polynomial basis

Noun

polynomial basis (plural polynomial bases)

1. (ring theory) 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).
   • If T(s)=M(s)D(s)⁻¹ is a RCMFD 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)=◌̅M(s)Q(s)D(s)⁻¹, where ◌̅M(s) is a least degree polynomial basis of 𝔛_t [15].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...
   • When dealing with various polynomial bases up to a certain degree and with corresponding moments, any moment (with respect to any basis) can be expressed as a function of moments of the same or fewer orders with respect to an arbitrary basis.[…]As we already saw in Chapter 1, OG^([orthogonal])...2009, Jan Flusser, Barbara Zitova, Tomas Suk, Moments and Moment Invariants in Pattern Recognition, Wiley, page 166
2. (field theory) Specifically, a basis, of the form {1, α, ..., αⁿ⁻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).
   • When using polynomial bases, the first stage in computing the product of two elements of #92;mathbbF#95;#123;2ⁿ#125; is the multiplication of two polynomials of degree at most n-1 in #92;mathbbF#95;2#91;x#93;.1999, I. Blake, G. Seroussi, N. Smart, Elliptic Curves in Cryptography, Cambridge University Press, page 20
   • In this case, the GF(2ᵐ) multiplication is implemented in hardware using three algorithms for polynomial basis (PB) and three for gaussian normal basis (GNB).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....