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 finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.
Normal basis theorem
Let be a Galois extension with Galois group . The classical normal basis theorem states that there is an element such that forms a basis of K, considered as a vector space over F. That is, any element can be written uniquely as for coefficients
A normal basis contrasts with a primitive element basis of the form , where is an element whose minimal polynomial has degree .
Case of finite fields
For finite fields this can be stated as follows: Let denote the field of q elements, where q = pm is a prime power, and let denote its extension field of degree n ≥ 1. Here the Galois group is with a cyclic group generated by the relative Frobenius automorphism with Then there exists an element β ∈ K such that
is a basis of K over F.
Proof for finite fields
In case the Galois group is cyclic as above, generated by with the Normal Basis Theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying ; then any distinct characters are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms thought of as mappings from the multiplicative group . Now as an F-vector space, so we may consider as an element of the matrix algebra since its powers are linearly independent (over K and a fortiori over F), its minimal polynomial[disambiguation needed] must have degree at least n, i.e. it must be . We conclude that the group algebra of G is a quotient of the polynomial ring F[X], and the F-vector space K is a module (or representation) for this algebra.
The second basic fact is the classification of modules over a PID such as F[G]. These are just direct sums of cyclic modules of the form where f(x) must be divisible by Xn − 1. (Here G acts by ) But since we can only have f(x) = Xn − 1, and
as F[G]-modules, namely the regular representation of G. (Note this is not an isomorphism of rings or F-algebras!) Now the basis on the right side of this isomorphism corresponds to a normal basis of K on the left.
Note that this proof would also apply in the case of a cyclic Kummer extension.
Consider the field over , with Frobenius automorphism . The proof above clarifies the choice of normal bases in terms of the structure of K as a representation of G (or F[G]-module). The irreducible factorization
means we have a direct sum of F[G]-modules (by the Chinese remainder theorem):
The first component is just , while the second is isomorphic as an F[G]-module to under the action (Thus as F[G]-modules, but not as F-algebras.)
The elements which can be used for a normal basis are precisely those outside either of the submodules, so that and . In terms of the G-orbits of K, which correspond to the irreducible factors of:
the elements of are the roots of , the nonzero elements of the submodule are the roots of , while the normal basis, which in this case is unique, is given by the roots of the remaining factor .
By contrast, for the extension field in which n = 4 is divisible by p = 2, we have the F[G]-module isomorphism
Here the operator is not diagonalizable, the module L has nested submodules given by generalized eigenspaces of , and the normal basis elements β are those outside the largest proper generalized eigenspace, the elements with .
Application to cryptography
The normal basis is frequently used in cryptographic applications based on the discrete logarithm problem, such as elliptic curve cryptography, since arithmetic using a normal basis is typically more computationally efficient than using other bases.
For example, in the field above, we may represent elements as bit-strings:
where the coefficients are bits Now we can square elements by doing a left circular shift, , since squaring β4 gives β8 = β. This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring.
Primitive normal basis
A primitive normal basis of an extension of finite fields E/F is a normal basis for E/F that is generated by a primitive element of E, that is a generator of the multiplicative group (Note that this is a more restrictive definition of primitive element than that mentioned above after the general Normal Basis Theorem: one requires powers of the element to produce every non-zero element of K, not merely a basis.) Lenstra and Schoof (1987) proved that every finite field extension possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport.
If K/F is a Galois extension and x in E generates a normal basis over F, then x is free in K/F. If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for K/KH, then x is said to be completely free in K/F. Every Galois extension has a completely free element.
- Nader H. Bshouty; Gadiel Seroussi (1989), Generalizations of the normal basis theorem of finite fields (PDF), p. 1; SIAM J. Discrete Math. 3 (1990), no. 3, 330–337.
- Dirk Hachenberger, Completely free elements, in Cohen & Niederreiter (1996) pp.97-107 Zbl 0864.11066
- Cohen, S.; Niederreiter, H., eds. (1996). Finite Fields and Applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11–14, 1995. London Mathematical Society Lecture Note Series. 233. Cambridge University Press. ISBN 978-0-521-56736-7. Zbl 0851.00052.
- Lenstra, H.W., jr; Schoof, R.J. (1987). "Primitive normal bases for finite fields". Mathematics of Computation. 48 (177): 217–231. doi:10.2307/2007886. JSTOR 2007886. Zbl 0615.12023.
- Menezes, Alfred J., ed. (1993). Applications of finite fields. The Kluwer International Series in Engineering and Computer Science. 199. Boston: Kluwer Academic Publishers. ISBN 978-0792392828. Zbl 0779.11059.