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Questions tagged [finite-field]

In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements.

In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements, called its order (the size of the underlying set). As with any field, a finite field is a set on which the operations of commutative multiplication, addition, subtraction and division (by anything except zero) have been defined.

Read more: http://en.wikipedia.org/wiki/Finite_field

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Incorrect evaluation of the irreducibility of the polynomial

in my function PolynomialIrreducibility() I'm evaluating if entered polynomial is irreducible or not over GF(prime_number). void PolynomialIrreducibility () { // Enter prime number ZZ prime_number; ZZ_pX polynom; do { cout…
martink
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A strange sequence of polynomials over $\mathbb{F}_{7}$

Let $P=X^{10} +5X^{5}+1$ and $Q=5X^{8}+6X^{3}$ in $\mathbb{F}{7}[X]$. How we can prove this strange relation of Euclid division $$U{2i}P-X^{4}U_{2i-1}^{7}=4.3^{2i-1}Q$$ $$U_{2i+1}P-X^{-4}U_{2i}^{7}=4.3^{2i}Q/X$$ with $U_{1}=X^{2}$?Note that…
Oussema
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Compounding CRC polynomials

I am trying to compound 2 CRC polynomials. I have a message for which I produce a CRC using one polynomial. I then CRC the result of the first CRC in order to obtain a second result. Is there any possible way to do this in one go? Example: Given the…
costin
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What is a basic explanation of Finite Fields?

I have zero background and I have never seen these symbols before. Can someone explain what is going on here?
santma
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How to do binary linear algebra on a sparse matrix in Matlab (or any other language)?

I have a sparse binary matrices whose properties I want to analyze over the binary field. The application is to analyze some sparse, binary error-correcting codes. The matrices themselves are too big to handle as full dense matrices, with sizes on…
Joe
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How to keep the value of multiplication within the finite field range? I am implementing GF(8) multiplication

I am implementing GF(8) Multiplication. Primitive Polynomial is x^3 + x + 1. I know the basics: if multiplication overflows, I can xor it with my primitive polynomial and bring it into the range of finite field. However the trouble occurs when the…
Saurabh Singh
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setsearch in GP/PARI: testing cubic residues

I am trying to write a program in GP/PARI which given a finite field (in this example, I will use the degree 2 field of 9 elements over p=3), computes the cube of all elements and stores it in a list (I understand this is very inefficient). Then, I…
Future
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Polynomial over final field library

I'm trying to find a C++ library that handles polynomials over some finite field GF(2^n) and have support of matrix representation with support for rank finding/inverse or even solving A=X*B. I'm trying to use Linbox, but there's very little…
G. Ko
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Computing the powers of a polynomial f(x) modulo irreducible polynomial h(x) in MATLAB

Suppose I have a polynomial f(x)= a_0 + a_1*x + a_2*x^2 +...+ a_(n-1)*x^(n-1) with a_i elements of F_q, q prime. How do I compute the powers f(x)^0, f(x)^1, f(x)^2, ..., f(x)^k modulo another polynomial h(x) of degree n for any positive integer k…
Simon Robert
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Python -- using __init__ with an inherited method for polynomials class

This is a class which will take in as input and then output a polynomial in string form (both ways same format). Some arithmetic is performed in the various methods. I've been trying to inherit this class into another class that will then use the…
stackuser
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Python --- multiplication in GF(2) field

This function is returning unusual values in the list g. It should return 32774, 65548, 1048768 but instead it's values are more like it's treating the entire binary like a big slinky and just moving the LSB's towards the MSB's instead of actually…
stackuser
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Python -- using regex on a class instance

I have a class that was taking in lists of 1's and 0's and performing GF(2) finite field arithmetic operations. It used to work until I tried to make it take the input in polynomial format. As for how the finite arithmetic will be done after fixing…
stackuser
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Python read binary polynomial into list

I'm trying to read in a polynomial in GF(2) finite field, which is basically only 1 or 0 for the coefficients or constants and the only number that really differentiates 1 part of polynomial from the other is the exponent. But the exponent ends up…
stackuser
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C Snippets for Algebra in Finite Prime Fields

I need to solve polynomials in finite prime fields on 16-bit CPUs. I have seen people using the fields GF((2^16)+1), GF((2^16)-15), and GF((2^32)-5). I guess these choices stem from the fact that there are several optimizations. However, apart from…
Konny
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VBA GF(256) log table

I'm working on a function to generate QR codes in vba. I'm following this tutorial. I am now working on generating the error correction words in this step. This requires GF(256) log/antilog tables, seen here. I really don't want to have to type in…
DasPete
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