mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

concept of

dual basis In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimension of V), the dual set of B is a set B^* of vectors in the dual space V^* with the same index set I such that B and ...

can be applied in the context of a

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

field extension In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...

''L''/''K'', by using the

field trace In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''. Definition Let ''K'' be a field and ''L'' a finite extension (and hence a ...

. This requires the property that the field trace ''Tr''_''L''/''K'' provides a

non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'') given by is not an isomorphism. An equivalent definition when ' ...

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

over ''K''. This can be guaranteed if the extension is separable; it is automatically true if ''K'' is a

perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has no multiple roots in any field extension ''F/k''. * Every irreducible polynomial over ''k'' has non-zero f ...

, and hence in the cases where ''K'' is

, or of characteristic zero. A dual basis () is not a concrete

basis Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to: Finance and accounting * Adjusted basis, the net cost of an asse ...

like the

polynomial basis In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely wri ...

or the

normal basis In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any ...

; 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 =

and :

B_2 =

then ''B''₂ can be considered a dual basis of ''B''₁ provided :

\operatorname(\alpha_i\cdot \gamma_j) = \begin
0, & \operatorname\ i \neq j
\\ 1, & \operatorname.
\end

Here the

trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album), by Nell Other uses in arts and entertainment * ...

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

\operatorname(\beta ) = \sum_^ \beta^

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

* , Definition 2.30, p. 54. {{DEFAULTSORT:Dual Basis In A Field Extension Linear algebra Field extensions Theory of cryptography