynomial to create the field | ||
| Line 34: | Line 39: | ||
Use Z[y[1..3]]; | Use Z[y[1..3]]; | ||
IntegeredGens := [BringIn(G)| G In IntegeredGens]; -- the generators in the integer representation | IntegeredGens := [BringIn(G)| G In IntegeredGens]; -- the generators in the integer representation | ||
| − | + | </example> | |
| − | + | If you have some polynomials in integer representation you can also compute their 'meaning', which is then the inverse of the transformation above. | |
| − | If you have some polynomials in integer representation you can also compute their 'meaning', which is then the inverse of above | ||
To achieve this, you could apply: | To achieve this, you could apply: | ||
| + | <example> | ||
Use Z[y[1..3]]; | Use Z[y[1..3]]; | ||
| Line 54: | Line 59: | ||
Gens := [ Sum([ Images[Mod(LC(M),2^Deg(Modulus))+1]*LogToTerm(Concat([0],Log(M))) | M In Monomials(G)]) |G In IntegeredGens]; | Gens := [ Sum([ Images[Mod(LC(M),2^Deg(Modulus))+1]*LogToTerm(Concat([0],Log(M))) | M In Monomials(G)]) |G In IntegeredGens]; | ||
| − | + | </example> | |
| − | + | </description> | |
| − | + | <key>finite</key> | |
| + | <key>field</key> | ||
| − | + | <wiki-category>ApCoCoA-1:Package_charP</wiki-category> | |
| + | </command> | ||
Latest revision as of 10:31, 7 October 2020
| This article is about a function from ApCoCoA-1. |
Representation of finite fields
Description
To represent a finite field of characteristic 2 in ApCoCoA an integer is used.
For example, let K = F_2[x]/(x^2+x+1).
So we have a field with 2^2 = 4 elements, namely one representation of F_4. The elements of the field are 0, 1, x, and x+1 (choosing their canonical representatives).
In the ring of integers, the elements 0,1,2 and 3 are chosen to represent the field. To map between both sets the substitution map, substituting x with 2 is chosen, so we have
Example
0 maps to 0 1 maps to 1 2 maps to x 3 maps to x+1.
A polynomial Ring Z[y[1..3]] can be used to represent F_4[y_1, y_2, y_3]. Therefore, we map Z[y[1..3]] to Z/(4)[y[1..3]] and interpret each integer via the above described map. So the polynomial 3y[1] + 1y[2] + 5y[3] - 1 y[1]y[2] is first mapped to 3y[1] + 1y[2] + 1y[3] +3 y[1]y[2] which means the polynomial (x+1)y_1 + y_2 + y_3 + (x_1)y_1 y_2.
To compute a Groebner basis in one of the implemented finite fields, polynomials with integer coefficients are used and sent to the ApCoCoAServer. These polynomials are then interpreted as described above. Then a Groebner basis is computed and the results were sent back, again as polynomials with integer coefficients.
In case you have represented your polynomials in the graphical user interface, using an additional indeterminate (e.g. x like above), you can use the Subst command of CoCoA and substitute x with 2 to compute a integer representation and send this to the server.
The corresponding CoCoAL code could be as follows:
Example
Use Z/(2)[x,y[1..3]]; Modulus := x^2 + x + 1; -- the polynomial to create the field Gens := [...]; -- your generators in CurrentRing()/(Modulus); Gens := [NR(G,[Modulus])| G In Gens]; -- first, reduce the generators, just to be sure. Use Z[x,y[1..3]]; IntegeredGens := [Sum([Subst(BringIn(M),[ [x,2 ] ]) |M In Monomials(G)]) |G In Gens]; Use Z[y[1..3]]; IntegeredGens := [BringIn(G)| G In IntegeredGens]; -- the generators in the integer representation
If you have some polynomials in integer representation you can also compute their 'meaning', which is then the inverse of the transformation above. To achieve this, you could apply:
Example
Use Z[y[1..3]]; IntegeredGens := [3y[1]]; --IntegeredGens := [...]; -- your generators in integer representation. Use Z/(2)[x,y[1..3]]; Modulus := x^2 + x + 1; Images := [0]; For I:=0 To Deg(Modulus)-1 Do Images :=Concat(Images, [x^I +Im | Im In Images]); EndFor; Gens := [ Sum([ Images[Mod(LC(M),2^Deg(Modulus))+1]*LogToTerm(Concat([0],Log(M))) | M In Monomials(G)]) |G In IntegeredGens];
