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Questions tagged [finite-fields]

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28 votes
6 answers

Alternative proofs of Schwartz–Zippel lemma

I'm only aware of two proofs of Schwartz–Zippel lemma. The first (more common) proof is described in the wikipedia entry. The second proof was discovered by Dana Moshkovitz. Are there any other ...
Dai Le's user avatar
  • 3,664
15 votes
2 answers

Status of Raghavendra's algorithm for solving linear systems in finite fields

In 2012, Lipton wrote a blog entry about a new algorithm for solving linear systems over finite fields by Prasad Raghavendra. The link to Raghavendra's draft paper on the topic is now dead, and I can'...
okintheory's user avatar
11 votes
1 answer

Binary vector $t$ in $span(S)$ over $\mathbb{Z}/q\mathbb{Z}$ for all prime powers $q$ $\Rightarrow$ $t$ in $span(S)$ over $\mathbb{Z}$?

I have a set of $n$ binary vectors $S = \{s_1, \ldots, s_n \} \subseteq \{0,1\}^k \setminus \{1^k\}$ and a target vector $t = 1^k$ which is the all-ones vector. Conjecture: If $t$ can be written as ...
Bart Jansen's user avatar
  • 5,275
7 votes
0 answers

How is a "low-degree polynomial" precisely defined in Algebrization?

I'm going through papers which present algebrization as a barrier and I'm trying to understand how "low-degree" polynomials are precisely defined, i.e. are they low with respect to the ...
Ari's user avatar
  • 285
6 votes
2 answers

Reference request for linear algebra over GF(2)

I have been looking for materials on the linear algebra over $GF(2)$ but so far I haven't found any substantial textbooks or notes on this subject. In fact in one of the notes I found the introduction ...
gen's user avatar
  • 290
6 votes
2 answers

How to find a non-zero point of a non-zero polynomial of low degree?

Given a circuit that computes a polynomial $P(x_1 \dots x_n)$ of low formal degree over some large field $\mathbb{F}$. Moreover, given a point $X \in \mathbb{F}^n$, such that $P(X) \neq 0$. Can one ...
ivmihajlin's user avatar
6 votes
1 answer

complexity of deciding whether there's a small polynomial with a given root

Let $f\in (\mathbb{Z}/p\mathbb{Z})^\ast$ be a nonzero element of a prime finite field. For $d, r\in \mathbb{N}$ consider the problem of deciding whether there is a nonzero polynomial $$P(x) = a_0 +...
Tim kinsella's user avatar
5 votes
0 answers

$\#$P hardness of computing weighted sum of degree $2$ polynomials

Consider polynomials $f: \{0, 1\}^{n} \rightarrow \{0, 1\}$ over $\mathbb{F}_2$ (addition and multiplication are taken modulo $2$.) Consider integers $x \in \{0,1\}^{n}$, written in binary. Let $\...
AngryLion's user avatar
  • 173
5 votes
0 answers

Algebraic dependence of roots of irreducibles over a finite field

I asked this question in Math SE too, but I have since modified it to make it more suited here. Also, in hindsight, the question itself was more algorithmic and was a better fit here. https://math....
BharatRam's user avatar
  • 393
5 votes
0 answers

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
Anurag's user avatar
  • 190
4 votes
2 answers

Hardness of finding roots of a degree $2$ polynomials over $\mathbb{F}_2$

Since every $3$-SAT instance with $n$ variables can be expressed as a degree-$3$ polynomial over $\mathbb{F}_2$ with $n$ unknowns, the NP-hardness of $3$-SAT directly translates to NP-hardness of ...
Vivek Bagaria's user avatar
4 votes
1 answer

Reference request: finite field computation over the Word-RAM model

Let $q = p^\ell$ be a positive integer power of a prime $p$, of size $q = \text{poly}(n)$. Over the Word-RAM model (with words of size $O(\log n)$), how quickly can we perform addition and ...
Naysh's user avatar
  • 686
4 votes
0 answers

Complexity of solving random underdetermined polynomial equations over finite fields

Consider a random system of degree-$d$ polynomials, with $n$ variables and $m$ equations, over some finite field $\mathbb{F}_q:$ $$\begin{align}\sum_{\substack{(\alpha_1,\dots,\alpha_n) \in \mathbb{Z}...
Quang Dao's user avatar
3 votes
1 answer

Constructing subfields of a finite field

Suppose we have a finite field $\mathbb{F}_{p^n} = \frac{\mathbb{F}_{p}[x]}{<f(x)>}$. I want a deterministic polynomial algorithm to compute all subfields of this field. I think we can do ...
xyz's user avatar
  • 243
3 votes
1 answer

Emptiness of complement of subspace arrangement

Given $k$ affine subspaces in $\{0,1\}^n$, consider the problem of testing whether their union covers all of $\{0,1\}^n$. What's the complexity of this problem? P.S.: It seems that this can be ...
arnab's user avatar
  • 7,010
2 votes
0 answers

Which invertible linear transformations can be computed reversibly without ancilla/garbage bits just as easily as they can be computed irreversibly?

Suppose that $L:F_{2}^{n}\rightarrow F_{2}^{n}$ is an invertible linear transformation. Then define $w(L)$ to be the gate count of the smallest reversible circuit on $n$ bits without ancilla/garbage ...
Joseph Van Name's user avatar
2 votes
0 answers

A random ensemble of sparse boundary operators

The following question arises from the study of quantum error correction, and high-dimensional expanders: Is there an algorithm that for given numbers $n>0,d≤n,r≤n$ samples uniformly a linear ...
Lior Eldar's user avatar
  • 1,224
2 votes
0 answers

Is berklemap-massey works on extention field

According to the articles that i read (also wikipedia [1]), they refer that the above algorithm works on finite field $GF(q)$. But I cannot find an evidence that it still works on extension field, i.e:...
user1387682's user avatar
2 votes
0 answers

Module Independence Graphs

Given $n, k \in \mathbb{N}$, we create a set $S \subseteq \mathbb{Z}_m^k$ by treating it as an $n\times k$ grid and choosing each $S_{ij}$ by sampling from $\mathbb{Z}_m$, such that $P(S_{ij} = z) = ...
Samuel Schlesinger's user avatar
1 vote
0 answers

BCH codes and polynomials with many values in a subfield

For points $P=\{x_1, \ldots, x_n\} \subset {\mathbb F}_{2^m}$ define $$\mathcal{C}(P, t) =\{(f(x_1), \ldots, f(x_n)) \mid \mbox{$f\in {\mathbb F}_{2^m}[X]$ has degree $t$}\}$$ and $\mathcal{C}'(P, t) =...
user6584's user avatar
  • 1,242
1 vote
0 answers

The curve used in Parvaresh-Vardy decoding

Consider the Parvaresh-Vardy list decoder. As I understand it, the idea is to decide on a curve over an extension field of the form $(f,f^h mod E, f^{h^2} mod E,\dots)$ and then evaluate each of ...
BharatRam's user avatar
  • 393
0 votes
0 answers

Solving Linear Equations over finite field $ Z_q $

Suppose we are given a linear equation $ Ax=b $, where $ A \in Z_q^{n \times m} $ and $ b \in Z_q^n $. Note that $ q $ is a prime here, and $ Rank(A)= Rank(A;b)=n<m $. I wonder whether the ...
zhen liu's user avatar
0 votes
0 answers

Looking for information about a problem of a least subset of vectors modulo 2 summing to another vector [duplicate]

I'm quite interested in the following algorithmic problem, on which I can't find any information. Phrased as a decision problem: Given a set of vectors $V$ in $\text{GF}(2)^n$, a vector $\mathbf u$ ...
Maja Trela's user avatar
0 votes
0 answers

Searchable finite field

Let $F$ be a large finite field, where the elements are strings of length $n$. We require, addition, multiplication, and division to be efficient (polynomial in $n$). We say that $F$ is searchable if ...
Christopher King's user avatar
0 votes
0 answers

On permanent mod $3^t$ computation

By $'$ I mean transpose. I gather the info here from We know that if $U\in\Bbb F_{3^t}^{n\times n}$ satisfies $UU'=I_n$ in $\Bbb F_{3^t}$ ...
Turbo's user avatar
  • 13k