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# Finite Fields - Science topic

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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?
I am not an expert in logic so I am not sure how to answer your question, nor if what I am saying is relevant. However, there are algorithms for factoring polynomials over the integers, so it would seem to me that the problem is decidable.
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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 Z23 i.e., {0,1}3, function is f(x1,x2)=1+x1 and range is Z2.
We can consider domain as products of finite fields. We are free to assume any degree polynomial and number of variables.
There is no general method to generate a formula for a given sequence of data.
Each sequence has a different representation; some are linear with constant coefficients such as Fibonacci type sequences, others are linear with variable coefficients beside the higher-order sequences.
Show the data, and then one can search the best sequence, and we are lucky if a closed formula exists.
Regards
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what it mean non-redundant value in the submitted image and how to find non-redundant value a matrix in finite field?
These articles might be resourceful, have a look:
Kind Regards
Qamar Ul Islam
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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).
can you using turbo pascal of simple programming..if have a problem can calling me or sending a message to help
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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?
To make the "projecto-morphism" statement correct, the name "division algebra" has to be interpreted as what is also called a "skew field" or "division ring".
There is a huge literature on the connection between geometrical and algebraic properties. See e.g., Ruth Moufang, Marshall Hall, the book "Latin Squares and their Applications" by Denes and Keedwell and its second edition by Keedwell and Denes.
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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 ?
Consider the linear map T(X):=AX-XB. The original equation has exactly one solution for each C iff T is injective, iff ker T=0, iff the only solution for AX-XB=0 is X=0. So suppose AX-XB=0, i.e., AX=XB for a fixed solution X. Then A^2X=AAX=AXB=XB^2, and by induction, for any polynomial p(x) we get p(A)X=Xp(B).
Now consider the characteristic polynomial p of B. Then p(A)X=Xp(B)=0 since p(B)=0. Thus p(A)X=0 as a matrix equation. Either p(A) is nonsingular and there is only one solution, or p(A) is singular and there are multiple solutions. But p is the characteristic polynomial of B, which splits over the algebraic closure of the base field as
p(x)=(x-L1·I)...(x-Ln·I),
where I is the identity matrix and the Lk are the eigenvalues of B.
If p(A)=(A-L1·I)...(A-Ln·I) is nonsingular, then A-Lk·I is nonsingular for each k, hence no Lk is an eigenvalue of A. Therefore the minimal polynomials of A and B must be coprime (otherwise they would share at least a linear factor for some eigenvalue over the algebraic closure).
If p(A) is singular, we must have A-Lk·I singular for some k, hence Lk is an eigenvalue of A for some k, so the linear factor x-Lk divides the gcd of the minimal polynomials of A and B over the algebraic closure, therefore the minimal polynomials over the original field have nontrivial gcd.
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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 !
in theses "Computational Aspects of Discrete Logarithms" by Rob Lambert,
Lanczos-type finite termination methods are considered for the solution
of linear systems over finite fields. Maybe it may help.
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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.
There are no challenges. An 8-bit finite field multiplier is about 200 gates. I think 65 nm has a gate density of about 400 kgates/mm2. That is 2000 8x8 multiplier per square mm. What are you after ?. What do you want to do with these multipliers ?. AES ?, BCH ECC ?. Reed-Solomon ECC ?
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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 ?
Dear Munir !
Thank you for your answer. The question is algorithmic: you are given a set (m matrices) of nXn matrices and you need to create a linear combination out of the given matrices such that the resulting matrix has the greatest rank that can be achieved in this way. The question is how to do it efficiently.
All The Best !
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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 ?
Any system of polynomial equations can be reduced to a system of quadratic equations in more variables. (This is well known.) For example, if a monomial m of degree d>=3 appears in an equation, write m=m_1*m_2
where m_1, m_2 are monomials of degree <d. Then add the variables z_1,z_2, replace the monomial m by the quadratic monomial z_1*z_2
and add the equations z_1=m_1 and z_2=m_2 of degree <d. By repeating this procedure we can get rid of all monomials of degree >=3 that appear in the equations.
So this shows that the fact that the equations are quadratic and not higher degree doesn't really help.
The 3-SAT problem could be expressed as a system of polynomial equations. This shows that solving a system of polynomial equations over a finite field is NP-complete. So one cannot expect a fast algorithm that works in general.
One approach would be using Buchberger's algorithm. If we are only interested in solutions in the finite field F_q (where q is a prime power)
then we can add the equations x_1^q-x_1,...,x_n^q-x_n where x_1,...,x_n are the variables. Of course this could be very slow. This could be doubly exponential in time.
Another approach could be just to check all possible vectors in F_q^n. This would be exponential in time.
Or, one could use a hybrid approach. Use Grobner basis with a lexicographic ordering, but compute a truncated version of it. With that I mean that we only do the computations in low degree and do not reduce S-polynomials whose degree is too high. This way, we might find some polynomials in fewer variables, say, the first k variables instead of all n variables. Then we can check for solutions in F_q^n using a back-tracking approach.
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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.
Dear Elahe,
Are you asking if there are types of footsteps which are considered normal and if you can diagnose conditions by measuring these parameters? In which case gait is very well studied and used for analysis of animal models with gait problems and also in human patients- briefly described here: https://emedicine.medscape.com/article/320160-overview#a1
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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.
@Aparna Sathya Murthy, what do you mean by "laws for large #s "? What is "#s" stands for? Please tell me little more details.
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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 ?
Recall the following fact.  Let A be a linear operator on a vector space V (over any field), and let v be a nonzero vector in V. Then A-cyclic subspace Lv is the smallest A-invariant subspace that contains v.
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I have no idea about solvability of this type of equations over finite fields.
Deterministic equation solving over finite fields
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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.
There is no need to appeal to Galois theory. Basic considerations on the separability of the finite extension E/F are sufficient . The proof of the so called "primitive element" theorem  (see any textbook) uses the following lemma : E is monogeneous, i.e. of the form F(a), if and only if  E/F admits only a finite number of intermediary subextensions. Then :
1) If E/F is separable,  E is monogeneous (the above lemma is actually a key step in the proof of this theorem)
2) If E/F is not separable, the number of subextensions can be infinite. Classical example: take a field k of non null characteristic p, E = k(X, Y) (rational fractions on 2 variables over k), F = k(X^p, Y^p). It is immediate that E/F has degree p^2. And it is an exercise to show that E/F admits an infinite number of  subextensions  (show that E/F is not monogeneous, or exhibit directly an infinite number of subextensions)
NB : This is probably the exercise in Lang's "Algebra" pointed out  by Giulio Peruginelli.
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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)?
Not every finite ring is a field. Finite rings with zero divisors are not fields.
Every finie field is a subring of an infinite field: if F is a fileld, it is a subfield of F(x), the field of rational functions over F. It is also a subfield of its own algebraic closure.
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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.)
Thanks for the 19_4 information. The question is for geometries over any field.  Type (0) is for general fields commutative of characteristic 0, type (1) is for general non-commutative fields (including the commutative ones), type (p) is for characteristic p commutative fields.  I do not know of examples of such n_k ctheorems that restrict to more than this: e.g. just to rationals.  Fano (or its matroid dual 7_4) is the only one I know that works only for a certain characteristic (2).  Instead of searching for n_k with n large and k fixed it could be more likely to find new ctheorems for k about n/2.  Maybe looking at the (p) case is easiest.  Perhaps Fano is the only such ctheorem.
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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?
Better news. The article:
by Zoe Chatzidakis if freely downloadable. At page 99 it is proven that there is a decision procedure if some sentence is true in all Z/pZ or not (without powers p^m!), and there they show also how to compute the exceptional primes for general formal sentences. It is this way which should be followed!
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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:
You should add to your question that t+k less than or equal to n. since it t+k>n, then the intersection not zero.
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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?
Well known combinations are the Joux protocol or identity based encryption.
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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
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Can we prove the  above result?
Thank You Thong sir for providing valuable suggestion of  this question.
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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 ?
If you want the answer in term of coordinates or in term of polynomial, you must fix the irreducible polynomial which defines the field $F_16$ over $F_2$. If you take the polynomial $x^4+x+1$  (which is primitive) I agree that x^13 (the inverse of x^2) can be written $1+x^2+x^3$, but if we take the  irreducible polynomial $x^4+x^3+1$, then $x^13$ can be written $x+x^2$. I agree that the Euclidean algorithm is better than the "power" algorithm. For example in $Z/pZ$ where $p$ is a prime, you can compute $a^{-1}$  by the Euclidean Algorithm or by the power algorithm (with the square and multiply algorithm) both are linear in the size of $a$ but  the first runs about four times faster than the second (with the first, the data size decreases for each iteration, which is not the case with the second).
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I need to find the inverse of generator of finite field ( F 24 ) with irreducible polynomial , f(x)=x4+x+1 i.e if g=0010 is the generator of this field then how to find g-1 ?
In general, Extended Euclid between the generator (viewed as a polynomial in x) and the polynomial defining the field.
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I am looking towards applying frequency analysis techniques on galois fields, specifically the GF(2^m),
I suggest you to take a look at the following book:
R. Lidl, H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, Cambridge, 1986.
Please see the attachment in the following:
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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 ?
Regarding your question I found this material concise and easy to follow.
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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?
I doubt RS over GF(256) can be decoded mentally. However there are well known papers of Vera Pless for instance about decoding Golay code by hand.
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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.
Hello everyone. Unfortunately, Bruce´s examples are incorrect. The pair (y,z) indeed is not an orthogonal basis, but the span (y,z) has other bases that are: (y,z+2y) in the first example and (z+y,z+2y) in the second.
In fact, as Maximiliano pointed out, you can use the Gram-Schmidt process (provided the characteristic is odd), as follows. Take any ordered basis of your space: v_1,...,v_n. Assuming v_1,...,v_{i-1} are already orthogonal to all basis vectors, consider v_i. If it is singular, try to add a linear combination of v_{i+1},...,v_n to it to make it nonsingular; if impossible, then v_i is already orthogonal to the whole space, move it to the end, and continue with v_{i+1}. If, on the contrary, v_i is nonsingular, make v_{i+1},...,v_n orthogonal to it by subtracting suitable multiples of v_i from them. Thus, we end up with an orthogonal basis, with a basis for the singular subspace (if any) at the end.
In char. 2, the correspondence between bilinear forms and quadratic forms breaks down, and this doesnt work, as Patrick´s example shows.
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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.
It depends of the matrix C. Over F_2, the equation is equivalent to X2+X+C=0. By multiplying with X from right and from left X2=X+C yelds the necessary condition XC = CX (they must commute!). This is already a system of linear equations with n2   equations and n2 unknowns, and if it has rank d, we make X already dependent of only n2 - d parameters. Now if you go back to the original equation, you get a system of quadratic equations where at most n2 - d many unknowns are really squared. Finally, we can replace the squared variables with new variables, solve againt the linear system (the solution is again dependent of some parameters) and finally replace the new unknowns with squared variables. Now it will become clear if the particular instance of the equation has solutions or not in M.
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Is it GL(n,q), SL(n,q), other?
Actually, you want to avoid these fuller finite matrix groups; For instance, a lot of the PSL(n,q) groups are simple! (Exceptions can occur for small n, and small finite fields q.)
Note that the group PSL(n,q) is just the quotient of SL(n,q) by it center (the diagonal matrices in SL(n,q)), so therefore the groups SL(n,q) are not far off from being simple.
Bogopolsky's book does a very nice job explaining in general terms how simplicity arises, and here are some hallmarks:
1) your group is highly transitive (say, at least two-transitive, for example) in its action of the vector space of the appropriate size,
2) your group has many elements of small support'' (given a point in the space you are acting on, there are elements in the group which only move this point and a few others).
3) your group is equal to its commutator subgroup.
In fact, these hallmarks were recognised by Higman in a paper going back to the 1950's, and then again in a well written short paper by D.B.A. Epstein (`The simplicity of certain groups of homeomorphisms'' Compositio Mathematica, 1970, pp 165--173), where in both cases, these hallmarks were looked for in infinite groups.
So, as any finite group can be found as a subgroup of permutation groups, and permutation groups are linear (embed any permutation group in a group of permutations of the major axes, which is a linear group). We see that there are lots of linear groups with lots of normal subgroups; just not really the ones you mentioned.
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Is this true or false?
Thanks Jeremy
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Why not any polynomial?
@Rama: in particular one nice property of a Galois ring is to be local, which would not be the case with a reducible polynomial
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