InnerNormalBasisFieldFunctions(GF)

ffdoms.spad line 2000 [edit on github]

• GF : FiniteFieldCategory

InnerNormalBasisFieldFunctions(GF) (unexposed): This package has functions used by every normal basis finite field extension domain.

* : (Vector(GF), Vector(GF)) -> Vector(GF)

x*y is undocumented See *

/ : (Vector(GF), Vector(GF)) -> Vector(GF)

x/y is undocumented See /

^ : (Vector(GF), Integer) -> Vector(GF)

x^n is undocumented See ^

basis : PositiveInteger -> Vector(Vector(GF))

basis(n) is undocumented See basis

dAndcExp : (Vector(GF), NonNegativeInteger, SingleInteger) -> Vector(GF)

dAndcExp(v, n, k) computes v^e interpreting v as an element of normal basis field. A divide and conquer algorithm similar to the one from D.R.Stinson, "Some observations on parallel Algorithms for fast exponentiation in GF(2^n)", Siam J. Computation, Vol.19, No.4, pp.711-717, August 1990 is used. Argument k is a parameter of this algorithm.

expPot : (Vector(GF), SingleInteger, SingleInteger) -> Vector(GF)

expPot(v, e, d) returns the sum from i = 0 to e - 1 of v^(q^i*d), interpreting v as an element of a normal basis field and where q is the size of the ground field. Note: for a description of the algorithm, see T.Itoh and S.Tsujii, "A fast algorithm for computing multiplicative inverses in GF(2^m) using normal bases", Information and Computation 78, pp.171-177, 1988.

index : (PositiveInteger, PositiveInteger) -> Vector(GF)

index(n, m) is a index function for vectors of length n over the ground field.

inv : Vector(GF) -> Vector(GF)

inv x is undocumented See inv

lookup : Vector(GF) -> PositiveInteger

lookup(x) is undocumented See lookup

minimalPolynomial : Vector(GF) -> SparseUnivariatePolynomial(GF)

minimalPolynomial(x) is undocumented See minimalPolynomial

norm : (Vector(GF), PositiveInteger) -> Vector(GF)

norm(x, n) is undocumented See norm

normal? : Vector(GF) -> Boolean

normal?(x) is undocumented See normal?

normalElement : PositiveInteger -> Vector(GF)

normalElement(n) is undocumented See normalElement

pol : Vector(GF) -> SparseUnivariatePolynomial(GF)

pol(v) turns the vector [v0, ..., vn] into the polynomial v0+v1*x+ ... + vn*x^n.

qPot : (Vector(GF), Integer) -> Vector(GF)

qPot(v, e) computes v^(q^e), interpreting v as an element of normal basis field, q the size of the ground field. This is done by a cyclic e-shift of the vector v.

random : PositiveInteger -> Vector(GF)

random(n) creates a vector over the ground field with random entries.

repSq : (Vector(GF), NonNegativeInteger) -> Vector(GF)

repSq(v, e) computes v^e by repeated squaring, interpreting v as an element of a normal basis field.

setFieldInfo : (Vector(List(Record(value : GF, index : SingleInteger))), GF) -> Void

setFieldInfo(m, p) initializes the field arithmetic, where m is the multiplication table and p is the respective normal element of the ground field GF.

trace : (Vector(GF), PositiveInteger) -> Vector(GF)

trace(x, n) is undocumented See trace

xn : NonNegativeInteger -> SparseUnivariatePolynomial(GF)

xn(n) returns the polynomial x^n-1.