Galois Fields¶
This section contains classes and functions for creating Galois field arrays.
Galois field class creation¶
Class factory functions
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Factory function to construct a Galois field array class for \(\mathrm{GF}(p^m)\). |
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Alias of |
Abstract base classes
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An array over \(\mathrm{GF}(p^m)\). |
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Defines a metaclass for all |
Pre-made Galois field classes
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An array over \(\mathrm{GF}(2)\). |
Prime field functions¶
Primitive roots
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Finds the smallest primitive root modulo \(n\). |
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Finds all primitive roots modulo \(n\). |
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Determines if \(g\) is a primitive root modulo \(n\). |
Extension field functions¶
Irreducible polynomials
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Returns a monic irreducible polynomial \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\). |
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Returns all monic irreducible polynomials \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\). |
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Determines whether the polynomial \(f(x)\) over \(\mathrm{GF}(p^m)\) is irreducible. |
Primitive polynomials
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Returns a monic primitive polynomial \(f(x)\) over \(\mathrm{GF}(q)\) with degree \(m\). |
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Returns all monic primitive polynomials \(f(x)\) over \(\mathrm{GF}(q)\) 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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Determines whether the polynomial \(f(x)\) over \(\mathrm{GF}(q)\) is primitive. |
Primitive elements
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Finds the smallest primitive element \(g(x)\) of the Galois field \(\mathrm{GF}(p^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\). |
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Finds all primitive elements \(g(x)\) of the Galois field \(\mathrm{GF}(p^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\). |
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Determines if \(g(x)\) is a primitive element of the Galois field \(\mathrm{GF}(p^m)\) with degree-\(m\) irreducible polynomial \(f(x)\) over \(\mathrm{GF}(p)\). |
Minimal polynomials
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Computes the minimal polynomial \(m_e(x) \in \mathrm{GF}(p)[x]\) of a Galois field element \(e \in \mathrm{GF}(p^m)\). |