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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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Algorithms better than meet-in-the-middle

I am given n = 256 64-bit binary vectors ∈ GF(2)^64. I am also given a k and a target vector T, and I am required to choose exactly k out of the n vectors without repetition such that their sum (mod 2) is T. Such a solution is guaranteed to…
Gareth Ma
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Julia ERROR: 'Can't Promote to Common Type' for multivariate polynomials in Nemo Library

I am new to Julia and have been trying to compute some polynomials using the Nemo library's multivariate polynomials. I have the following code: R = GF(2); # create finite field S, (z, x) = PolynomialRing(R, ["z", "x"]); # Multivariate…
Sputn1k
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How to perform addition and multiplication in F_{2^8}

I want to perform addition and multiplication in F_{2^8} I currently have this code which seems to work for add but doesn't work for multiply; the issue seems to be that when I modulo by 100011011 (which represents x^8 + x^4 + x^3 + x + 1), it…
harlee53
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Is there a better way to do modulo in a finite field when directly working on polynomials rather than binary numbers?

So currently I am trying to implement finite fields using only polynomials. So like, I don't want to be operating on binary numbers using operations such as AND. Instead I want to make the whole thing with polynomials. I have got very far with this,…
Evelyn
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pyfinite gives wrong result for multiplication in field GF(2^8)

I am using pyfinte to calculate multiplication for AES over the field it uses which is F(2^8) but when I do as following: from pyfinite import ffield a = 0xbf b = 0x03 F = ffield.FField(8) c = F.Multiply(a, b) print(hex(c)) the actual result is…
mark
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Python - Algorithm for finding order of cyclic group generator

I wanted to find the order of a generator g chosen from a cyclic group G = Z*q where q is a very large (hundreds of bits long) number. I have tried the following code from Rosetta Code but it is taking too long: def gcd(a, b): while b != 0: …
user7091463
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Finite fields: Compute the inverse of a matrix

I am working with finite fields in Python. I have a matrix containing polynomials, each polynomial is represented as an integer. For example, the polynomial x^3 + x + 1 is represented as 11, because: x^3 + x + 1 ==> (1,0,1,1) ==> (8,0,2,1) ==> 8 + 0…
dimitris93
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Create and invert large Galois field matrix

I have a matrix of size 128x128. Each entry is a binary field element (in my use case, only 0 and 1). I try to invert this matrix in matlab. I find some functions in matlab that does finite field matrix inversion here…
drdot
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square root of a finite field element in C++

Is there any implementation of a method to obtain the square root of an element from a finite field. Programmed in C++ I was using NTL but the do not provide a method to do that. Thanks in advance
DaWNFoRCe
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Sympy symbol and the cls parameter

I have seen f = sympy.symbols('f', cls=Function) but not any documentation. Python does not like x = sympy.symbols('x', cls=FF(8)), it complains about raise CoercionFailed("expected an integer, got %s" % a) CoercionFailed: expected an integer, got…
Johan
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GF(256) finite field multiplication function in C#

I'm implementing AES in C# and at some point (MixColumns function) I have to multiply two Bytes over the GF(2^8) finite field. So, I have three options: Use a default function that dotNet has (does it have something like that?) Write a custom…
Cezar D.
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Solving a large system of linear equations over the finite field F2

I have 10163 equations and 9000 unknowns, all over finite fields, like this style: Of course my equation will be much larger than this, I have 10163 rows and 9000 different x. Presented in the form of a matrix is AX=B. A is a 10163x9000 coefficient…
Abraham
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Julia: Adding a multivariate polynomial to a univariate polynomial

I am new Julia and I am trying to add a multivariate polynomial to a univariate polynomial. In essence my problem is summed up in the following code: R = GF(2); S, z = PolynomialRing(R, z); a = Array{gfp_poly}(undef, 1, 5); a = [z*0 z*0 z^1 z^2…
Mors
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fast and efficient matrix multiplication in GF(2) field - python

I want to perform matrix multiplication over the GF(2) field. (in other words, '+' maps to XOR, and '×' maps to AND) I usually do matrix multiplication in the Real field followed by a mod(2) operation. A simple python example: import numpy as…
Mohammad Hossein Ashoori
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Is there an efficient algorithm to compute the Jacobsthal matrix or quadratic character in GF(q)?

Is there an efficient algorithm to compute the Jacobsthal matrix [WP] or equivalently the quadratic character χ in GF(q), J [ i, j ] = χ ( i - j ) = 0 if i = j else 1 if i - j is a square in GF(q) else -1, where i, j run over the elements of…
Max
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