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@ tmonteil What is the solution then?

I am facing problem during arithmetic operation in $ GF(4091^2)$. But everything working fine in $ GF(13^2)$ as

Var('x')
F.<x> = GF(13^2)
a=F.fetch_int(150)+F.fetch_int(97)
print a

How can I fix this?

Let us consider an example:

Var('x')
F.<x> = GF(13^2)

We consider two elements 2*x + 11, 3*x + 2 corresponding to the integers 37 and 41 respectively. Now we can operate them as

a=F.fetch_int(37)+F.fetch_int(41)
m=F.fetch_int(37)*F.fetch_int(41)

We have a=4*x + 10 , m= 4*x + 10

In reverse way we can get the corresponding integers as :

ai=a.integer_representation()
mi=m.integer_representation()

That is, ai=65, mi=62

How can I do the same for the field GF(251) using the same function?

I would like to generate and solve a system of equations in both GF(251) and GF(13^2) using the same function.

Let us illustrate it in $GF(251)~ [or~ GF(13^2)]$,

Equation construction part:
$\phi(x)=a_0+a_1x+a_2x^2$, where $a_0,a_1,a_2\in GF(251) ~[or~ GF(13^2)]$.

Solve system of equations:
$a_0+a_1.\alpha_1++a_2.\alpha_1^2=\phi(\alpha_1)$
$a_0+a_1.\alpha_2++a_2.\alpha_2^2=\phi(\alpha_2)$
$a_0+a_1.\alpha_3++a_2.\alpha_3^2=\phi(\alpha_3)$
where $\alpha_1,\alpha_2,\alpha_3\in GF(251) ~[or~ GF(13^2)]$.

MWE: Case -I : $GF(251)$
Choose $(a_0,a_1,a_2)=(5,123,49)\in GF(251)$. $(\alpha_1,\alpha_2,\alpha_3)=(1,2,3)\in GF(251)$. Then
$\phi(x)=5+123x+49x^2$, and the system of equations becomes
$a_0+a_1.1++a_2.1^2=\phi(1)=177$
$a_0+a_1.2++a_2.2^2=\phi(2)=196$
$a_0+a_1.3++a_2.3^2=\phi(3)=62$

Case -II : $GF(13^2)$
Choose $(a_0,a_1,a_2)=(5,123,49)=(5, 9x + 6, 3x + 10) \in GF(13^2)$. $(\alpha_1,\alpha_2,\alpha_3)=(1,2,3)\in GF(13^2)$. Then
$\phi(x)=5+(9x + 6)x+(3x + 10)x^2$ and the system of equations becomes
$a_0+a_1.1++a_2.1^2=\phi(1)=12x + 8$
$a_0+a_1.2++a_2.2^2=\phi(2)=4x + 5$
$a_0+a_1.3++a_2.3^2=\phi(2)=2*x + 9$

How can I do the above using the same function for the both filed $GF(251)$ and $GF(13^2)$?

I would like to assign an integer corresponding to element of a finite field $GF(p^m)$, where $p^m\in[ {13^2,3^5, 131,137,139,251}] $
MWE:

 F.<x> = GF(3^5, impl='givaro')

THe elements of GF(3^5) are 0,1,2,x,x^2 etc, we would like to indexing each element such as $0-->0, 1-->1,2-->2,x-->3, x^2--4$ etc. Not only that, if I call any element for example if I call x^2 it should rerun 4 and conversely.

This process should work for the field order prime $p^m, m=1$ also. How can I do this?

@ JohnPalmieri I did according to you, but shows access denied: Access to the file was deniedThe file at file:///home/mks/.local/share/jupyter/runtime/nbserver-9493-open.html is not readable. It may have been removed, moved, or file permissions may be preventing access. ERR_ACCESS_DENIED

@ EmmanuelCharpentier BUt it does not solve my problem how to fix it?

SageMath is not working on Ubuntu 20.04. When I am going to open notebook interface it show the following errors:

https://imgur.com/a/aQQkUSQ

How can I fix this?

@ tmonteil Is there any package instead of "givaro" which has no bug?

I would like to solve a system of equations in a finite field of prime order $p$ (illustrated below with $p = 229$).

The system consists in four equations and has four unknowns $a_0$, $a_1$, $a_2$, $a_3$.

It depends on parameters $\alpha_i$, $b_i$, all in $F(p)$, for $i = 1, 2, 3, 4$.

The four equations are

$$a_0 + a_1 \alpha_1 + a_2 \alpha_1^2 + a_3 \alpha_1^3 = b_1$$ $$a_0 + a_1 \alpha_2 + a_2 \alpha_2^2 + a_3 \alpha_2^3 = b_2$$ $$a_0 + a_1 \alpha_3 + a_2 \alpha_3^2 + a_3 \alpha_3^3 = b_3$$ $$a_0 + a_1 \alpha_4 + a_2 \alpha_4^2 + a_3 \alpha_4^3 = b_4$$

To do this I have tried with the following examples:

pm = 229
bp = 229
F.<x> = GF(pm, impl='givaro')
R.<a0, a1, a2, a3> = PolynomialRing(F)

def NP(a):
    return F(ZZ(a).digits(bp))  # integer to polynomial

eqns = [a0 + a1*NP(2) + a2*NP(2)^2 + a3*NP(2)^3 - NP(78),
        a0 + a1*NP(3) + a2*NP(3)^2 + a3*NP(3)^3 - NP(136),
        a0 + a1*NP(4) + a2*NP(4)^2 + a3*NP(4)^3 - NP(179),
        a0 + a1*NP(5) + a2*NP(5)^2 + a3*NP(5)^3 - NP(166)]
A = matrix(F, [[eqn.coefficient(b) for b in R.gens()] for eqn in eqns])
b = vector(F, [-eqn.constant_coefficient() for eqn in eqns])
X = A.solve_right(b)
print(X)

But it shows erros:

Unhandled SIGSEGV: A segmentation fault occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
------------------------------------------------------------------------
/usr/share/sagemath/bin/sage-python: line 2:  7655 Segmentation fault      (core dumped) sage -python "$@"

How can I fix this?

When field size is of the form $p^m$, where $p$ is a prime and $m>1$ is a positive integer, there is no problem. For example

F.<x> = GF(5^2)
print F.fetch_int(4)

It outputs : 4.

But if we take field size is in the form $p^1$, then the above code does not working. For example

F.<x> = GF(5^1)
print F.fetch_int(4)

It gives errors. How can I fix this using the same function in both the cases?

I would like to solve the following system of equations over $\mathrm{GF}(17^2)$:

var('x')
F.<x> = GF(17^2, name='x', modulus=x^2 + x + 1)
R.<a0,a1> = PolynomialRing(F)

eqns = [a0+a1*x^2 - 7, a0+a1*x^3 - (7*x+14)]
A = matrix(F, [[eqn.coefficient(b) for b in R.gens()] for eqn in eqns])
b = vector(F, [-eqn.constant_coefficient() for eqn in eqns])
Y=A.solve_right(b)
print Y

It produces : (16x + 15, 8x + 16)
But the correct solution is (3x+10,7x+7).
How can I fix this problem?

A circle is a section of a sphere by a plane, the equation \begin{equation} x^2+y^2+z^2+2gx+2fy+2hz+d=0\tag{1} \end{equation} and \begin{equation} ax+by+cz=p\tag{2} \end{equation} together represent a circle. How do I plot like these diagram for given $g,f,h,a,b,c,p$. How to input by using @interact for the parameter $a,b,c,p$ to move the plane through the sphere ?