Questions tagged [finite-fields]
The finite-fields tag has no summary.
54 questions
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Why is rule matching FiniteField not working? [closed]
I am introducing a rule:
$Post = # /. x_?NumericQ FiniteField[q_?PrimeQ] [s_] :> FiniteField[q] [x] FiniteField[q] [s] &;
But for some reason, upon ...
- 3,862
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2
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236
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FiniteField elements beyond the field size
FiniteField[2][2] should give FiniteField[2][0] in my opinion. Instead it gives ...
- 3,862
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0
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110
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Strange incorrect output message for modular r-th root computation with PowerMod
In this [Wolfram reference page] we read that:
PowerMod[a,1/r,m] should find a modular $r$-th root of $a$ modulo $m$.
However, with Engine 14.1.0 I get a claim that ...
- 381
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100
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Nullspace etc using a different inner product over finite fields [closed]
In the context of coding theory, linear codes over finite fields make repeated use of kernels/nullspaces for creating e.g. parity check matrices.
However, often the "dot" product between two ...
- 11
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101
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Is it possible to use the Finite Fields package to define the elements of GF(4) in terms of the irreducible polynomial $P$?
I am new to the Finite Fields package and am finding the package tutorial confusing.
I am wondering, if I am working over GF[4], is there a way of finding the elements of GF[4] in terms of the ...
- 997
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271
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Solving matrix equations over a finite field
I'm afraid this question is trivial, but I have spent the last couple of hours trying to figure out what goes wrong without any luck.
Given the following:
...
- 143
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137
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Compute all bivariate polynomials over GF(2) of degree d or less and evaluate them at certain polynomial input
I want to compute all $<=d$ degree bivariate polynomials of form $f_1(x)g_1(y) + f_2(x)g_2(y)$, over field $GF(2)$, and evaluate them at a certain polynomial input
for eg $d = 1$, evaluation at $(p^...
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1
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157
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How do I get the span of a list of vectors from a finite field of prime order
I have a list of vectors from a vector space over a finite field of order p for some prime p. For example: I have: {{1,0,1},{0,1,2}} where the vectors are in $\mathbb{F}_3^3$. I want a list of the ...
- 897
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3
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696
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Reducing Polynomial Functions over a Finite Field
Over $\mathbb{F}_{4}$, for example, I am looking for a function that will, for example, reduce the polynomial function (not an element of the finite field itself) $$x^5 + 6x^4 +x^3 + 1 \rightarrow x^2 ...
- 53
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187
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Inverse/Division in finite field?
Think of multiplictavie group of finite field F[p,n], where p is a prime number and n is a positive integer.
The whole elements of F[p,n] can be represented as
p^n-p^(n-1) positive integers in the ...
- 4,435
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302
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Fields in Mathematica
Suppose $F=\{0,1,w,w^2\}$ is a field of $4$ elements, where $w$ is a root of $x^2+x+1$. (This means: $w^2=w+1,w=w^2+1, 1=w+w^2,w^3=1,2w=2w^2=1+1=0$). In order to find out the remainder $$\frac{(1+w ...
- 634
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80
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Does Mathematica limit the size of a set of permutation group generators?
I tried to generate general and special linear groups PGL2 and PSL2 over finite fields using straightforward algorithms over finite fields: The generators are permutations defined as the additive ...
- 1,822
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375
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Root finding over finite field extension
I'd like to know if there exists any method on Mathematica, third-party coded resource or library that can compute roots of a polynomial over an extension $\mathbb{E}$ where $E=F_p[x]/f(x)$ and $f(x)$ ...
- 203
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245
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How to compute the multiplicative order of an element in a finite field
I want to find the multiplicative order of say the element {0,1,0} in GF[8].
I made a list:
b = Map[GF[8][#] &, Tuples[{0, 1}, 3]]
I tried:
NestWhileList[b[[3]]*# &, b[[3]], # != b[[5]] &...
- 897
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2
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485
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Formatting results of a polynomial long division (Extension for finite fields)
I've just read Formatting results of a polynomial long division with great interest. For my teaching purposes it would be great to enable the code given in that post to treat polynomial division with ...
- 43
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271
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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 ...
- 175
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1
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246
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Parametric solution of a system of polynomial equations
I have the following system of equations,
1+x+y+z==0, 1+x*y+y*z+x*z==0
which I want to solve in the extension field of GF(2), the algebraic closure of GF(2) for ...
- 690
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601
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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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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 ...
- 233
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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 <...
- 233
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247
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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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311
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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 ...
- 690
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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 ...
- 1,731
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467
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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 $p\times p$ ...
- 11
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871
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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 $\mathbb F_5$.
...
- 101
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2
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330
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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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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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340
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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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323
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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}...
- 195
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139
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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 ...
- 121
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345
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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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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 ...
- 21
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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 ...
- 1,171
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210
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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:...
- 389
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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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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 ...
- 1,731
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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 ...
- 1,731
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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 ...
- 1,171
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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 ...
- 7,931
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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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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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284
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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 ...
- 13
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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 ...
- 1,237
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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 ...
- 4,142
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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,
...
- 4,142
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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 ...
- 175
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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 ...
- 121
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319
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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:
...
- 1,171
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1k
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Polynomial factorization over finite fields with non-prime order
One can easily factor a polynomial over finite fields of prime order, using Factor command:
...
- 433
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3
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731
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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 ...
