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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra ffnb.spad}
\author{Johannes Grabmeier, Alfred Scheerhorn}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\begin{verbatim}
-- 28.01.93: AS and JG: setting of initlog?, initmult?, and initelt? flags in
-- functions initializeLog, initializeMult and initializeElt put at the
-- end to avoid errors with interruption.
-- factorsOfCyclicGroupSize() changed.
-- 12.05.92: JG: long lines
-- 25.02.92: AS: parametrization of FFNBP changed, compatible to old
-- parametrization. Along with this some changes concerning
-- global variables and deletion of impl. of represents.
-- 25.02.92: AS: parameter in implementation of FFNB,FFNBX changed:
-- Extension now generated by
-- createLowComplexityNormalBasis(extdeg)$FFF(GF)
-- 25.02.92: AS added following functions in FFNBP: degree,
-- linearAssociatedExp,linearAssociatedLog,linearAssociatedOrder
-- 19.02.92: AS: FFNBP trace + norm added.
-- 18.02.92: AS: INBFF normalElement corrected. The old one returned a wrong
-- result for a FFNBP(FFNBP(..)) domain.
\end{verbatim}
\section{package INBFF InnerNormalBasisFieldFunctions}
<<package INBFF InnerNormalBasisFieldFunctions>>=
)abbrev package INBFF InnerNormalBasisFieldFunctions
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 26.03.1991
++ Date Last Updated: 31 March 1991
++ Basic Operations:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: finite field, normal basis
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ D.R.Stinson: Some observations on parallel Algorithms for fast
++ exponentiation in GF(2^n), Siam J. Comp., Vol.19, No.4, pp.711-717,
++ August 1990
++ T.Itoh, S.Tsujii: A fast algorithm for computing multiplicative inverses
++ in GF(2^m) using normal bases, Inf. and Comp. 78, pp.171-177, 1988
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ InnerNormalBasisFieldFunctions(GF) (unexposed):
++ This package has functions used by
++ every normal basis finite field extension domain.
InnerNormalBasisFieldFunctions(GF): Exports == Implementation where
GF : FiniteFieldCategory -- the ground field
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
I ==> Integer
SI ==> SingleInteger
SUP ==> SparseUnivariatePolynomial
VGF ==> Vector GF
M ==> Matrix
V ==> Vector
L ==> List
OUT ==> OutputForm
TERM ==> Record(value:GF,index:SI)
MM ==> ModMonic(GF,SUP GF)
Exports ==> with
setFieldInfo: (V L TERM,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.
random : PI -> VGF
++ random(n) creates a vector over the ground field with random entries.
index : (PI,PI) -> VGF
++ index(n,m) is a index function for vectors of length n over
++ the ground field.
pol : VGF -> SUP GF
++ pol(v) turns the vector \spad{[v0,...,vn]} into the polynomial
++ \spad{v0+v1*x+ ... + vn*x**n}.
xn : NNI -> SUP GF
++ xn(n) returns the polynomial \spad{x**n-1}.
dAndcExp : (VGF,NNI,SI) -> VGF
++ dAndcExp(v,n,k) computes \spad{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.
repSq : (VGF,NNI) -> VGF
++ repSq(v,e) computes \spad{v**e} by repeated squaring,
++ interpreting v as an element of a normal basis field.
expPot : (VGF,SI,SI) -> VGF
++ expPot(v,e,d) returns the sum from \spad{i = 0} to
++ \spad{e - 1} of \spad{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.
qPot : (VGF,I) -> VGF
++ qPot(v,e) computes \spad{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.
-- the semantic of the following functions is obvious from the finite field
-- context, for description see category FAXF
** :(VGF,I) -> VGF
++ x**n \undocumented{}
++ See \axiomFunFrom{**}{DivisionRing}
* :(VGF,VGF) -> VGF
++ x*y \undocumented{}
++ See \axiomFunFrom{*}{SemiGroup}
/ :(VGF,VGF) -> VGF
++ x/y \undocumented{}
++ See \axiomFunFrom{/}{Field}
norm :(VGF,PI) -> VGF
++ norm(x,n) \undocumented{}
++ See \axiomFunFrom{norm}{FiniteAlgebraicExtensionField}
trace :(VGF,PI) -> VGF
++ trace(x,n) \undocumented{}
++ See \axiomFunFrom{trace}{FiniteAlgebraicExtensionField}
inv : VGF -> VGF
++ inv x \undocumented{}
++ See \axiomFunFrom{inv}{DivisionRing}
lookup : VGF -> PI
++ lookup(x) \undocumented{}
++ See \axiomFunFrom{lookup}{Finite}
normal? : VGF -> Boolean
++ normal?(x) \undocumented{}
++ See \axiomFunFrom{normal?}{FiniteAlgebraicExtensionField}
basis : PI -> V VGF
++ basis(n) \undocumented{}
++ See \axiomFunFrom{basis}{FiniteAlgebraicExtensionField}
normalElement:PI -> VGF
++ normalElement(n) \undocumented{}
++ See \axiomFunFrom{normalElement}{FiniteAlgebraicExtensionField}
minimalPolynomial: VGF -> SUP GF
++ minimalPolynomial(x) \undocumented{}
++ See \axiomFunFrom{minimalPolynomial}{FiniteAlgebraicExtensionField}
Implementation ==> add
-- global variables ===================================================
sizeGF:NNI:=size()$GF
-- the size of the ground field
multTable:V L TERM:=new(1,nil()$(L TERM))$(V L TERM)
-- global variable containing the multiplication table
trGen:GF:=1$GF
-- controls the imbedding of the ground field
logq:List SI:=[0,10::SI,16::SI,20::SI,23::SI,0,28::SI,_
30::SI,32::SI,0,35::SI]
-- logq.i is about 10*log2(i) for the values <12 which
-- can match sizeGF. It's used by "**"
expTable:L L SI:=[[],_
[4::SI,12::SI,48::SI,160::SI,480::SI,0],_
[8::SI,72::SI,432::SI,0],_
[18::SI,216::SI,0],_
[32::SI,480::SI,0],[],_
[72::SI,0],[98::SI,0],[128::SI,0],[],[200::SI,0]]
-- expT is used by "**" to optimize the parameter k
-- before calling dAndcExp(..,..,k)
-- functions ===========================================================
-- computes a**(-1) = a**((q**extDeg)-2)
-- see reference of function expPot
inv(a) ==
b:VGF:=qPot(expPot(a,(#a-1)::NNI::SI,1::SI)$$,1)$$
erg:VGF:=inv((a *$$ b).1 *$GF trGen)$GF *$VGF b
-- "**" decides which exponentiation algorithm will be used, in order to
-- get the fastest computation. If dAndcExp is used, it chooses the
-- optimal parameter k for that algorithm.
a ** ex ==
e:NNI:=positiveRemainder(ex,sizeGF**((#a)::PI)-1)$I :: NNI
zero?(e)$NNI => new(#a,trGen)$VGF
one?(e)$NNI => copy(a)$VGF
-- inGroundField?(a) => new(#a,((a.1*trGen) **$GF e))$VGF
e1:SI:=(length(e)$I)::SI
sizeGF >$I 11 =>
q1:SI:=(length(sizeGF)$I)::SI
logqe:SI:=(e1 quo$SI q1) +$SI 1$SI
10::SI * (logqe + sizeGF-2) > 15::SI * e1 =>
-- print("repeatedSquaring"::OUT)
repSq(a,e)
-- print("divAndConquer(a,e,1)"::OUT)
dAndcExp(a,e,1)
logqe:SI:=((10::SI *$SI e1) quo$SI (logq.sizeGF)) +$SI 1$SI
k:SI:=1$SI
expT:List SI:=expTable.sizeGF
while (logqe >= expT.k) and not zero? expT.k repeat k:=k +$SI 1$SI
mult:I:=(sizeGF-1) *$I sizeGF **$I ((k-1)pretend NNI) +$I_
((logqe +$SI k -$SI 1$SI) quo$SI k)::I -$I 2
(10*mult) >= (15 * (e1::I)) =>
-- print("repeatedSquaring(a,e)"::OUT)
repSq(a,e)
-- print(hconcat(["divAndConquer(a,e,"::OUT,k::OUT,")"::OUT])$OUT)
dAndcExp(a,e,k)
-- computes a**e by repeated squaring
repSq(b,e) ==
a:=copy(b)$VGF
one? e => a
odd?(e)$I => a * repSq(a*a,(e quo 2) pretend NNI)
repSq(a*a,(e quo 2) pretend NNI)
-- computes a**e using the divide and conquer algorithm similar to the
-- one from D.R.Stinson,
-- "Some observations on parallel Algorithms for fast exponentiation in
-- GF(2^n)", Siam J. Computation, Vol.19, No.4, pp.711-717, August 1990
dAndcExp(a,e,k) ==
plist:List VGF:=[copy(a)$VGF]
qk:I:=sizeGF**(k pretend NNI)
for j in 2..(qk-1) repeat
if positiveRemainder(j,sizeGF)=0 then b:=qPot(plist.(j quo sizeGF),1)$$
else b:=a *$$ last(plist)$(List VGF)
plist:=concat(plist,b)
l:List NNI:=nil()
ex:I:=e
while not(ex = 0) repeat
l:=concat(l,positiveRemainder(ex,qk) pretend NNI)
ex:=ex quo qk
if first(l)=0 then erg:VGF:=new(#a,trGen)$VGF
else erg:VGF:=plist.(first(l))
i:SI:=k
for j in rest(l) repeat
if j~=0 then erg:=erg *$$ qPot(plist.j,i)$$
i:=i+k
erg
a * b ==
e:SI:=(#a)::SI
erg:=zero(#a)$VGF
for t in multTable.1 repeat
for j in 1..e repeat
y:=t.value -- didn't work without defining x and y
x:=t.index
k:SI:=addmod(x,j::SI,e)$SI +$SI 1$SI
erg.k:=erg.k +$GF a.j *$GF b.j *$GF y
for i in 1..e-1 repeat
for j in i+1..e repeat
for t in multTable.(j-i+1) repeat
y:=t.value -- didn't work without defining x and y
x:=t.index
k:SI:=addmod(x,i::SI,e)$SI +$SI 1$SI
erg.k:GF:=erg.k +$GF (a.i *$GF b.j +$GF a.j *$GF b.i) *$GF y
erg
lookup(x) ==
erg:I:=0
for j in (#x)..1 by -1 repeat
erg:=(erg * sizeGF) + (lookup(x.j)$GF rem sizeGF)
erg=0 => (sizeGF**(#x)) :: PI
erg :: PI
-- computes the norm of a over GF**d, d must devide extdeg
-- see reference of function expPot below
norm(a,d) ==
dSI:=d::SI
r:=divide((#a)::SI,dSI)
not(r.remainder = 0) => error "norm: 2.arg must divide extdeg"
expPot(a,r.quotient,dSI)$$
-- computes expPot(a,e,d) = sum form i=0 to e-1 over a**(q**id))
-- 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
expPot(a,e,d) ==
deg:SI:=(#a)::SI
e=1 => copy(a)$VGF
k2:SI:=d
y:=copy(a)
if bit?(e,0) then
erg:=copy(y)
qpot:SI:=k2
else
erg:=new(#a,inv(trGen)$GF)$VGF
qpot:SI:=0
for k in 1..length(e) repeat
y:= y *$$ qPot(y,k2)
k2:=addmod(k2,k2,deg)$SI
if bit?(e,k) then
erg:=erg *$$ qPot(y,qpot)
qpot:=addmod(qpot,k2,deg)$SI
erg
-- computes qPot(a,n) = a**(q**n), q=size of GF
qPot(e,n) ==
ei:=(#e)::SI
m:SI:= positiveRemainder(n::SI,ei)$SI
zero?(m) => e
e1:=zero(#e)$VGF
for i in m+1..ei repeat e1.i:=e.(i-m)
for i in 1..m repeat e1.i:=e.(ei+i-m)
e1
trace(a,d) ==
dSI:=d::SI
r:=divide((#a)::SI,dSI)$SI
not(r.remainder = 0) => error "trace: 2.arg must divide extdeg"
v:=copy(a.(1..dSI))$VGF
sSI:SI:=r.quotient
for i in 1..dSI repeat
for j in 1..sSI-1 repeat
v.i:=v.i+a.(i+j::SI*dSI)
v
random(n) ==
v:=zero(n)$VGF
for i in 1..n repeat v.i:=random()$GF
v
xn(m) == monomial(1,m)$(SUP GF) - 1$(SUP GF)
normal?(x) ==
gcd(xn(#x),pol(x))$(SUP GF) = 1 => true
false
x:VGF / y:VGF == x *$$ inv(y)$$
setFieldInfo(m,n) ==
multTable:=m
trGen:=n
void()$Void
minimalPolynomial(x) ==
dx:=#x
y:=new(#x,inv(trGen)$GF)$VGF
m:=zero(dx,dx+1)$(M GF)
for i in 1..dx+1 repeat
dy:=#y
for j in 1..dy repeat
for k in 0..((dx quo dy)-1) repeat
qsetelt_!(m,j+k*dy,i,y.j)$(M GF)
y:=y *$$ x
v:=first nullSpace(m)$(M GF)
pol(v)$$
basis(n) ==
bas:(V VGF):=new(n,zero(n)$VGF)$(V VGF)
for i in 1..n repeat
uniti:=zero(n)$VGF
qsetelt_!(uniti,i,1$GF)$VGF
qsetelt_!(bas,i,uniti)$(V VGF)
bas
normalElement(n) ==
v:=zero(n)$VGF
qsetelt_!(v,1,1$GF)
v
-- normalElement(n) == index(n,1)$$
index(degm,n) ==
m:I:=n rem$I (sizeGF ** degm)
erg:=zero(degm)$VGF
for j in 1..degm repeat
erg.j:=index((sizeGF+(m rem sizeGF)) pretend PI)$GF
m:=m quo sizeGF
erg
pol(x) ==
+/[monomial(x.i,(i-1)::NNI)$(SUP GF) for i in 1..(#x)::I]
@
\section{domain FFNBP FiniteFieldNormalBasisExtensionByPolynomial}
<<domain FFNBP FiniteFieldNormalBasisExtensionByPolynomial>>=
import NonNegativeInteger
import Matrix
)abbrev domain FFNBP FiniteFieldNormalBasisExtensionByPolynomial
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 26.03.1991
++ Date Last Updated: 08 May 1991
++ Basic Operations:
++ Related Constructors: InnerNormalBasisFieldFunctions, FiniteFieldFunctions,
++ Also See: FiniteFieldExtensionByPolynomial,
++ FiniteFieldCyclicGroupExtensionByPolynomial
++ AMS Classifications:
++ Keywords: finite field, normal basis
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM .
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteFieldNormalBasisExtensionByPolynomial(GF,uni) implements a
++ finite extension of the ground field {\em GF}. The elements are
++ represented by coordinate vectors with respect to. a normal basis,
++ i.e. a basis
++ consisting of the conjugates (q-powers) of an element, in this case
++ called normal element, where q is the size of {\em GF}.
++ The normal element is chosen as a root of the extension
++ polynomial, which MUST be normal over {\em GF} (user responsibility)
FiniteFieldNormalBasisExtensionByPolynomial(GF,uni): Exports == _
Implementation where
GF : FiniteFieldCategory -- the ground field
uni : Union(SparseUnivariatePolynomial GF,_
Vector List Record(value:GF,index:SingleInteger))
PI ==> PositiveInteger
NNI ==> NonNegativeInteger
I ==> Integer
SI ==> SingleInteger
SUP ==> SparseUnivariatePolynomial
V ==> Vector GF
M ==> Matrix GF
OUT ==> OutputForm
TERM ==> Record(value:GF,index:SI)
R ==> Record(key:PI,entry:NNI)
TBL ==> Table(PI,NNI)
FFF ==> FiniteFieldFunctions(GF)
INBFF ==> InnerNormalBasisFieldFunctions(GF)
Exports ==> FiniteAlgebraicExtensionField(GF) with
getMultiplicationTable: () -> Vector List TERM
++ getMultiplicationTable() returns the multiplication
++ table for the normal basis of the field.
++ This table is used to perform multiplications between field elements.
getMultiplicationMatrix:() -> M
++ getMultiplicationMatrix() returns the multiplication table in
++ form of a matrix.
sizeMultiplication:() -> NNI
++ sizeMultiplication() returns the number of entries in the
++ multiplication table of the field.
++ Note: the time of multiplication
++ of field elements depends on this size.
Implementation ==> add
-- global variables ===================================================
Rep:= V -- elements are represented by vectors over GF
alpha :=new()$Symbol :: OutputForm
-- get a new Symbol for the output representation of the elements
initlog?:Boolean:=true
-- gets false after initialization of the logarithm table
initelt?:Boolean:=true
-- gets false after initialization of the primitive element
initmult?:Boolean:=true
-- gets false after initialization of the multiplication
-- table or the primitive element
extdeg:PI :=1
defpol:SUP(GF):=0$SUP(GF)
-- the defining polynomial
multTable:Vector List TERM:=new(1,nil()$(List TERM))
-- global variable containing the multiplication table
if uni case (Vector List TERM) then
multTable:=uni :: (Vector List TERM)
extdeg:= (#multTable) pretend PI
vv:V:=new(extdeg,0)$V
vv.1:=1$GF
setFieldInfo(multTable,1$GF)$INBFF
defpol:=minimalPolynomial(vv)$INBFF
initmult?:=false
else
defpol:=uni :: SUP(GF)
extdeg:=degree(defpol)$(SUP GF) pretend PI
multTable:Vector List TERM:=new(extdeg,nil()$(List TERM))
basisOutput : List OUT :=
qs:OUT:=(q::Symbol)::OUT
append([alpha, alpha **$OUT qs],_
[alpha **$OUT (qs **$OUT i::OUT) for i in 2..extdeg-1] )
facOfGroupSize :=nil()$(List Record(factor:Integer,exponent:Integer))
-- the factorization of the cyclic group size
traceAlpha:GF:=-$GF coefficient(defpol,(degree(defpol)-1)::NNI)
-- the inverse of the trace of the normalElt
-- is computed here. It defines the imbedding of
-- GF in the extension field
primitiveElt:PI:=1
-- for the lookup the primitive Element computed by createPrimitiveElement()
discLogTable:Table(PI,TBL):=table()$Table(PI,TBL)
-- tables indexed by the factors of sizeCG,
-- discLogTable(factor) is a table with keys
-- primitiveElement() ** (i * (sizeCG quo factor)) and entries i for
-- i in 0..n-1, n computed in initialize() in order to use
-- the minimal size limit 'limit' optimal.
-- functions ===========================================================
initializeLog: () -> Void
initializeElt: () -> Void
initializeMult: () -> Void
coerce(v:GF):$ == new(extdeg,v /$GF traceAlpha)$Rep
represents(v) == v::$
degree(a: %): PositiveInteger ==
d:PI:=1
b:= qPot(a::Rep,1)$INBFF
while (b~=a) repeat
b:= qPot(b::Rep,1)$INBFF
d:=d+1
d
linearAssociatedExp(x,f) ==
xm:SUP(GF):=monomial(1$GF,extdeg)$(SUP GF) - 1$(SUP GF)
r:= (f * pol(x::Rep)$INBFF) rem xm
vectorise(r,extdeg)$(SUP GF)
linearAssociatedLog(x) == pol(x::Rep)$INBFF
linearAssociatedOrder(x) ==
xm:SUP(GF):=monomial(1$GF,extdeg)$(SUP GF) - 1$(SUP GF)
xm quo gcd(xm,pol(x::Rep)$INBFF)
linearAssociatedLog(b,x) ==
zero? x => 0
xm:SUP(GF):=monomial(1$GF,extdeg)$(SUP GF) - 1$(SUP GF)
e:= extendedEuclidean(pol(b::Rep)$INBFF,xm,pol(x::Rep)$INBFF)$(SUP GF)
e = "failed" => "failed"
e1:= e :: Record(coef1:(SUP GF),coef2:(SUP GF))
e1.coef1
getMultiplicationTable() ==
if initmult? then initializeMult()
multTable
getMultiplicationMatrix() ==
if initmult? then initializeMult()
createMultiplicationMatrix(multTable)$FFF
sizeMultiplication() ==
if initmult? then initializeMult()
sizeMultiplication(multTable)$FFF
trace(a:$) == retract trace(a,1)
norm(a:$) == retract norm(a,1)
generator() == normalElement(extdeg)$INBFF
basis(n:PI) ==
(extdeg rem n) ~= 0 => error "argument must divide extension degree"
[Frobenius(trace(normalElement,n),i) for i in 0..(n-1)]::(Vector $)
a:GF * x:$ == a *$Rep x
x:$/a:GF == x/coerce(a)
-- x:$ / a:GF ==
-- a = 0$GF => error "division by zero"
-- x * inv(coerce(a))
coordinates(x:$) == x::Rep
Frobenius(e) == qPot(e::Rep,1)$INBFF
Frobenius(e,n) == qPot(e::Rep,n)$INBFF
retractIfCan(x) ==
inGroundField?(x) =>
x.1 *$GF traceAlpha
"failed"
retract(x) ==
inGroundField?(x) =>
x.1 *$GF traceAlpha
error("element not in ground field")
-- to get a "normal basis like" output form
coerce(x:$):OUT ==
l:List OUT:=nil()$(List OUT)
n : PI := extdeg
one? n => (x.1) :: OUT
for i in 1..n for b in basisOutput repeat
if not zero? x.i then
mon : OUT :=
one? x.i => b
((x.i)::OUT) *$OUT b
l:=cons(mon,l)$(List OUT)
null(l)$(List OUT) => (0::OUT)
r:=reduce("+",l)$(List OUT)
r
initializeElt() ==
facOfGroupSize := factors factor(size()$GF**extdeg-1)$I
-- get a primitive element
primitiveElt:=lookup(createPrimitiveElement())
initelt?:=false
void()$Void
initializeMult() ==
multTable:=createMultiplicationTable(defpol)$FFF
setFieldInfo(multTable,traceAlpha)$INBFF
-- reset initialize flag
initmult?:=false
void()$Void
initializeLog() ==
if initelt? then initializeElt()
-- set up tables for discrete logarithm
limit:Integer:=30
-- the minimum size for the discrete logarithm table
for f in facOfGroupSize repeat
fac:=f.factor
base:$:=index(primitiveElt)**((size()$GF**extdeg -$I 1$I) quo$I fac)
l:Integer:=length(fac)$Integer
n:Integer:=0
if odd?(l)$I then n:=shift(fac,-$I (l quo$I 2))$I
else n:=shift(1,l quo$I 2)$I
if n <$I limit then
d:=(fac -$I 1$I) quo$I limit +$I 1$I
n:=(fac -$I 1$I) quo$I d +$I 1$I
tbl:TBL:=table()$TBL
a:$:=1
for i in (0::NNI)..(n-1)::NNI repeat
insert_!([lookup(a),i::NNI]$R,tbl)$TBL
a:=a*base
insert_!([fac::PI,copy(tbl)$TBL]_
$Record(key:PI,entry:TBL),discLogTable)$Table(PI,TBL)
initlog?:=false
-- tell user about initialization
--print("discrete logarithm table initialized"::OUT)
void()$Void
tableForDiscreteLogarithm(fac) ==
if initlog? then initializeLog()
tbl:=search(fac::PI,discLogTable)$Table(PI,TBL)
tbl case "failed" =>
error "tableForDiscreteLogarithm: argument must be prime _
divisor of the order of the multiplicative group"
tbl :: TBL
primitiveElement() ==
if initelt? then initializeElt()
index(primitiveElt)
factorsOfCyclicGroupSize() ==
if empty? facOfGroupSize then initializeElt()
facOfGroupSize
extensionDegree(): PositiveInteger == extdeg
sizeOfGroundField() == size()$GF pretend NNI
definingPolynomial() == defpol
trace(a,d) ==
v:=trace(a::Rep,d)$INBFF
erg:=v
for i in 2..(extdeg quo d) repeat
erg:=concat(erg,v)$Rep
erg
characteristic == characteristic$GF
random() == random(extdeg)$INBFF
x:$ * y:$ ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
x::Rep *$INBFF y::Rep
1 == new(extdeg,inv(traceAlpha)$GF)$Rep
0 == zero(extdeg)$Rep
size() == size()$GF ** extdeg
index(n:PI) == index(extdeg,n)$INBFF
lookup(x:$) == lookup(x::Rep)$INBFF
basis() ==
a:=basis(extdeg)$INBFF
vector([e::$ for e in entries a])
x:$ ** e:I ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
(x::Rep) **$INBFF e
normal?(x) == normal?(x::Rep)$INBFF
-(x:$) == -$Rep x
x:$ + y:$ == x +$Rep y
x:$ - y:$ == x -$Rep y
x:$ = y:$ == x =$Rep y
n:I * x:$ == x *$Rep (n::GF)
representationType() == "normal"
minimalPolynomial(a) ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
minimalPolynomial(a::Rep)$INBFF
-- is x an element of the ground field GF ?
inGroundField?(x) ==
erg:=true
for i in 2..extdeg repeat
not(x.i =$GF x.1) => erg:=false
erg
x:$ / y:$ ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
x::Rep /$INBFF y::Rep
inv(a) ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
inv(a::Rep)$INBFF
norm(a,d) ==
if initmult? then initializeMult()
setFieldInfo(multTable,traceAlpha)$INBFF
norm(a::Rep,d)$INBFF
normalElement() == normalElement(extdeg)$INBFF
@
\section{domain FFNBX FiniteFieldNormalBasisExtension}
<<domain FFNBX FiniteFieldNormalBasisExtension>>=
)abbrev domain FFNBX FiniteFieldNormalBasisExtension
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 26.03.1991
++ Date Last Updated:
++ Basic Operations:
++ Related Constructors: FiniteFieldNormalBasisExtensionByPolynomial,
++ FiniteFieldPolynomialPackage
++ Also See: FiniteFieldExtension, FiniteFieldCyclicGroupExtension
++ AMS Classifications:
++ Keywords: finite field, normal basis
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteFieldNormalBasisExtensionByPolynomial(GF,n) implements a
++ finite extension field of degree n over the ground field {\em GF}.
++ The elements are represented by coordinate vectors with respect
++ to a normal basis,
++ i.e. a basis consisting of the conjugates (q-powers) of an element, in
++ this case called normal element. This is chosen as a root of the extension
++ polynomial, created by {\em createNormalPoly} from
++ \spadtype{FiniteFieldPolynomialPackage}
FiniteFieldNormalBasisExtension(GF,extdeg):_
Exports == Implementation where
GF : FiniteFieldCategory -- the ground field
extdeg: PositiveInteger -- the extension degree
NNI ==> NonNegativeInteger
FFF ==> FiniteFieldFunctions(GF)
TERM ==> Record(value:GF,index:SingleInteger)
Exports ==> FiniteAlgebraicExtensionField(GF) with
getMultiplicationTable: () -> Vector List TERM
++ getMultiplicationTable() returns the multiplication
++ table for the normal basis of the field.
++ This table is used to perform multiplications between field elements.
getMultiplicationMatrix: () -> Matrix GF
++ getMultiplicationMatrix() returns the multiplication table in
++ form of a matrix.
sizeMultiplication:() -> NNI
++ sizeMultiplication() returns the number of entries in the
++ multiplication table of the field. Note: the time of multiplication
++ of field elements depends on this size.
Implementation ==> FiniteFieldNormalBasisExtensionByPolynomial(GF,_
createLowComplexityNormalBasis(extdeg)$FFF)
@
\section{domain FFNB FiniteFieldNormalBasis}
<<domain FFNB FiniteFieldNormalBasis>>=
)abbrev domain FFNB FiniteFieldNormalBasis
++ Authors: J.Grabmeier, A.Scheerhorn
++ Date Created: 26.03.1991
++ Date Last Updated:
++ Basic Operations:
++ Related Constructors: FiniteFieldNormalBasisExtensionByPolynomial,
++ FiniteFieldPolynomialPackage
++ Also See: FiniteField, FiniteFieldCyclicGroup
++ AMS Classifications:
++ Keywords: finite field, normal basis
++ References:
++ R.Lidl, H.Niederreiter: Finite Field, Encyclopedia of Mathematics and
++ Its Applications, Vol. 20, Cambridge Univ. Press, 1983, ISBN 0 521 30240 4
++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM.
++ AXIOM Technical Report Series, ATR/5 NP2522.
++ Description:
++ FiniteFieldNormalBasis(p,n) implements a
++ finite extension field of degree n over the prime field with p elements.
++ The elements are represented by coordinate vectors with respect to
++ a normal basis,
++ i.e. a basis consisting of the conjugates (q-powers) of an element, in
++ this case called normal element.
++ This is chosen as a root of the extension polynomial
++ created by \spadfunFrom{createNormalPoly}{FiniteFieldPolynomialPackage}.
FiniteFieldNormalBasis(p,extdeg):_
Exports == Implementation where
p : PositiveInteger
extdeg: PositiveInteger -- the extension degree
NNI ==> NonNegativeInteger
FFF ==> FiniteFieldFunctions(PrimeField(p))
TERM ==> Record(value:PrimeField(p),index:SingleInteger)
Exports ==> FiniteAlgebraicExtensionField(PrimeField(p)) with
getMultiplicationTable: () -> Vector List TERM
++ getMultiplicationTable() returns the multiplication
++ table for the normal basis of the field.
++ This table is used to perform multiplications between field elements.
getMultiplicationMatrix: () -> Matrix PrimeField(p)
++ getMultiplicationMatrix() returns the multiplication table in
++ form of a matrix.
sizeMultiplication:() -> NNI
++ sizeMultiplication() returns the number of entries in the
++ multiplication table of the field. Note: The time of multiplication
++ of field elements depends on this size.
Implementation ==> FiniteFieldNormalBasisExtensionByPolynomial(PrimeField(p),_
createLowComplexityNormalBasis(extdeg)$FFF)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<package INBFF InnerNormalBasisFieldFunctions>>
<<domain FFNBP FiniteFieldNormalBasisExtensionByPolynomial>>
<<domain FFNBX FiniteFieldNormalBasisExtension>>
<<domain FFNB FiniteFieldNormalBasis>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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