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# Finite Fields - Science topic

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Questions related to Finite Fields

There are a number of criteria for determining whether a polynomial with integral coefficients is irreducible over rational numbers (the traditional ones being Eisenstein criterion and irreducibility over a prime finite field).

I was wondering if the decision problem of "Given an arbitrary polynomial with integral coefficients is irreducible over rational numbers or not" is decidable or undecidable?

What is the best modelling of sequence so we can get any binary elements in our sequences?

For example:

to get sequence 1,0,1,0,1,0,1,0 we may assume our domain is Z

_{2}^{3}i.e., {0,1}^{3}, function is f(x_{1},x_{2})=1+x_{1 }and range is Z_{2}.We can consider domain as products of finite fields. We are free to assume any degree polynomial and number of variables.

what it mean non-redundant value in the submitted image and how to find non-redundant value a matrix in finite field?

In field theory, a

**primitive element**of a finite field GF(*q*) is a generator of the multiplicative group of the field. In other words,*α*∈ GF(*q*) is called a primitive element if it is a primitive (*q*− 1)th root of unity in GF(*q*); this means that each non-zero element of GF(*q*) can be written as*αi*for some integer*i*.For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7).

A projective space is "projecto-morphic" to a space of rays over a division algebra if and only if the conclusion of Desargues theorem holds.

The underlying division algebra is commutative if and only if the conclusion of Pappos theorem holds.

The three centers of a quadrangle are co-linear if and only if the underlying field is of characteristic two. Fano plane is an example of this.

Are there any other theorems that link geometric facts (or configurations) to the properties of the underlying algebraic structure (the division algebra)?

What is the closest thing to a geometric proof of the Wedderburn theorem (finite -> commutative)?

How projective spaces over fields of nonzero characteristic differ from our experiences born of real numbers?

Let A,B nxn matrices over finite field F.

One can prove that if the minimal polynomials of A,B are co-prime then the Sylvester equation AX-AB=C has uniqe solution for any given matrix C over F.

Is the opposite also true, i.e. if AX-AB=C has uniqe solution X for any given matrix C over F, does the minimal polynomials of A,B are co-prime ?

In the article "Fast Parallel Computation of Characteristic Polynomial by Leverrier's Power Sum Method Adapted to Fields of Finite Characteristic" by Arnold Schonhage 1993, the author represets an algorithm for computing the characteristic polynomial of nxn matrix A over finite field from which the invertibility of A can be detected and the inverse of A can be calculated.

Is it the best parallel algorithm for matrix inversion over finite fields ?

Is there any refference where I can find this algorithm in more details and more explanations ?

Has anyone implemented this algorithm in some liberary or is there any open source implementation of it ?

Many Thanks !

The low power is important in electronics industry especially in Very Large Scale Integration (VLSI) field. The multiplier utilize multiplication process with 70nm Complementary Metal Oxide Semiconductor (CMOS) technology with a clock period of 2 GHz.

Papers:

A. Karimi, A. Rezai, M.M. Hajhashemkhani.2018. A novel design for ultra-low power pulse-triggered D-Flip-Flop with optimized leakage power, Integration, the VLSI Journal, 60: 160-166.

O.S. Fadl, M.F. Abu-Elyazeed, M.B. Abdelhalim, H.H. Amer, A.H. Madian. 2016. Accurate dynamic power estimation for CMOS combinational logic circuits with real gate delay model, Journal of advanced research, 7 : 89-94.

Given $n\times n$ matrices $A_{1},\ldots,A_{m}$ over a finite field, how can one construct a matrix $\sum_{j}^{m}a_{j}A_{j}$ with the maximal possible rank ?

Given an overdetermined set of quadratic multivariable polynomial equations over some finite field F_{q}, what is the best known algorithm to solve the problem and what is its complexity ?

What are the underlying assumptions leading to the given complexity ? The same questions over F_{2} ?

Is there any result regarding sets of such equations that have some structure ?

Is it true that when the number of equations m>n, where n is the number of variables, when n tends to infinity, the number of solutions tends to be 0 or 1 ?

Is there any proof or any explanation to this phenomena or is it only from observations ?

How bigger m should be in order to see this phenomena in experiments ?

if we can find the pattern of normal walking ,we can compare any pattern of different people with normal one and diagnosis problem in neuromuscular system.it is the way just like electrocardiogram that we use it for diagnostic situations.

I am using GMP[https://gmplib.org] for implementation of cryptographic algorithms, especially for big integer/finite field arithmetic. However, I got the comment that GMP is not constant time, therefore it is not a good choice in for cryptographic implementations. But I wonder

1. Why GMP is not constant time?

2. And what makes the difference than any constant time bignum library?

3. Is there any general purpose constant time big number library for C/C++ especially targeted for Cryptographic use?

My understanding is addition, subtraction, multiplication, and division implemented in C do not depends on the size of the input. Therefore, such operations on any platform are constant time. Now it seems I'm completely wrong.

Let F denote a finite field and let A denote a n times n matrix over F. How can one compute efficiently all the invariant subspaces of A with dimension less than or equal to n_{0} (where n_{0}<<n). Also, is there any method to build (and cover all) these subspaces from smaller subspaces ?

I have no idea about solvability of this type of equations over finite fields.

Let F be a field. K be an field extension of F such that [K : F]=n for some $n\in \mathbb{N}$. (where [K : F] is dimension of K over F) Let L be an intermediate field between F and K. How many are of them are possible? meaning finite or infinite.

1. Consider an infinite ring. Can it be embedded into a finite field? Can it have a finite quotient field?

2. Consider, on the other hand, an infinite field. Does it always contain infinite subrings?

3. Can we improvise a local ring to generate a field(finite or infinite)?

An n_k ctheorem (configurational theorem) is a set of n points and n hyperplanes with k points on each hyperplane and k hyperplanes through each point, all embedded in (k-1)-dimensional space. (The type of space could be e.g. a projective (or affine) space over a general commutative field (type (0)), over a general possibly non-commutative field (type (1)), or over a general field of prime characteristic p (type (p)). If the existence of n-1 hyperplanes implies the existence of the n'th hyperplane then it is called a "ctheorem".

Those known are:

Desargues 10_3 (type (1)) discovered about 1650 CE

Pappus 9_3 (type (0)) discovered about 300 CE

Moebius 8_4 (type (0)) discovered 1828 by A.F. Mobius

Glynn 8_4 (type (0)) discovered 2010 by D.G. Glynn (Theorems of points and planes ...)

Glynn 9_4 (type (1)) discovered 2010 by D.G. Glynn (same paper)

Fano 7_3 (type (2)) known to geometers in the late19th century (the matroid dual is a 7_4 ctheorem of type (2) also) Could be called "anti-Fano" since Fano's axiom proscribed it in the geometry. It is also called PG(2,2), the projective geometry of dimension 2 over the finite field GF(2).

Note that the matroid dual of an n_k ctheorem is also a ctheorem n_{n-k} (if type (0) or (p)), so Pappus gives a ctheorem in 5-d space. The two 8_4's (they are unique) are self-matroid-dual. (Sometimes the matroid dual is a bit degenerate, as in the cases 10_3 and 9_4.)

Assume that Q(x) is a polynomial with integer coefficients. Is there a prime number p such that the equation Q(x)=0 has all its root in the finite field Z/pZ?

In an n-dimensional vector space W over a finite field F, what is the number of k-dimensional subspaces that intersect in zero a fixed t-dimensional subspace V?

UPD: the answer to this and a more general question may be found here:

Article Intersection Graph of a Module

ECC (Elliptical curve cryptography) which is based on algebraic structures of elliptical curves over finite field, is used to design public key cryptography. PBC (pairing based cryptography) which is based on bilinear pairings over elliptical curves is used to design asymmetric cryptography.

So can we combine them together to make a hybrid scheme?

i am trying to compute FFCT matrix (8*8)using modular arithmetic operation now i got correct matrix element in odd no of rows but my even no of rows all getting wrong ans .please any one knows how to compute this help me soon

I want to find the inverse of generator of finite field F 2^4 i.e if g=0010 is the generator of this field then how to find g^-1 ?

I need to find the inverse of generator of finite field ( F

_{2}^{4}) with irreducible polynomial , f(x)=x^{4}+x+1 i.e if g=0010 is the generator of this field then how to find g^{-1}?I am looking towards applying frequency analysis techniques on galois fields, specifically the GF(2^m),

Is there any good reference to the conditions on matrices A,B,C,D such that

the equation of the form:

XCX+XD-AX-B=0

have any solution over a general finite field F ?

If affirmative, is there any description of all the solutions ?

Also, is there any formula for the number of solutions ?

Specifically, are there any known conditions for a unique solution and a formula for the unique solution ?

How can Vedic mathematics solve the problems associated with error-correcting codes?

Is there any application in finite fields to solve the 'Key Equation solver' block of RS code?

By self-orthogonal basis, I mean a basis B where <u, v> = 0 for every distinct vectors u and v in B. The <.,.> is the standard scalar product, i.e. <u,v> = u_1v_1 + ... + u_nv_n.

Consider a set M of all possible square matrices of dimension k over a finite field Fp. Consider a map f defined on M as f(X)=X^2+C where X in M and C is an arbitrary fixed matrix from the set M.

It is worth mentioning that the operations addition and multiplication on M are over finite field Fp.

How to determine analytically the fixed points and periodic points of different period.

Answers even in case of p=2 is highly appreciated.

Is it GL(n,q), SL(n,q), other?

Recall that the period of a polynomial p(x) is the smallest n such that p divides x^n-1.The average is known to be of the order of q^n . I am interested in order of magnitude of the tail of the distribution...Say how many polys have period at most Kn for some fixed constant K.