Polynomial basis

English

Noun

1. Template:Lb A basis of a polynomial ring (said ring being viewed either as a vector space over the field of coefficients or as a free module over the ring of coefficients).
   • 2007, Nicos Karcanias, Efstathios Milonidis, 2: Structural Methods for Linear Systems: An Introduction, Matthew C. Turner, Declan G. Bates (editors), Mathematical Methods for Robust and Nonlinear Control: EPSRC Summer School, Springer, Lecture Notes in Control and Information Sciences 367, page 89,

     If $T(s)=M(s)D(s{)}^{-1}$ is a RCMFD^{[right coprime matrix factor description]} of $T(s)$, then $M(s)$ is a polynomial basis for ${𝔛}_t$. If $Q(s)$ is a greatest right divisor of $M(s)$ then $T(s)=\bar{M}(s)Q(s)D(s{)}^{-1}$, where $\bar{M}(s)$ is a least degree polynomial basis of ${𝔛}_t$ [15].

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2. Template:Lb Specifically, a basis, of the form {1, α, ..., α^n-1}, of a finite extension F_qⁿ of a Galois field F_q, where α is a primitive element of F_qⁿ (i.e., a root of a degree-n primitive polynomial over F_q).
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   • 2010, Vladimir Tujillo-Olaya, Jaime Velasco-Medina, Hardware Architectures for Elliptic Curve Cryptoprocessors Using Polynomial and Gaussian Normal Basis Over GF(2²³³), Marina L. Gravrilova, C. J. Kenneth Tan, Edward David Moreno (editors), Transactions on Computational Science XI: Special Issue on Security in Computing, Part 2, Springer, Template:W 6480, page 79,

     In this case, the GF(2^m) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB).

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Hyponyms

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See also

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Further reading

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