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{{Unreferenced|date=December 2009}}
In [[mathematics]], the [[linear algebra]] concept of [[dual basis]] can be applied in the context of a [[finite extension]] ''L''/''K'', by using the [[field trace]]. This requires the property that the field trace ''Tr''<sub>''L''/''K''</sub>  provides a [[non-degenerate]] [[quadratic form]] over ''K''. This can be guaranteed if the extension is [[separable extension|separable]]; it is automatically true if ''K'' is a [[perfect field]], and hence in the cases where ''K'' is finite, or of characteristic zero.

A '''dual basis''' isn't a concrete [[basis (linear algebra)|basis]] like the [[polynomial basis]] or the [[normal basis]]; rather it provides a way of using a second basis for computations.  

Consider two bases for elements in a [[finite field]], GF(''p''<sup>''m''</sup>):

:<math>B_1 = {\alpha_0, \alpha_1, \ldots, \alpha_{m-1}}</math>

and

:<math>B_2 = {\gamma_0, \gamma_1, \ldots, \gamma_{m-1}}</math>

then ''B''<sub>2</sub> can be considered a dual basis of ''B''<sub>1</sub> provided

:<math>\operatorname{Tr}(\alpha_i\cdot \gamma_j) = \left\{\begin{matrix} 0, & \operatorname{if}\ i \neq j\\ 1, & \operatorname{otherwise} \end{matrix}\right. </math>

Here the [[field trace|trace]] of a value in GF(''p''<sup>''m''</sup>) can be calculated as follows:

:<math>\operatorname{Tr}(\beta ) = \sum_{i=0}^{m-1} \beta^{p^i}</math>

Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than having to explicitly convert between bases using the [[change of bases]] formula.  Furthermore, if a dual basis is implemented then conversion from an element in the original basis to the dual basis can be accomplished with a multiplication by the multiplicative identity (usually 1).

{{DEFAULTSORT:Dual Basis In A Field Extension}}
[[Category:Linear algebra]]
[[Category:Field extensions]]
[[Category:Theory of cryptography]]