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Questions tagged [finite-fields]

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Is linear algebra over $\mathbb{F}_2$ or a finite field possible? [closed]

I was interested in using Mathematica to do linear algebra over $\mathbb{F}_2$ in the context of solving for the code space of linear codes given a parity check matrix in $\mathbb{F}_2$. I've searched ...
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1answer
170 views

How to construct addition and multiplication table for finite field?

For example, choose $p(X) = X^4 + X + 1$, how to construct addition and multiplication tables for $\mathbb{F}_{2}[X] \mod{X^4 + X + 1} $ ?
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1answer
61 views

How to generate addition table for $GF(2)[x] \mod x^3 + 1 = 0$ [closed]

I have been playing with Mathematica for a while. I tried generating addition table for a simple ring $R$ such that $R = \mathbb{Z}_{15}$ as asked in my last question. However, I am completely ...
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1answer
63 views

How to generate addition table for ring $\mathbb Z_{15}$?

How do I generate an addition table for ring R such that $R = \mathbb{Z}_{15}$ or generally speaking, how to generate an addition table for any polynomial ring <...
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1answer
62 views

Plotting over a finite field

Is there a way to use a finite field for plotting functions in Mathematica? I couldn't find any options in the documentation.
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93 views

singular value decomposition of a matrix over a field K

Can I do singular value decomposition of a matrix which has entries from a field K, in my case $Z_m$, in Mathematica? For example, I have a matrix over a field Z2, with entries only 0 and 1. Using an ...
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1answer
65 views

What is the number of (binary) 4 x 4 matrices over GF(2) (the field with 2 elements) that do not have an eigenvalue of 0 or 1

My code below returns 4032. The OEIS sequence A002820 says there are 5824 such matrices. Is there something wrong with my code. I am assuming that the eigenvalues of a matrix are precisely the ...
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1answer
137 views

Checking matrix-invertibility in finite fields

Suppose the finite field $F=GF(2^d)$ contains all $p$-th roots of unity, where $p$ is a prime and $\omega\in F$ is the principal $p$-th root of unity. Let $V=(\omega^{ij})$ be a $p\times p$ ...
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0answers
310 views

How to compute the eigenvalue of a matrix over a finite field

How to compute the eigenvalue of a matrix over a finite field? I would like to compute the eigenvalues of the following matrix over $\mathbb F_5$. ...
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2answers
135 views

Construct Matrices from Coefficient of f[x] in Finite Fields?

I'm working on a problem at the moment using Mathematica and the Finite Fields package, and I've ran into some difficulty converting the problem to a Matrix, so I'm hoping somebody here would know how ...
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1answer
158 views

Matrix operations “Modulus->non-prime” problem

I am using Mathematica 10.4.1.0. When I specify the "Modulus" option to be a non-prime, the functions RowReduce and MatrixRank return the error ...
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0answers
136 views

display the elements of a quotient ring

How can I display the elements (polynomials) of the ring $$\mathbb Z_3[x]/\langle x^3+2x^2+1\rangle$$ Is there a built-in function that displays them?
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1answer
115 views

How to plot a polynomial modulo a prime

I want to plot graphs of polynomials, and the solutions to equations, modulo a prime. So for example, for modulus 5 plot the graph of $x^2$, or the solutions to $x^2+y^2=2$, as subsets of $\mathbb{F}...
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0answers
65 views

Finite fields package doesn't simplify

I'm trying to do arithmetic using the FiniteFields package. Supposing a is a generator of a Galois field (say GF(2^5)) I want to be able to simplify things like a^32 and (a^8)^-1. I don't care about ...
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1answer
90 views

How to perform an euclidian division in a Finite Field?

I am quite new to mathematica, I am looking for a function similar to QuotientRemainder[] but which works with finite fields as implemented by the ...
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1answer
375 views

How can I make general linear group over finite field?

I'm newbie in mathematica. I would like to construct a conjugacy class of an involution in GL over a finite field, but have no idea how to make the group and a matrix over the field. I'm sure that ...
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1answer
333 views

How can we do LDU decomposition modulo $p$?

If we have an $n \times n$ matrix, with all entries taken modulo $p$, how can we output the three matrixes in LDU decomposition, with all entries again modulo $p$? We can assume the input matrix is ...
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1answer
77 views

Why is While not showing any output?

I'm trying to use the Finite Fields package to show that $x$ is not a primitive element of $\mathbb{Z}_3[x]/\langle x^3 + 2x + 2 \rangle \cong GF(3^3)$. The idea is to take the following piece of code:...
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0answers
52 views

PolynomialExtendedGCD in 2 variables

Consider a field $R$ and the ring $A=R[y]$. Consider two polynomials $g,h\in A[x]$. I want to obtain $d=\gcd(g,h)\in A[x]$ and two polynomials $s,t\in A[x]$ satisfying the Bézout relation: $sg+th=d$. ...
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0answers
76 views

How can I express Sqrt[2] as an element in the field Q(2^(1/4))?

If I input: ToNumberField[Sqrt[2], 2^(1/4)] Mathematica returns: AlgebraicNumber[Root[-2 + #1^4 &, 2], {0, 0, 1, 0}] How ...
3
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1answer
134 views

How do I convert Subscript[{4, 5, 1}, 7] to the integer 4*1 + 5*7 + 1*7^2 = 88?

I am working with the finite fields package in Mathematica 9. If I input: GF[7, {-2, 0, 0, 1}][{3, 4, 1}]^2 Mathematica gives me $\{4,5,1\}_7$. I want to ...
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2answers
109 views

How do I find a polynomial in a field?

If I have a polynomial: $$f(x) = c_0 x^0 + c_1 x^1 + c_2 x^2 + \dots + c_n x^n$$ How can I find the polynomial, modulo a prime number $p$? In other words, I want to take all of the coefficients ...
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1answer
1k views

Fast calculation of discrete logarithms

Does Mathematica have any built-in fast algorithms for calculating discrete logarithms over $(\mathbb{Z}_p)^\times$ (the group of integers modulo $p$)? Essentially, for a fixed large prime ...
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1answer
142 views

Primitive polynomials of a field [closed]

I would like to know how I can get the primitive polynomials to generate the points of the fields GF(7) and GF(9) using Mathematica. Any help is appreciated.
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1answer
135 views

updating matrix elements based on a condition

I am (still) fairly new to Mathematica and trying to perform some operation on the matrix elements, say for the matrix ...
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0answers
71 views

Sum of zeros in finite Galois field

In the following code: << FiniteFields` GF[2][{0}] + GF[2][{0}] == GF[2][{0}] GF[2][{0}] + GF[2][{0}] + GF[2][{0}] == GF[2][{0}] Why does the second line ...
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0answers
185 views

Package FiniteFields

In a Mathematica session, I evaluate: Needs["FiniteFields`"]; fld = GF[2]; Now, for example, I'd like to compute $a^3+a^5$ for $a \in{Z_2}$. The result I'd like ...
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1answer
315 views

Computing Smith normal form of a matrix with $\bmod p$ coefficients

I would like to compute the Smith normal form of a matrix with coefficients in $GF(p)$. In particular, I am interested in $GF(2)$. I have used the Smith normal form packages for integer and ...
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0answers
962 views

Elliptic curve cryptography in Mathematica

I can find no resources for doing elliptic curve cryptography. I have used the finite field package, but I find it cumbersome and it does not seem to have any builtin methods for ECC. How can I get ...
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1answer
174 views

Square root of a value defined in a finite field?

I am trying to find the right way to compute the square root of a number defined in a finite field. For example, ...
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1answer
805 views

How to enter and solve this equation in finite fields?

Is there a way in Mathematica 9 to enter and solve the following equation $p(x) = r^x$ where $p(x)$ is a polynomial whose coefficients are drawn from a finite field, and $r$ is a primitive root of ...
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1answer
571 views

How can I calculate all irreducible polynomials of 31 degree in $\mathbb Z_2[x]$?

How can I calculate all binary irreducible polynomials of degree 31? or how i calculate all irreducible $f$ in $\mathbb Z_2[x]$? (The irreducible polynomial in $\mathbb Z_2[x]$ and $\mathbb R$ are ...
2
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1answer
164 views

How can we use RowReduce with a modulous AND variables?

We can use RowReduce with a field. For example, we state RowReduce[{{1,3,5},{0,1,2}},Modulous->23] ...which then returns: ...
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0answers
640 views

Polynomial factorization over finite fields with non-prime order

One can easily factor a polynomial over finite fields of prime order, using Factor command: ...
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3answers
444 views

How to evaluate all the essentially distinct polynomials in 4 variables over $F_2$ on points of $F_2 ^ 4$

I am a beginner with Mathematica. For my research purpose I would like to get a list of all the polynomials in $F_2[x,y,z,w]$ and for each polynomial I would like to know the result that it gives then ...
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1answer
260 views

Interpolating a Bivariate Polynomial over a Finite Field

Let $F=GF(p)$ be a finite field, $p$ prime and write $F^\times=\{x_1,\ldots,x_n\}$. I'm trying to implement an earlier version of Sudan's list-decoding algorithm for Reed Solomon Codes $$RS(d+1,n+1)...
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3answers
1k views

How do I make Reduce yield all solutions explicitly?

Say I want to do the following: Reduce[ x+y+z==1, {x,y,z}, Modulus -> 7 ] then I get a solution with parameters, C[1] : <...
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1answer
1k views

How to do the polynomial stuff over finite fields extensions fast?

This question is raised from the problem of package FiniteFields being very slow (please, see the corresponding question): I have had an evidence that Mathematica ...
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2answers
789 views

FiniteFields package is very slow. Any fast substitute for Mathematica?

I want to compute the inverse of matrix, say with dimensions $100 \times 100$, defined over a large finite field extension such as $GF(2^{120})$. I am using the package FiniteFields, but Mathematica's ...