Polynomials over Galois Fields¶
This section contains classes and functions for creating polynomials over Galois fields.
Polynomial classes¶
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Create a polynomial \(f(x)\) over \(\mathrm{GF}(p^m)\). |
Polynomial functions¶
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Finds the greatest common divisor of two polynomials \(a(x)\) and \(b(x)\) over \(\mathrm{GF}(q)\). |
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Finds the polynomial multiplicands of \(a(x)\) and \(b(x)\) such that \(a(x)s(x) + b(x)t(x) = \mathrm{gcd}(a(x), b(x))\). |
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Efficiently exponentiates a polynomial \(f(x)\) to the power \(k\) reducing by modulo \(g(x)\), \(f(x)^k\ \textrm{mod}\ g(x)\). |
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Factors the polynomial \(f(x)\) into a product of \(n\) irreducible factors \(f(x) = g_0(x)^{k_0} g_1(x)^{k_1} \dots g_{n-1}(x)^{k_{n-1}}\) with \(k_0 \le k_1 \le \dots \le k_{n-1}\). |
Special polynomial creation¶
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Returns a monic irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Returns all monic irreducible polynomials \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Returns a monic primitive polynomial \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Returns all monic primitive polynomials \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Returns the Conway polynomial \(C_{p,m}(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Returns Matlab’s default primitive polynomial \(f(x)\) over \(\mathrm{GF}(p)\) with degree \(m\). |
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Computes the minimal polynomial \(m_e(x) \in \mathrm{GF}(p)[x]\) of a Galois field element \(e \in \mathrm{GF}(p^m)\). |
Polynomial tests¶
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Determines whether the polynomial is monic, i.e. having leading coefficient equal to 1. |
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Checks whether the polynomial \(f(x)\) over \(\mathrm{GF}(p)\) is irreducible. |
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Checks whether the polynomial \(f(x)\) over \(\mathrm{GF}(p)\) is primitive. |