All Questions
12
questions
1
vote
0
answers
72
views
Is $\{x^3, x+b\}$ a generating set of $\mathrm{Sym}(\mathbb F_q)$?
Let $q=2^n$ where $n$ is a sufficiently large odd number. Consider the fintie field $\mathbb F_q$ and the symmetric group $\mathrm{Sym}(\mathbb F_q)$ over it.
I use $x^3$ to denote the permutation $x \...
- 21
1
vote
1
answer
151
views
How to calculate modulo of two polynomials
I am having issues with calculating things like this:
$3x^2 + 2 \mod x^2 + 3$
How would I got about this? I somehow couldn't find information anywhere.
- 11
0
votes
1
answer
71
views
True or False questions regarding $π½_9$ with the irreducible polynomial $x^2 +2x+2$
Let $π½_9$ be constructed with the irreducible polynomial $x^2 +2x+2$.
For $a,b \in π½_3$ we write $ax+b \in π½_9$ for $ab$.
In our exam we had to find out whether the following are true or false.
I ...
- 931
1
vote
1
answer
52
views
Why is $x^4+x^2+1$ over $π½_2$ a reducible polynomial? What do I misunderstand?
I don't quite understand when a polynomial is irreducible and when it's not.
Take $x^2 +1$ over $π½_3$.
As far as I know, I have to do the following:
0 1 2 using $x \in π½_3$
1 2 2 using $p(x)$
I ...
- 931
1
vote
1
answer
296
views
Number of solutions of a polynomial over finite fields
Consider in $\mathbb{F}_q[x_1,\dots,x_n]$, where $r$ is a positive integer dividing $n$, the polynomial
$$
f(x_1,\dots,x_n)=x_1x_2\dots x_r+x_{r+1}x_{r+2}\dots x_{2r}+\dots+x_{n-r+1}x_{n-r+2}x_{n}.
$$
...
- 71
1
vote
2
answers
424
views
Non-cyclic Codes
We are now studying all about cyclic codes. We determine when does a code C cyclic. My teacher give an easy example which is the Hamming Code. Then, he gave this question if there a non-cyclic hamming ...
- 33
1
vote
1
answer
355
views
Extended Hamming code to cyclic code
Is there any way to present [8, 4] extended Hamming code as a cyclic code?
Empirically, it seems not possible; however, I cannot prove or disprove it.
0
votes
1
answer
302
views
Find the number of monic square-free polynomials of degree j over finite field [duplicate]
Find the number of monic square-free polynomials of degree j >=1 over the finite field GF(q) ?
I have no idea how to approach this. I was thinking if there was a way to write a monic polynomial ...
- 860
1
vote
1
answer
164
views
Discrete Logarithm Problem in $GF(p^m)$
I have question regarding DLP in $GF(p^m)$
I know the algorithms for solving the DLP in $GF(p)$ like Baby Step-Giant Step, Pohlig-Hellman etc...
But what if we move into the $GF(p^m)$ and are ...
- 131
2
votes
1
answer
274
views
polynomials over finite field with irreducible factors of odd degrees
It is well-known that the number of monic $n$-degree polynomials over a finite field of size $q$ is $q^n$. How many such degree-$n$ polynomials can be completely factored into only irreducible ...
- 943
2
votes
2
answers
926
views
Definition of a primitive polynomial
I understand there are already some questions (A, B) on primitive polynomials. But none of these clears my confusion.
In page 84 of Handbook of Applied Cryptography, primitive polynomial has been ...
- 1,811
4
votes
5
answers
2k
views
Primitive polynomials
I am revising for a discrete mathematics exam and as quite stuck on this question.
Show that the polynomial $f = x^2 + 2 x + 3 \in \mathbb{Z}_5[x]$ is primitive. How many monic primitive quadratic ...
- 41