All Questions
Tagged with finite-fields nt.number-theory
220 questions
0
votes
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80
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Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
- 119
-1
votes
0
answers
73
views
Why is there in theory no morphism/isogenies when enlarging a prime field sharing a common suborder/subgroup? [closed]
Simple question : I have a prime field having modulus $p$ where $p−1$ contains $O$ as prime factor, and I have a larger prime field $q$ also having $O$ as its suborder/subgroup. Why are there no ...
- 87
8
votes
1
answer
366
views
Why do we have fewer distinct Gauss sums over a field of characteristic $2$?
Let $p$ be a prime number and $q$ be a power of $p$. Fix a non-trivial additive character $\psi\colon \mathbb{F}_q\to \mathbb{C}^\times$ and, for each non-trivial multiplicative character $\chi\colon \...
- 711
2
votes
0
answers
120
views
Looking at versions of Implicit Function Theorem (IFT) on rings
$ \let \ovr \overline
\def \Z {\mathbb Z}
\def \C {\mathbb C}
\def \F {\mathbb F}
\def \P {\mathcal P}
\def \x {\boldsymbol x}
\def \a {\boldsymbol a} $
Let $ \P = \{ p _ i ( \x , y ) \} _ { i = 1 } ^ ...
- 346
0
votes
0
answers
122
views
Is it in theory possible to create a subexponential algorithm for solving discrete logarithms in multiplicative subgroups or within an Integer range?
As far I understand, when it comes to finite fields, Pollard rho and Pollard’s lambda are still the best algorithm for solving discrete logarithms in a multiplicative subgroup/suborder…
Index calculus ...
- 87
8
votes
1
answer
827
views
Integer solutions for a simple cubic
What are the integer solutions to $x^3 - 7xy + y + 1 = 0$? A computation only finds $(0, -1), (-1, 0), (-6, 5), (-49, 342)$. This is surprisingly few.
Are these all of them? Is there an algebraic ...
- 83
3
votes
0
answers
174
views
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 $x: \text{Spec}(k)\to X$ ...
- 667
0
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0
answers
61
views
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 ...
- 87
4
votes
1
answer
338
views
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 ...
- 153
3
votes
1
answer
334
views
Sum of products of Fourier coefficients in finite field
Let $\mathbb{F_q}$ be some finite field and let $f,g: \mathbb{F_q} \to \mathbb{C}$.
By $\hat{f}, \hat{g}$ let's denote the Fourier coefficients of $f,g$ with respect to the additive characters of the ...
- 153
6
votes
0
answers
176
views
Fundamental lemma of sieve theory in function fields
Is there any literature concerning the fundamental lemma of sieve theory in $\mathbb{F}_q[T]$?
In integers there are various versions of the lemma (bases on different sieves); I would be happy with ...
- 14.6k
1
vote
0
answers
121
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Solution formula in an explicit equation over $\mathbb{F}_p^3$
I'm looking into a formula involving prime numbers $p \geq 7$ and an equation's solutions. The equation in question is:
$$z^2 = (x^2 - 4x)(y^2 - 4y)((x + 1 - y)^2 - 4x),$$
where $(x,y,z)\in \mathbb{F}...
- 71
9
votes
0
answers
261
views
Who was the first to prove that the automorphism group of a finite field is cyclic and is generated by the Frobenius automorphism?
$\DeclareMathOperator\Aut{Aut}$It is well-known that the automorphism group $\Aut(F)$ of a finite field $F$ of characteristic $p$ is cyclic of order $n$ where $|F|=p^n$. Moreover, the cyclic group $\...
- 41.8k
4
votes
0
answers
163
views
When is $q$ invertible mod $m$, mod its order mod $m$, mod its order mod its order mod $m$, ad infinitum?
Fix an [edit: positive] integer $q$. Let me say that an [edit: positive] integer $m$ is IK over $q$ if $q$ and $m$ are coprime and the (multiplicative) order of $q$ mod $m$ is IK over $q$. Note that ...
- 54.6k
3
votes
0
answers
118
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 563
2
votes
0
answers
124
views
Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 18.4k
2
votes
2
answers
444
views
Defining a sign of square roots in GF(p)
$\DeclareMathOperator\GF{GF}$Consider the following expression:
$$
\sqrt{a_1} \pm \sqrt{a_2} \pm \dots \pm \sqrt{a_n} = 0,
$$
where $a_1, \dots, a_n$ are positive integers. We want to find the number ...
- 1,171
1
vote
0
answers
71
views
Bias of $a^k / q$ modulo $q$?
Let $q$ be a prime. Let $0< a < q$ be an integer so that it is primitive modulo $q$. Let $k$ be a random integer up to $q-1$. Consider
$$a^k = b_k + q * c_k$$
as $k$ varies modulo $q^2$. So $b_k$...
- 591
0
votes
1
answer
195
views
Trying to solve for the remainder of $a^q$ modulo $q$
Let $q$ be a prime and $a$ be a number from $0$ to $q-1$ (not an equivalence class).
The elements $a^q$ are exactly the elements of order $q-1$ modulo $q^2$.
I'm trying to solve the equation:
$$a+2*\...
- 591
4
votes
0
answers
175
views
Intrinsic maps between complex integers modulo $p$ and integers modulo $p+2$
$\DeclareMathOperator\GF{GF}$Let $p$ and $p+2$ be twin primes. Let's assume that $-1$ is not a quadratic residue modulo $p$ (and therefore is a Q.R. modulo $p+2$).
Consider the complex numbers $a+bi$ ...
- 591
3
votes
0
answers
73
views
Is the discrete logarithm equivalent to solving polynomial discrete logarithms?
Suppose we can quickly solve the discrete logarithm modulo $p$. Let's say $2$ is a generator so we can quickly find $l$ for which $2^l =h$ for any given target $h$.
An interesting observation is that ...
- 591
1
vote
1
answer
259
views
On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 563
0
votes
0
answers
92
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 563
1
vote
0
answers
91
views
Functions that take quadratic residues to non quadratic residues
Let $p$ be prime and $Q$ be the set of integers $x$ mod $p$ so that $x^2-1$ is a quadratic residue. Let $Q^c$ be the complement of $Q$. If we don't consider $x = 1$ then these two sets have the same ...
- 591
4
votes
0
answers
263
views
Cosine Modulo $p$?
Consider the integers modulo a prime $p$. I'm looking for a nice polynomial function that acts as a sort of "cosine" on the integers modulo $p$. Specifically, I'm looking for solutions to ...
- 591
4
votes
1
answer
334
views
GCD in $\mathbb{F}_3[T]$ with powers of linear polynomials
This is a continuation of my previous question on $\gcd$s of polynomials of type $f^n - f$.
Let us call $n > 1$ simple at a prime $p$ when $p-1 \mid n-1$ but $p^k - 1 \not\mid n-1$ for all $k > ...
- 63.1k
2
votes
0
answers
145
views
What is the periodicity of $((a^n \text{ modulo } p) \text{ modulo } q)$
This feels like it should be elementary but it came up in my research and I was not able to solve it.
We can ask this question for any $p$ and $q$ but,let $p$ and $q$ be primes for simplicity. The ...
- 591
5
votes
1
answer
388
views
Fermat cubic hypersurfaces over finite fields
Consider the Fermat cubic
$$
X = \{x_0^3+\dots +x_n^3 = 0\}\subset\mathbb{P}^n_{\mathbb{F}_{q}}
$$
over a finite field $\mathbb{F}_{q}$ with $q$ elements.
If $q \equiv 2 \mod 3$ then the projection $\...
- 8,998
4
votes
1
answer
237
views
Points on affine hypersurface over finite field
I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by
$$
X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\}
$$
over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
- 8,998
2
votes
0
answers
191
views
Factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$ [closed]
Is anything known about the factorization of the polynomial $x^k + x^{k-1} + x^{k-2} + \cdots + x + 1$ in $\mathbb{F}_2[x]$?
When can it be factored, what are the irreducible factors, what are the ...
- 219
2
votes
1
answer
154
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 563
4
votes
1
answer
334
views
Polynomial that is not always a square over $\mathbb{Z}_p$
Let $p > 3$ be prime. Is is true that there exists $x \in \mathbb{Z}_p$ such that
$$
(1+x^2)^3-1
$$
is not a square in $\mathbb{Z}_p$? In particular, when $-1$ is not a square in $\mathbb{Z}_p$, ...
- 43
0
votes
0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user126352
1
vote
1
answer
468
views
Artin's conjecture for polynomials and rational functions over finite fields
Artin's conjecture on primitive roots over the integers states that a given integer
$0\ne h\in \mathbb{Z}$ that is neither a square number nor $-1$ is a primitive root modulo infinitely many primes $p$...
- 11
0
votes
0
answers
122
views
Will an integer program to deterministically factor integers help derandomize $\mathbb F_q[x]$ factoring?
There are many analogies between the objects $\mathbb F_q[x]$ and $\mathbb Z$.
Supposing there is a fixed (say $10^9$) dimension linear integer program (describable without any objective function) in ...
- 13.9k
3
votes
0
answers
91
views
Equirepartition of sums for large multisets in subsets of finite fields
Let
$p$ be a prime number and let $\mathcal A$ be a subset of $a\leq p$ distinct
elements in $\mathbb F_p$.
We denote by $\mathcal M_k(\mathcal A)$ the set of all ${k+a-1\choose k}$
multisets ...
- 17.9k
1
vote
0
answers
254
views
Reductions of a system of equations at various primes
Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^...
- 685
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0
answers
253
views
Cardinality of a polynomial image $\pmod{p^n}$
Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...
- 61
0
votes
1
answer
269
views
Product of subspace and its inverse
$\DeclareMathOperator\GF{GF}$Let $R=\GF(q)$ be a finite field with $q=p^r$ elements, where $p$ is a prime number, $S=\GF(q^n)$ be an extension of $R$, where $n\in \mathbb{N}$, $n\geq 2$ and let $K=\GF(...
7
votes
2
answers
463
views
Sum of two $n$th powers in finite fields
Let $q$ be a prime power, let $n$ be a positive integer and let $\mathbb{F}_q$ be the finite field of cardinality $q$. I have some computational evidence that the set $$\{x^n+(-1)^nay^n:x,y\in\mathbb{...
- 934
3
votes
1
answer
175
views
Largest subset of quadratic residues with no pair of elements differing by 1
In $\mathbb{F_p}$, $p$ prime what is the larget subset $S$ of quadratic residues with no pair of elements differing by 1?
In this related question Seva gives an example:
"...assuming $p\equiv\...
- 4,862
8
votes
1
answer
355
views
The distribution of certain Galois groups
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Let $G_p^+$ be the Galois group of the polynomial $f(x)-y$ over $\overline{\mathbb{F}}_p(y)$ and $G_p$ be the Galois group of the ...
- 7,193
3
votes
0
answers
106
views
Is there a bijection between elements in algebraic closure of F2 and all bi-infinite periodic sequences made of 0 and 1, filling the properties below?
(And if so, how can I describe the "multiplication" on the sequence?)
We consider a bijection, denoted by $f$, from the algebraic closure of $\mathbb{F}_2$, named $\bar{\mathbb{F}_2}$, to ...
- 31
2
votes
4
answers
751
views
How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?
How to prove without using advanced theorems that quaternions algebra $H = \genfrac(){}{}{-1,-1}{\mathbb{Z}_p}$, where $p$ is prime that $H \cong\operatorname{Mat}_2({\mathbb{Z}_p})$?
My ideas: I ...
1
vote
1
answer
285
views
A "semi-genetic" definition of addition and multiplication in the field $\operatorname{On}_p$?
Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$.
Recursive definition of addition:
$$x \oplus y := ((x+y) \...
- 3,064
2
votes
0
answers
121
views
When $\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))$?
Let's $P,Q\in\mathbb K[x]$, with $\mathbb K$ a finite field.
On what necessary and sufficient condition on $R \in \mathbb K[x]$ is it :
$\gcd(P(x),Q(x))\bmod R(x)=\gcd(P(x) \bmod R(x),Q(x) \bmod R(x))...
- 4,074
8
votes
2
answers
580
views
Zero trace elements in finite fields
There is so much literature on the relation between the multiplicative structure of a finite field and elements having zero trace, that I am hoping that the following is known.
Let $q$ be a prime ...
- 934
3
votes
1
answer
339
views
Number of certain elements in a finite field having zero trace
I have a question concerning certain elements having zero trace in a finite field extension and I do have the feeling that additive characters should play a role, but I am not sure how. I am stating ...
- 934
9
votes
0
answers
462
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
- 15.6k
7
votes
1
answer
501
views
Subgroups of the multiplicative group of a finite field satisfying a certain additive property
Let $G \subseteq \mathbb F_p^*$ be a subgroup. Then $G$ is called almost trivial if $G \cap (2-G)$ consists of the element 1.
Then I am wondering how big $G$ can be in terms of $p$. If $G$ is a random ...
- 2,224