0
votes
0answers
71 views

Multivariable irreducible polynomials over finite fields

It is not difficult to prove the following result, and it seems that it should be already proved. I would appreciate it if someone offer me some reference to it. For any $f(x_1,\dots, x_n)=\sum ...
0
votes
2answers
35 views

Subgroups of Galois groups of finite fields

According to the notion of Galois group, for $E=GF(2^n)$ as an extension of the field $F=GF(2)$, the Galois group $Gal(E/F)$ is a cyclic group of order $n$. Now my question is: for finding the ...
0
votes
0answers
104 views

Order of orthogonal groups over finite field

The wikipedia article gives a formula for calculating the order of an orthogonal group over finite filed $O(n,q)$: I don't see how I can get such formula. Can one come up with some references?
0
votes
0answers
69 views

How can commuting with Frobenius imply the order of an element in the inertia group.

In this video, one asserts, in the beginning, that, for $\tau\in \mathbb V_0$ such that $\tau$ generates $V_0/V_1$ in the quotient group, and $\sigma\in \mathbb Z$ which is a Frobenius in $\mathbb ...
1
vote
0answers
61 views

The mathematics behind Sobol sequences

I am using Sobol sequence as random number generator in a computer program. Beyond just making the program work, I would like to learn the mathematics behind the Sobol sequence (and other ...
3
votes
1answer
68 views

Generators for $\mathbb F_p^*$

Let $p$ be prime, then it is a well-known fact that $\mathbb F_p^*= \mathbb F_p -\{0\}$ is a cyclic group under multiplication. Are there any methods to determine the generators of this cyclic or any ...
2
votes
1answer
124 views

Products of linearly independent sets in finite fields

Let $\mathbb F_q$ be the finite field with $q$ elements, $q=2^n$. This is a vector space over $\mathbb F_2$. My question is rather general: given two linearly independent sets of vectors of the same ...
11
votes
2answers
466 views

Whats the probability a subset of an $\mathbb F_2$ vector space is a spanning set?

Let $V$ be an $n$-dimensional $\mathbb F_2$ vector space. Note that $V$ has $2^n$ elements and $\mathcal P(V)$ has $2^{2^n}$. I'm interested in the probability (under a uniform distribution) that an ...
2
votes
1answer
176 views

Transition between field representation

In a number of papers related to efficient implementation of the AES Sbox, people are computing stuff (the multiplicative inverse for instance) in GF(($2^4$)$^2$) instead of GF($2^8$). In some cases ...