Johan's profile - activity

When running sagemath 2.7 in virtual box and export a notebook (using jupyter) to pdf I get an error message that pdflatex could not be found. By installing texlive, pdflatex was found but not all the latex packages.

How can I use the latex install on my window machine when export the notebook (in jupyter) to pdf?

Almost, v.apply_map(ff.embedding()) give an element in VectorSpace(f16,2). I want an element in f16, an isomorphism from VectorSpace(f4,2) to VectorSpace(f16,2). The purpose will be to verify an implementation of a finite field (f16) using a subfield (f4) for its computations.

@ slelievre, Yes

Hallo

I want to perform finite field multiplication in by using operation in one of its sub fields.

f4 = GF(4, 'x')
f16 =  GF(16, 'y')
ff = RelativeFiniteFieldExtension(f16, f4)

Now I am stuck:

1. How do I map element from ff to a vector over f4
2. After performing the computation in f4 map the element back in f?

The ideal will be that the mapping to ff just concatenate two element in f4.

I run pycharm on a windows machine and sage in a linux server. How can I use a python interpreter or pycharm on windows do execute my scripts in sage?

It would also be use full if pycharm does not indicate sage packages/classes/fusctions as syntax errors.

I have a similar problem where my version of bb is not so simple and (sms == sms.subs(bb)) is true. Does a different method exist to solve the problem?

If the system p is 3D, can I fix one parameter to get a 2D system without start a new instance of MixedIntegerLinearProgram()?

Solving a linear programming problem there input constrains and addition constraints my be generated during solving the problem.

I am only interested in constrains of the form x <= y.

Does sage provide a method to the all the constrains between the variables? How easy is it to represent the constrain visually?

Hallo,

how can I include a script in more than one notebook?

Regards.

@Luca, yes was supposed to be additive.

What type of code do I need to write to be able to write something like f = sympy.Function('f') y + f(y^6) + f(x^2) + x where y is an element in a finite field and x is a variable/Symbol. If I was working in c++ I would at least to overload the + operator between the FiniteField sympy.Function can sympy.Symbol types.

H_u was suppose to the a multiplicative subgroup of Kn_1 and now can't get away to generate a subgroup in sage. The reason I want to use a morphism is that I am only interested in the Range and Domain and the mapping. This will also help met to learn a functionality of sage.

My requirement is to be able to use unknowns and function (not fully defined) in equations over a specific finite field. The only knowledge about the unknowns is that the unknown is in the field. Also the range and domain of the function f is the field. In the end I want to use sage as a calculator to perform arithmetic in a finite field to simplify a set of equations. These equation is not linear and that where the function f comes in, it will be used to represent a non-linear function.

@slelievre Thanx it was suppose to be x^(2^8), I have updated the question

The curly brackets got lost in the definition of H_u. The function $l(x)=x^(2^i)+x$ is linear over GF(2^(n+1)) and H_u is image of $l(x)$ giving that H_u is a subgroup of GF and can also be seen as a vector space.

Hallo,

I am interested in creating morphims between $GF(2^{n+1}) \stackrel{h_1}{\longrightarrow} H_u \stackrel{h_2}\longrightarrow GF(2^n)$ where $H_u={x^{2^i} + x : x \in GF(2^{n+1})}$ where $gcd(i,n+1) =1$. I am interested in constructing $h_1$ and $h_2$,

My attempt to create $h_1$ is:

n = 4
i = 1
f = lambda x : x^(2^i) + x
Kn_1 = GF(2^(n+1),'x')
H_u = map(f, Kn_1)
hs = Hom(Kn_1,H_u)
pi = SetMorphism(Hom(Kn_1,Kn), lambda y : y^(2^i) + y)
pi(Kn_1.random_element()) in Kn
 

now $hs$ is a set of morphism and $h_1 \in hs$. Can I get a (the) specific $h_1$ from $hs$? My attempt is $pi$ but 'pi(Kn_1.random_element()) in Kn' fails.

To construct $h_2$ I have more success.

n = 4
i = 2
f = lambda x: x^(2^i) + x

Kn_1 = GF(2^(n+1),'x1')
Vn_1 = Kn_1.vector_space()
Sn_1 = Vn_1.subspace([Vn_1(f(u)) for u in Kn_1])

Kn = GF(2^n,'x')
Vn = Kn.vector_space()

h_a = Sn_1.basis_matrix().transpose()
 

Now $h_a : Vn\rightarrow Sn_1$ where $Vn$ and $Sn_1$ is vector space representation of $GF(2^n)$ and $H_u$. To get from $GF(2^n)$ to $Vn$ and back I am good with But to create $h_2$ I have no luck. My attempts to use

MatrixMorphism(Hom(Vn,Sn_1), Sn_1.basis_matrix().transpose())
 

but get errors with regards to the dimensions of the matrix.

Regards

Hallo

I am interested in symbolic computations in a file field. Working in the field $GF(2^8)$ with $x$ as a generator and a variable $y$, also there is a function $f:GF(2^8)\rightarrow GF(2^8)$ of which the exact definition is not given. Can sage do the following:

$(x + y) ^ {2^8} \mapsto x + y$

$(x + y) ^ {2^6} \mapsto x ^ {2^6} + y ^ {2^6}$

$f(y) + f(y) \mapsto 0\cdot x$

If sage cannot do it does there exist another program which can?

Below is my question updated:

For the function I have a partial solution but I cannot mix it with an element of an finite field

import sympy
f = sympy.Function('f')
y = var('y')
sympy.expand((f(y)+f(y)), modulus=2)
 

When I want to add an element of a finite field

import sympy
f = sympy.Function('f')
x = GF(2^8,'x').gen()
f(x)
f(x) + x
 

the statement $f(x) +x$ give me an error that makes sense ... TypeError: unsupported operand parent(s) for '+': 'f' and 'Finite Field in g of size 2^8'

To create a variable in a finite field I decided to use a Polynomial ring

import sympy
f = sympy.Function('f')
R.< x > = PolynomialRing(GF(2))
f(x)
f(x) + x
 

both $f(x)$ and $f(x) + x$ fails with a very long Trace message.

Regards

@tmonteil, thanx the map was suppose to be $x\mapsto x\cdot a^8 + x^8 \cdot a$. By adding the code f = lambda x: x * a^8 + a * x^8 S = V.subspace([f(x) for x in K]) S.basis() I got a solution. What will the most effective method to create a map (isomorphism) from $S$ to $GF(2^n)$?