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 \(\mathrm{GF}(p^m)\). |
Special polynomials¶
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 \(m\). |
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 \(m\). |
Minimal polynomials
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Computes the minimal polynomial \(m_e(x) \in \mathrm{GF}(p)[x]\) of a Galois field element \(e \in \mathrm{GF}(p^m)\). |
Polynomial functions¶
Divisibility
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Finds the greatest common divisor of \(a\) and \(b\). |
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Finds the multiplicands of \(a\) and \(b\) such that \(a s + b t = \mathrm{gcd}(a, b)\). |
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Computes the least common multiple of the arguments. |
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Computes the product of the arguments. |
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Determines if the arguments are pairwise coprime. |
Congruences
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Efficiently performs modular exponentiation. |
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Solves the simultaneous system of congruences for \(x\). |
Polynomial factorization
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Computes the prime factors of a positive integer or the irreducible factors of a non-constant, monic polynomial. |
Factors the monic polynomial \(f(x)\) into a product of square-free polynomials. |
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Factors the monic, square-free polynomial \(f(x)\) into a product of polynomials whose irreducible factors all have the same degree. |
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Factors the monic, square-free polynomial \(f(x)\) of degree \(rd\) into a product of \(r\) irreducible factors with degree \(d\). |
Polynomial tests
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Determines whether the polynomial is monic, i.e. having leading coefficient equal to 1. |
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Determines whether the polynomial \(f(x)\) over \(\mathrm{GF}(p^m)\) is irreducible. |
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Determines whether the polynomial \(f(x)\) over \(\mathrm{GF}(q)\) is primitive. |
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Determines if the positive integer or the non-constant, monic polynomial is square-free. |