Questions tagged [finite-fields]
A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
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On the sheaves-functions dictionary
Let $X$ be a variety over a finite field $k$. Let $\pi_{1}(X)$ be the arithmetic etale fundamental group of $X$, and $\rho:\pi_{1}(X)\to k^{\times}$ a continuous character. If $x: \text{Spec}(k)\to X$ ...
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Given a random point on a curve defined over a prime field, is it possible to compute 2 different scalar integers that will lead to the same result? [closed]
Simple question : given a randomly selected point $P$ belonging on a Edwards curve defined on a prime field, does 2 scalars $S1$ $S2$ exist such as :
$packed(S1\cdot P)= packed(S2\cdot P$) (where ...
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Are there conditions for an elliptic curve to have a quadratic $\mathbb{F}_q$-cover of the line without ramification $\mathbb{F}_q$-points?
Consider an elliptic curve $E: y^2 = f(x) := x^3 + ax + b$ over a finite field $\mathbb{F}_q$ of characteristic $> 3$. Obviously, the projection to $x$ is a quadratic $\mathbb{F}_q$-cover of the ...
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What do we know about efficiently finding a solution to a system of multivariate polynomials over finite fields?
Consider the following (NP-complete) problem:
Given a system of polynomials $f_1, f_2, \ldots, f_m \in \mathbb{F}_q[x_1, x_2, \ldots, x_n]$ of total degree at most $d$, find an $\mathbb{F}_q$-rational ...
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ECDLP : what’s the discrete logarithm between self or identical points? [closed]
Let it be a random point $X$ on a given well-known elliptic curve (be it a Weierstrass or Edwards or Montgomery curve) defined over a prime field $P$, then is there a way to efficiently compute 1 or ...
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Is generating semirandom blake256 hashes until packed points is on the curve, a safe algorithm to avoid the discrete log between the generated points?
I know there are more robust methods, but I wanted to know about this specific one
For any distinct said randomly generated point : $P_i,P_j\in \{P_1,...,P_k\}$ it should be hard to find $s$ such that ...
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Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
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Solving sparse bilinear systems with a relatively large number of variables
I'm trying to solve a bilinear system of equations over a finite field. (More specifically: I'm trying to find a single solution, if one exists.) The system consists of equations of the form
$$y^T A_i ...
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Asymptotic size of largest subset in $\mathbb F_p^2$ defining only lines of different slopes
Suppose that all lines defined by pairs of distinct elements in a subset of $\mathbb F_p^2$ have different slopes. How large can such a subset be asymptotically (for primes $p\rightarrow \infty$)?
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Under row operations and column permutations a matrix A can be put in the non-unique form ( I | X ), what is known about the set of possible X?
Given a full row-rank matrix $A$, this can be put into a unique reduced row echelon form via elementary row operations. Allow column permutations (no column addition / multiplication) and this can be ...
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Algebraic independence and substitution for quadratics
Let $f_{1},...,f_{n-1} \in \mathbb{F}[x_1,...,x_n]$ such that $\{ f_1,..., f_{n-1},x_n \}$ is algebraically independent over $\mathbb{F}$. Let $G \in \mathbb{F}[x_1,...,x_n,y_1,...,y_{n-1}]\...
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Relation between minimality and algebraic independence for binomials?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
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Minimality implies algebraic independence?
$\DeclareMathOperator\supp{supp}$Given $f_1,...,f_n \in \mathbb{F}[x_1,...,x_n]$ such that
$f_1 = x_1 + q_1$
$f_2 = x_2 + q_2$
$\cdot \cdot \cdot$
$f_{n-1} = x_{n-1} + q_{n-1}$
$f_{n} = q_n$
such that ...
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Are these kinds of bases for $\mathbb{F}_2^q$ seen as a vector space studied?
In the context of my research, I have to work with sets of vectors $\left\{y_i\right\}_{i\in[n+1]}$ of $\mathbb{F}_2^n$ such that the following property is true:
$$\forall i\in[n+1], \left\{y_i\oplus ...
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Counting zero-sum subsets of a finite field with a particular form
Let $\mathbb{F}$ be a finite prime field of characteristic different than $2$ and $\beta \in \mathbb{F}$ a generator of the $2$-power order multiplicative subgroup of order $2^k$, so $\beta^{2^{k-1}} =...
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