# Tagged Questions

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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 ...
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 ...
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] : ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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: ...
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 ...
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 ...
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 ...
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: ...
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 ...
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.
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, ...
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 ...
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 ...
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:...
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)...
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 Bézout relation: $sg+th=d$. ...
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 ...
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 $\mathrm{GF}(2)$?
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.