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22
votes
1answer
351 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 ...
12
votes
2answers
644 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 ...
8
votes
3answers
724 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] : ...
4
votes
1answer
721 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 ...
4
votes
1answer
187 views

Computing Smith Normal form of a matrix with mod 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 polynomial ...
3
votes
1answer
113 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 ...
3
votes
1answer
90 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 ...
2
votes
2answers
72 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 ...
2
votes
1answer
231 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
votes
1answer
116 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: ...
1
vote
0answers
51 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 ...
1
vote
0answers
74 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 ...
1
vote
0answers
417 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 ...
1
vote
0answers
392 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: ...
0
votes
3answers
321 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 ...
0
votes
1answer
65 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.
0
votes
1answer
94 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, ...
0
votes
1answer
303 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 ...
0
votes
1answer
50 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 ...
0
votes
1answer
45 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:...
0
votes
1answer
194 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)...
0
votes
0answers
28 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 B├ęzout relation: $sg+th=d$. ...
0
votes
0answers
55 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 ...
0
votes
1answer
316 views

Finite Field matrix rank calculation

How does one define a matrix over $\mathrm{GF}(p^r)$ in Mathematica in order to compute rank? I am working with $\mathrm{GF}(2)$?
0
votes
0answers
39 views

Equality of Galois Fields (GF[5][{4}]^4 = GF[5][{3}]^4)?

under FiniteFields, how can i test if, for example, if GF[5][{3}]^4 == GF[5][{2}]^4 Using regular == won't work.