Dual basis in a field extension explained

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_L/K provides a non-degenerate quadratic form over K. This can be guaranteed if the extension is 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 is not a concrete 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^m):

B _{1} = {\alpha _{0,} \alpha _{1,} \ldots, \alpha _{m-1} }

and

B _{2} = {\gamma _{0,} \gamma _{1,} \ldots, \gamma _{m-1} }

then B₂ can be considered a dual basis of B₁ provided

\operatorname{Tr}(\alpha _{i ⋅} \gamma _{j)} = \left\{\begin{matrix} 0 , & \operatorname{if} i ≠ j\ 1 , & \operatorname{otherwise} \end{matrix}\right.

Here the trace of a value in GF(p^m) can be calculated as follows:

\operatorname{Tr}(\beta ) = \sum ^{m -1} _{i =0} \beta ^{p ^{i}}

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 multiplication by the multiplicative identity (usually 1).

References

• Book: Lidl, Rudolf. 10.1017/cbo9781139172769. Introduction to finite fields and their applications. Niederreiter. Harald. 1994. Cambridge University Press. 9781139172769. Cambridge., Definition 2.30, p. 54.