4
votes
0answers
148 views

# roots of polynomials over GF(2)

Consider a polynomial $p(x)$ with of degree $d$ with coefficients in $GF(2)$. How many roots can it have in $GF(2^m)$? The intent here is that $d \ll 2^m$. The trivial bound is of course $\leq d$. ...
4
votes
1answer
222 views

Full-rank rectangular matrices over GF(2)

Given positive integers $k$, $m$, $n$, let $A$ be an $m \times n$ matrix over $GF(2)$ constructed as follows. Let $X_1, \ldots, X_m$ be independent random subsets of $\{1,\ldots,n\}$ with cardinality ...
10
votes
3answers
640 views

Probability of coprime polynomials

Given positive integer $N$, we choose $m_1, m_2, n_1, n_2$ independently and with equal probabilities from $\{0,1,\ldots,N\}$, and let $f_1 = x^{m_1} + (1+x)^{n_1}$ and $f_2 = x^{m_2} + (1+x)^{n_2}$ ...
0
votes
1answer
93 views

Probability of summing products of irreducible polynomials in a finite field to zero

Let $f(x), g(x), h(x)$ be randomly chosen irreducible polynomials over the finite field $GF(2^n)$. What would the probability be for $\sum_{(i,j,k:i,j,k\in\mathbb{N},i+j+k=C)} f^i(x)g^j(x)h^k(x)=0$, ...
2
votes
0answers
172 views

Small geometric progression modulo N

An problem related to integer factorization using the General Number Field Sieve is the following: Let $N$ be a composite. Must there exist a 5-term geometric progression $\lbrace ...
5
votes
1answer
362 views

Probability of a set of random vectors over finite field being a spanning set

Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, ...