\documentclass{article} \usepackage{axiom} \begin{document} \title{\$SPAD/src/algebra ffcat.spad} \author{Johannes Grabmeier, Alfred Scheerhorn, Barry Trager, James Davenport} \maketitle \begin{abstract} \end{abstract} \eject \tableofcontents \eject \begin{verbatim} -- 28.01.93: AS and JG:another Error in discreteLog(.,.) in FFIEDLC corrected. -- 08.05.92: AS Error in discreteLog(.,.) in FFIEDLC corrected. -- 03.04.92: AS Barry Trager added package FFSLPE and some functions to FFIELDC -- 25.02.92: AS added following functions in FAXF: impl.of mrepresents, -- linearAssociatedExp,linearAssociatedLog, linearAssociatedOrder -- 18.02.92: AS: more efficient version of degree added, -- first version of degree in FAXF set into comments -- 18.06.91: AS: general version of minimalPolynomial added -- 08.05.91: JG, AS implementation of missing functions in FFC and FAXF -- 04.05.91: JG: comments -- 04.04.91: JG: old version of charthRoot in FFC was dropped -- Fields with finite characteristic \end{verbatim} \section{category FPC FieldOfPrimeCharacteristic} <>= )abbrev category FPC FieldOfPrimeCharacteristic ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 10 March 1991 ++ Date Last Updated: 31 March 1991 ++ Basic Operations: _+, _* ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: field, finite field, prime characteristic ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. ++ Description: ++ FieldOfPrimeCharacteristic is the category of fields of prime ++ characteristic, e.g. finite fields, algebraic closures of ++ fields of prime characteristic, transcendental extensions of ++ of fields of prime characteristic. FieldOfPrimeCharacteristic:Category == _ Join(Field,CharacteristicNonZero) with order: $ -> OnePointCompletion PositiveInteger ++ order(a) computes the order of an element in the multiplicative ++ group of the field. ++ Error: if \spad{a} is 0. discreteLog: ($,$) -> Union(NonNegativeInteger,"failed") ++ discreteLog(b,a) computes s with \spad{b**s = a} if such an s exists. primeFrobenius: $ -> $ ++ primeFrobenius(a) returns \spad{a ** p} where p is the characteristic. primeFrobenius: ($,NonNegativeInteger) -> $ ++ primeFrobenius(a,s) returns \spad{a**(p**s)} where p ++ is the characteristic. add primeFrobenius(a) == a ** characteristic() primeFrobenius(a,s) == a ** (characteristic()**s) @ \section{category XF ExtensionField} <>= )abbrev category XF ExtensionField ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 10 March 1991 ++ Date Last Updated: 31 March 1991 ++ Basic Operations: _+, _*, extensionDegree, algebraic?, transcendent? ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: field, extension field ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. ++ Description: ++ ExtensionField {\em F} is the category of fields which extend ++ the field F ExtensionField(F:Field) : Category == Join(Field,RetractableTo F,VectorSpace F) with if F has CharacteristicZero then CharacteristicZero if F has CharacteristicNonZero then FieldOfPrimeCharacteristic algebraic? : $ -> Boolean ++ algebraic?(a) tests whether an element \spad{a} is algebraic with ++ respect to the ground field F. transcendent? : $ -> Boolean ++ transcendent?(a) tests whether an element \spad{a} is transcendent ++ with respect to the ground field F. inGroundField?: $ -> Boolean ++ inGroundField?(a) tests whether an element \spad{a} ++ is already in the ground field F. degree : $ -> OnePointCompletion PositiveInteger ++ degree(a) returns the degree of minimal polynomial of an element ++ \spad{a} if \spad{a} is algebraic ++ with respect to the ground field F, and \spad{infinity} otherwise. extensionDegree : () -> OnePointCompletion PositiveInteger ++ extensionDegree() returns the degree of the field extension if the ++ extension is algebraic, and \spad{infinity} if it is not. transcendenceDegree : () -> NonNegativeInteger ++ transcendenceDegree() returns the transcendence degree of the ++ field extension, 0 if the extension is algebraic. -- perhaps more absolute degree functions if F has Finite then FieldOfPrimeCharacteristic Frobenius: $ -> $ ++ Frobenius(a) returns \spad{a ** q} where q is the \spad{size()$F}. Frobenius: ($,NonNegativeInteger) -> $ ++ Frobenius(a,s) returns \spad{a**(q**s)} where q is the size()$F. add algebraic?(a) == not infinite? (degree(a)@OnePointCompletion_ (PositiveInteger))$OnePointCompletion(PositiveInteger) transcendent? a == infinite?(degree(a)@OnePointCompletion _ (PositiveInteger))$OnePointCompletion(PositiveInteger) if F has Finite then Frobenius(a) == a ** size()$F Frobenius(a,s) == a ** (size()$F ** s) @ \section{category FAXF FiniteAlgebraicExtensionField} <>= )abbrev category FAXF FiniteAlgebraicExtensionField ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 11 March 1991 ++ Date Last Updated: 31 March 1991 ++ Basic Operations: _+, _*, extensionDegree, ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: field, extension field, algebraic extension, finite extension ++ References: ++ R.Lidl, H.Niederreiter: Finite Field, Encycoldia 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: ++ FiniteAlgebraicExtensionField {\em F} is the category of fields ++ which are finite algebraic extensions of the field {\em F}. ++ If {\em F} is finite then any finite algebraic extension of {\em F} is finite, too. ++ Let {\em K} be a finite algebraic extension of the finite field {\em F}. ++ The exponentiation of elements of {\em K} defines a Z-module structure ++ on the multiplicative group of {\em K}. The additive group of {\em K} ++ becomes a module over the ring of polynomials over {\em F} via the operation ++ \spadfun{linearAssociatedExp}(a:K,f:SparseUnivariatePolynomial F) ++ which is linear over {\em F}, i.e. for elements {\em a} from {\em K}, ++ {\em c,d} from {\em F} and {\em f,g} univariate polynomials over {\em F} ++ we have \spadfun{linearAssociatedExp}(a,cf+dg) equals {\em c} times ++ \spadfun{linearAssociatedExp}(a,f) plus {\em d} times ++ \spadfun{linearAssociatedExp}(a,g). ++ Therefore \spadfun{linearAssociatedExp} is defined completely by ++ its action on monomials from {\em F[X]}: ++ \spadfun{linearAssociatedExp}(a,monomial(1,k)\$SUP(F)) is defined to be ++ \spadfun{Frobenius}(a,k) which is {\em a**(q**k)} where {\em q=size()\$F}. ++ The operations order and discreteLog associated with the multiplicative ++ exponentiation have additive analogues associated to the operation ++ \spadfun{linearAssociatedExp}. These are the functions ++ \spadfun{linearAssociatedOrder} and \spadfun{linearAssociatedLog}, ++ respectively. FiniteAlgebraicExtensionField(F : Field) : Category == _ Join(ExtensionField F, RetractableTo F) with -- should be unified with algebras -- Join(ExtensionField F, FramedAlgebra F, RetractableTo F) with basis : () -> Vector $ ++ basis() returns a fixed basis of \$ as \spad{F}-vectorspace. basis : PositiveInteger -> Vector $ ++ basis(n) returns a fixed basis of a subfield of \$ as ++ \spad{F}-vectorspace. coordinates : $ -> Vector F ++ coordinates(a) returns the coordinates of \spad{a} with respect ++ to the fixed \spad{F}-vectorspace basis. coordinates : Vector $ -> Matrix F ++ coordinates([v1,...,vm]) returns the coordinates of the ++ vi's with to the fixed basis. The coordinates of vi are ++ contained in the ith row of the matrix returned by this ++ function. represents: Vector F -> $ ++ represents([a1,..,an]) returns \spad{a1*v1 + ... + an*vn}, where ++ v1,...,vn are the elements of the fixed basis. minimalPolynomial: $ -> SparseUnivariatePolynomial F ++ minimalPolynomial(a) returns the minimal polynomial of an ++ element \spad{a} over the ground field F. definingPolynomial: () -> SparseUnivariatePolynomial F ++ definingPolynomial() returns the polynomial used to define ++ the field extension. extensionDegree : () -> PositiveInteger ++ extensionDegree() returns the degree of field extension. degree : $ -> PositiveInteger ++ degree(a) returns the degree of the minimal polynomial of an ++ element \spad{a} over the ground field F. norm: $ -> F ++ norm(a) computes the norm of \spad{a} with respect to the ++ field considered as an algebra with 1 over the ground field F. trace: $ -> F ++ trace(a) computes the trace of \spad{a} with respect to ++ the field considered as an algebra with 1 over the ground field F. if F has Finite then FiniteFieldCategory minimalPolynomial: ($,PositiveInteger) -> SparseUnivariatePolynomial $ ++ minimalPolynomial(x,n) computes the minimal polynomial of x over ++ the field of extension degree n over the ground field F. norm: ($,PositiveInteger) -> $ ++ norm(a,d) computes the norm of \spad{a} with respect to the field of ++ extension degree d over the ground field of size. ++ Error: if d does not divide the extension degree of \spad{a}. ++ Note: norm(a,d) = reduce(*,[a**(q**(d*i)) for i in 0..n/d]) trace: ($,PositiveInteger) -> $ ++ trace(a,d) computes the trace of \spad{a} with respect to the ++ field of extension degree d over the ground field of size q. ++ Error: if d does not divide the extension degree of \spad{a}. ++ Note: \spad{trace(a,d) = reduce(+,[a**(q**(d*i)) for i in 0..n/d])}. createNormalElement: () -> $ ++ createNormalElement() computes a normal element over the ground ++ field F, that is, ++ \spad{a**(q**i), 0 <= i < extensionDegree()} is an F-basis, ++ where \spad{q = size()\$F}. ++ Reference: Such an element exists Lidl/Niederreiter: Theorem 2.35. normalElement: () -> $ ++ normalElement() returns a element, normal over the ground field F, ++ i.e. \spad{a**(q**i), 0 <= i < extensionDegree()} is an F-basis, ++ where \spad{q = size()\$F}. ++ At the first call, the element is computed by ++ \spadfunFrom{createNormalElement}{FiniteAlgebraicExtensionField} ++ then cached in a global variable. ++ On subsequent calls, the element is retrieved by referencing the ++ global variable. normal?: $ -> Boolean ++ normal?(a) tests whether the element \spad{a} is normal over the ++ ground field F, i.e. ++ \spad{a**(q**i), 0 <= i <= extensionDegree()-1} is an F-basis, ++ where \spad{q = size()\$F}. ++ Implementation according to Lidl/Niederreiter: Theorem 2.39. generator: () -> $ ++ generator() returns a root of the defining polynomial. ++ This element generates the field as an algebra over the ground field. linearAssociatedExp:($,SparseUnivariatePolynomial F) -> $ ++ linearAssociatedExp(a,f) is linear over {\em F}, i.e. ++ for elements {\em a} from {\em \$}, {\em c,d} form {\em F} and ++ {\em f,g} univariate polynomials over {\em F} we have ++ \spadfun{linearAssociatedExp}(a,cf+dg) equals {\em c} times ++ \spadfun{linearAssociatedExp}(a,f) plus {\em d} times ++ \spadfun{linearAssociatedExp}(a,g). Therefore ++ \spadfun{linearAssociatedExp} is defined completely by its action on ++ monomials from {\em F[X]}: ++ \spadfun{linearAssociatedExp}(a,monomial(1,k)\$SUP(F)) is defined to ++ be \spadfun{Frobenius}(a,k) which is {\em a**(q**k)}, ++ where {\em q=size()\$F}. linearAssociatedOrder: $ -> SparseUnivariatePolynomial F ++ linearAssociatedOrder(a) retruns the monic polynomial {\em g} of ++ least degree, such that \spadfun{linearAssociatedExp}(a,g) is 0. linearAssociatedLog: $ -> SparseUnivariatePolynomial F ++ linearAssociatedLog(a) returns a polynomial {\em g}, such that ++ \spadfun{linearAssociatedExp}(normalElement(),g) equals {\em a}. linearAssociatedLog: ($,$) -> Union(SparseUnivariatePolynomial F,"failed") ++ linearAssociatedLog(b,a) returns a polynomial {\em g}, such that the ++ \spadfun{linearAssociatedExp}(b,g) equals {\em a}. ++ If there is no such polynomial {\em g}, then ++ \spadfun{linearAssociatedLog} fails. add I ==> Integer PI ==> PositiveInteger NNI ==> NonNegativeInteger SUP ==> SparseUnivariatePolynomial DLP ==> DiscreteLogarithmPackage represents(v) == a:$:=0 b:=basis() for i in 1..extensionDegree()@PI repeat a:=a+(v.i)*(b.i) a transcendenceDegree() == 0$NNI dimension() == (#basis()) ::NonNegativeInteger::CardinalNumber extensionDegree():OnePointCompletion(PositiveInteger) == (#basis()) :: PositiveInteger::OnePointCompletion(PositiveInteger) degree(a):OnePointCompletion(PositiveInteger) == degree(a)@PI::OnePointCompletion(PositiveInteger) coordinates(v:Vector $) == m := new(#v, extensionDegree(), 0)$Matrix(F) for i in minIndex v .. maxIndex v for j in minRowIndex m .. repeat setRow_!(m, j, coordinates qelt(v, i)) m algebraic? a == true transcendent? a == false extensionDegree() == (#basis()) :: PositiveInteger -- degree a == degree(minimalPolynomial a)$SUP(F) :: PI trace a == b := basis() abs : F := 0 for i in 1..#b repeat abs := abs + coordinates(a*b.i).i abs norm a == b := basis() m := new(#b,#b, 0)$Matrix(F) for i in 1..#b repeat setRow_!(m,i, coordinates(a*b.i)) determinant(m) if F has Finite then linearAssociatedExp(x,f) == erg:$:=0 y:=x for i in 0..degree(f) repeat erg:=erg + coefficient(f,i) * y y:=Frobenius(y) erg linearAssociatedLog(b,x) == x=0 => 0 l:List List F:=[entries coordinates b] a:$:=b extdeg:NNI:=extensionDegree()@PI for i in 2..extdeg repeat a:=Frobenius(a) l:=concat(l,entries coordinates a)$(List List F) l:=concat(l,entries coordinates x)$(List List F) m1:=rowEchelon transpose matrix(l)$(Matrix F) v:=zero(extdeg)$(Vector F) rown:I:=1 for i in 1..extdeg repeat if qelt(m1,rown,i) = 1$F then v.i:=qelt(m1,rown,extdeg+1) rown:=rown+1 p:=+/[monomial(v.(i+1),i::NNI) for i in 0..(#v-1)] p=0 => messagePrint("linearAssociatedLog: second argument not in_ group generated by first argument")$OutputForm "failed" p linearAssociatedLog(x) == linearAssociatedLog(normalElement(),x) :: SparseUnivariatePolynomial(F) linearAssociatedOrder(x) == x=0 => 0 l:List List F:=[entries coordinates x] a:$:=x for i in 1..extensionDegree()@PI repeat a:=Frobenius(a) l:=concat(l,entries coordinates a)$(List List F) v:=first nullSpace transpose matrix(l)$(Matrix F) +/[monomial(v.(i+1),i::NNI) for i in 0..(#v-1)] charthRoot(x):Union($,"failed") == (charthRoot(x)@$)::Union($,"failed") -- norm(e) == norm(e,1) pretend F -- trace(e) == trace(e,1) pretend F minimalPolynomial(a,n) == extensionDegree()@PI rem n ^= 0 => error "minimalPolynomial: 2. argument must divide extension degree" f:SUP $:=monomial(1,1)$(SUP $) - monomial(a,0)$(SUP $) u:$:=Frobenius(a,n) while not(u = a) repeat f:=f * (monomial(1,1)$(SUP $) - monomial(u,0)$(SUP $)) u:=Frobenius(u,n) f norm(e,s) == qr := divide(extensionDegree(), s) zero?(qr.remainder) => pow := (size()-1) quo (size()$F ** s - 1) e ** (pow::NonNegativeInteger) error "norm: second argument must divide degree of extension" trace(e,s) == qr:=divide(extensionDegree(),s) q:=size()$F zero?(qr.remainder) => a:$:=0 for i in 0..qr.quotient-1 repeat a:=a + e**(q**(s*i)) a error "trace: second argument must divide degree of extension" size() == size()$F ** extensionDegree() createNormalElement() == characteristic() = size() => 1 res : $ for i in 1.. repeat res := index(i :: PI) not inGroundField? res => normal? res => return res -- theorem: there exists a normal element, this theorem is -- unknown to the compiler res normal?(x:$) == p:SUP $:=(monomial(1,extensionDegree()) - monomial(1,0))@(SUP $) f:SUP $:= +/[monomial(Frobenius(x,i),i)$(SUP $) _ for i in 0..extensionDegree()-1] gcd(p,f) = 1 => true false degree a == y:$:=Frobenius a deg:PI:=1 while y^=a repeat y := Frobenius(y) deg:=deg+1 deg @ \section{package DLP DiscreteLogarithmPackage} <>= )abbrev package DLP DiscreteLogarithmPackage ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 12 March 1991 ++ Date Last Updated: 31 March 1991 ++ Basic Operations: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: discrete logarithm ++ References: ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. ++ Description: ++ DiscreteLogarithmPackage implements help functions for discrete logarithms ++ in monoids using small cyclic groups. DiscreteLogarithmPackage(M): public == private where M : Join(Monoid,Finite) with "**": (M,Integer) -> M ++ x ** n returns x raised to the integer power n public ==> with shanksDiscLogAlgorithm:(M,M,NonNegativeInteger)-> _ Union(NonNegativeInteger,"failed") ++ shanksDiscLogAlgorithm(b,a,p) computes s with \spad{b**s = a} for ++ assuming that \spad{a} and b are elements in a 'small' cyclic group of ++ order p by Shank's algorithm. ++ Note: this is a subroutine of the function \spadfun{discreteLog}. I ==> Integer PI ==> PositiveInteger NNI ==> NonNegativeInteger SUP ==> SparseUnivariatePolynomial DLP ==> DiscreteLogarithmPackage private ==> add shanksDiscLogAlgorithm(logbase,c,p) == limit:Integer:= 30 -- for logarithms up to cyclic groups of order limit a full -- logarithm table is computed p < limit => a:M:=1 disclog:Integer:=0 found:Boolean:=false for i in 0..p-1 while not found repeat a = c => disclog:=i found:=true a:=a*logbase not found => messagePrint("discreteLog: second argument not in cyclic group_ generated by first argument")$OutputForm "failed" disclog pretend NonNegativeInteger l:Integer:=length(p)$Integer if odd?(l)$Integer then n:Integer:= shift(p,-(l quo 2)) else n:Integer:= shift(1,(l quo 2)) a:M:=1 exptable : Table(PI,NNI) :=table()$Table(PI,NNI) for i in (0::NNI)..(n-1)::NNI repeat insert_!([lookup(a),i::NNI]$Record(key:PI,entry:NNI),_ exptable)$Table(PI,NNI) a:=a*logbase found := false end := (p-1) quo n disclog:Integer:=0 a := c b := logbase ** (-n) for i in 0..end while not found repeat rho:= search(lookup(a),exptable)_ $Table(PositiveInteger,NNI) rho case NNI => found := true disclog:= n * i + rho pretend Integer a := a * b not found => messagePrint("discreteLog: second argument not in cyclic group_ generated by first argument")$OutputForm "failed" disclog pretend NonNegativeInteger @ \section{category FFIELDC FiniteFieldCategory} <>= )abbrev category FFIELDC FiniteFieldCategory ++ Author: J. Grabmeier, A. Scheerhorn ++ Date Created: 11 March 1991 ++ Date Last Updated: 31 March 1991 ++ Basic Operations: _+, _*, extensionDegree, order, primitiveElement ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: field, extension field, algebraic extension, finite field ++ Galois field ++ References: ++ D.Lipson, Elements of Algebra and Algebraic Computing, The ++ Benjamin/Cummings Publishing Company, Inc.-Menlo Park, California, 1981. ++ J. Grabmeier, A. Scheerhorn: Finite Fields in AXIOM. ++ AXIOM Technical Report Series, ATR/5 NP2522. ++ Description: ++ FiniteFieldCategory is the category of finite fields FiniteFieldCategory() : Category ==_ Join(FieldOfPrimeCharacteristic,Finite,StepThrough,DifferentialRing) with -- ,PolynomialFactorizationExplicit) with charthRoot: $ -> $ ++ charthRoot(a) takes the characteristic'th root of {\em a}. ++ Note: such a root is alway defined in finite fields. conditionP: Matrix $ -> Union(Vector $,"failed") ++ conditionP(mat), given a matrix representing a homogeneous system ++ of equations, returns a vector whose characteristic'th powers ++ is a non-trivial solution, or "failed" if no such vector exists. -- the reason for implementing the following function is that we -- can implement the functions order, getGenerator and primitive? on -- category level without computing the, may be time intensive, -- factorization of size()-1 at every function call again. factorsOfCyclicGroupSize:_ () -> List Record(factor:Integer,exponent:Integer) ++ factorsOfCyclicGroupSize() returns the factorization of size()-1 -- the reason for implementing the function tableForDiscreteLogarithm -- is that we can implement the functions discreteLog and -- shanksDiscLogAlgorithm on category level -- computing the necessary exponentiation tables in the respective -- domains once and for all -- absoluteDegree : $ -> PositiveInteger -- ++ degree of minimal polynomial, if algebraic with respect -- ++ to the prime subfield tableForDiscreteLogarithm: Integer -> _ Table(PositiveInteger,NonNegativeInteger) ++ tableForDiscreteLogarithm(a,n) returns a table of the discrete ++ logarithms of \spad{a**0} up to \spad{a**(n-1)} which, called with ++ key \spad{lookup(a**i)} returns i for i in \spad{0..n-1}. ++ Error: if not called for prime divisors of order of ++ multiplicative group. createPrimitiveElement: () -> $ ++ createPrimitiveElement() computes a generator of the (cyclic) ++ multiplicative group of the field. -- RDJ: Are these next lines to be included? -- we run through the field and test, algorithms which construct -- elements of larger order were found to be too slow primitiveElement: () -> $ ++ primitiveElement() returns a primitive element stored in a global ++ variable in the domain. ++ At first call, the primitive element is computed ++ by calling \spadfun{createPrimitiveElement}. primitive?: $ -> Boolean ++ primitive?(b) tests whether the element b is a generator of the ++ (cyclic) multiplicative group of the field, i.e. is a primitive ++ element. ++ Implementation Note: see ch.IX.1.3, th.2 in D. Lipson. discreteLog: $ -> NonNegativeInteger ++ discreteLog(a) computes the discrete logarithm of \spad{a} ++ with respect to \spad{primitiveElement()} of the field. order: $ -> PositiveInteger ++ order(b) computes the order of an element b in the multiplicative ++ group of the field. ++ Error: if b equals 0. representationType: () -> Union("prime","polynomial","normal","cyclic") ++ representationType() returns the type of the representation, one of: ++ \spad{prime}, \spad{polynomial}, \spad{normal}, or \spad{cyclic}. add I ==> Integer PI ==> PositiveInteger NNI ==> NonNegativeInteger SUP ==> SparseUnivariatePolynomial DLP ==> DiscreteLogarithmPackage -- exported functions differentiate x == 0 init() == 0 nextItem(a) == zero?(a:=index(lookup(a)+1)) => "failed" a order(e):OnePointCompletion(PositiveInteger) == (order(e)@PI)::OnePointCompletion(PositiveInteger) conditionP(mat:Matrix $) == l:=nullSpace mat empty? l or every?(zero?, first l) => "failed" map(charthRoot,first l) charthRoot(x:$):$ == x**(size() quo characteristic()) charthRoot(x:%):Union($,"failed") == (charthRoot(x)@$)::Union($,"failed") createPrimitiveElement() == sm1 : PositiveInteger := (size()$$-1) pretend PositiveInteger start : Integer := -- in the polynomial case, index from 1 to characteristic-1 -- gives prime field elements representationType = "polynomial" => characteristic()::Integer 1 found : Boolean := false for i in start.. while not found repeat e : $ := index(i::PositiveInteger) found := (order(e) = sm1) e primitive? a == -- add special implementation for prime field case zero?(a) => false explist := factorsOfCyclicGroupSize() q:=(size()-1)@Integer equalone : Boolean := false for exp in explist while not equalone repeat -- equalone := one?(a**(q quo exp.factor)) equalone := ((a**(q quo exp.factor)) = 1) not equalone order e == e = 0 => error "order(0) is not defined " ord:Integer:= size()-1 -- order e divides ord a:Integer:= 0 lof:=factorsOfCyclicGroupSize() for rec in lof repeat -- run through prime divisors a := ord quo (primeDivisor := rec.factor) -- goon := one?(e**a) goon := ((e**a) = 1) -- run through exponents of the prime divisors for j in 0..(rec.exponent)-2 while goon repeat -- as long as we get (e**ord = 1) we -- continue dividing by primeDivisor ord := a a := ord quo primeDivisor -- goon := one?(e**a) goon := ((e**a) = 1) if goon then ord := a -- as we do a top down search we have found the -- correct exponent of primeDivisor in order e -- and continue with next prime divisor ord pretend PositiveInteger discreteLog(b) == zero?(b) => error "discreteLog: logarithm of zero" faclist:=factorsOfCyclicGroupSize() a:=b gen:=primitiveElement() -- in GF(2) its necessary to have discreteLog(1) = 1 b = gen => 1 disclog:Integer:=0 mult:Integer:=1 groupord := (size() - 1)@Integer exp:Integer:=groupord for f in faclist repeat fac:=f.factor for t in 0..f.exponent-1 repeat exp:=exp quo fac -- shanks discrete logarithm algorithm exptable:=tableForDiscreteLogarithm(fac) n:=#exptable c:=a**exp end:=(fac - 1) quo n found:=false disc1:Integer:=0 for i in 0..end while not found repeat rho:= search(lookup(c),exptable)_ $Table(PositiveInteger,NNI) rho case NNI => found := true disc1:=((n * i + rho)@Integer) * mult c:=c* gen**((groupord quo fac) * (-n)) not found => error "discreteLog: ?? discrete logarithm" -- end of shanks discrete logarithm algorithm mult := mult * fac disclog:=disclog+disc1 a:=a * (gen ** (-disc1)) disclog pretend NonNegativeInteger discreteLog(logbase,b) == zero?(b) => messagePrint("discreteLog: logarithm of zero")$OutputForm "failed" zero?(logbase) => messagePrint("discreteLog: logarithm to base zero")$OutputForm "failed" b = logbase => 1 not zero?((groupord:=order(logbase)@PI) rem order(b)@PI) => messagePrint("discreteLog: second argument not in cyclic group _ generated by first argument")$OutputForm "failed" faclist:=factors factor groupord a:=b disclog:Integer:=0 mult:Integer:=1 exp:Integer:= groupord for f in faclist repeat fac:=f.factor primroot:= logbase ** (groupord quo fac) for t in 0..f.exponent-1 repeat exp:=exp quo fac rhoHelp:= shanksDiscLogAlgorithm(primroot,_ a**exp,fac pretend NonNegativeInteger)$DLP($) rhoHelp case "failed" => return "failed" rho := (rhoHelp :: NNI) * mult disclog := disclog + rho mult := mult * fac a:=a * (logbase ** (-rho)) disclog pretend NonNegativeInteger FP ==> SparseUnivariatePolynomial($) FRP ==> Factored FP f,g:FP squareFreePolynomial(f:FP):FRP == squareFree(f)$UnivariatePolynomialSquareFree($,FP) factorPolynomial(f:FP):FRP == factor(f)$DistinctDegreeFactorize($,FP) factorSquareFreePolynomial(f:FP):FRP == f = 0 => 0 flist := distdfact(f,true)$DistinctDegreeFactorize($,FP) (flist.cont :: FP) * (*/[primeFactor(u.irr,u.pow) for u in flist.factors]) gcdPolynomial(f:FP,g:FP):FP == gcd(f,g)$EuclideanDomain_&(FP) @ \section{FFIELDC.lsp BOOTSTRAP} {\bf FFIELDC} depends on a chain of files. We need to break this cycle to build the algebra. So we keep a cached copy of the translated {\bf FFIELDC} category which we can write into the {\bf MID} directory. We compile the lisp code and copy the {\bf FFIELDC.o} file to the {\bf OUT} directory. This is eventually forcibly replaced by a recompiled version. Note that this code is not included in the generated catdef.spad file. <>= (|/VERSIONCHECK| 2) (SETQ |FiniteFieldCategory;AL| (QUOTE NIL)) (DEFUN |FiniteFieldCategory| NIL (LET (#:G83129) (COND (|FiniteFieldCategory;AL|) (T (SETQ |FiniteFieldCategory;AL| (|FiniteFieldCategory;|)))))) (DEFUN |FiniteFieldCategory;| NIL (PROG (#1=#:G83127) (RETURN (PROG1 (LETT #1# (|Join| (|FieldOfPrimeCharacteristic|) (|Finite|) (|StepThrough|) (|DifferentialRing|) (|mkCategory| (QUOTE |domain|) (QUOTE (((|charthRoot| (|$| |$|)) T) ((|conditionP| ((|Union| (|Vector| |$|) "failed") (|Matrix| |$|))) T) ((|factorsOfCyclicGroupSize| ((|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|)))))) T) ((|tableForDiscreteLogarithm| ((|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|))) T) ((|createPrimitiveElement| (|$|)) T) ((|primitiveElement| (|$|)) T) ((|primitive?| ((|Boolean|) |$|)) T) ((|discreteLog| ((|NonNegativeInteger|) |$|)) T) ((|order| ((|PositiveInteger|) |$|)) T) ((|representationType| ((|Union| "prime" "polynomial" "normal" "cyclic"))) T))) NIL (QUOTE ((|PositiveInteger|) (|NonNegativeInteger|) (|Boolean|) (|Table| (|PositiveInteger|) (|NonNegativeInteger|)) (|Integer|) (|List| (|Record| (|:| |factor| (|Integer|)) (|:| |exponent| (|Integer|)))) (|Matrix| |$|))) NIL)) |FiniteFieldCategory|) (SETELT #1# 0 (QUOTE (|FiniteFieldCategory|))))))) (MAKEPROP (QUOTE |FiniteFieldCategory|) (QUOTE NILADIC) T) @ \section{FFIELDC-.lsp BOOTSTRAP} {\bf FFIELDC-} depends on {\bf FFIELDC}. We need to break this cycle to build the algebra. So we keep a cached copy of the translated {\bf FFIELDC-} category which we can write into the {\bf MID} directory. We compile the lisp code and copy the {\bf FFIELDC-.o} file to the {\bf OUT} directory. This is eventually forcibly replaced by a recompiled version. Note that this code is not included in the generated catdef.spad file. <>= (|/VERSIONCHECK| 2) (DEFUN |FFIELDC-;differentiate;2S;1| (|x| |$|) (|spadConstant| |$| 7)) (DEFUN |FFIELDC-;init;S;2| (|$|) (|spadConstant| |$| 7)) (DEFUN |FFIELDC-;nextItem;SU;3| (|a| |$|) (COND ((SPADCALL (LETT |a| (SPADCALL (|+| (SPADCALL |a| (QREFELT |$| 11)) 1) (QREFELT |$| 12)) |FFIELDC-;nextItem;SU;3|) (QREFELT |$| 14)) (CONS 1 "failed")) ((QUOTE T) (CONS 0 |a|)))) (DEFUN |FFIELDC-;order;SOpc;4| (|e| |$|) (SPADCALL (SPADCALL |e| (QREFELT |$| 17)) (QREFELT |$| 20))) (DEFUN |FFIELDC-;conditionP;MU;5| (|mat| |$|) (PROG (|l|) (RETURN (SEQ (LETT |l| (SPADCALL |mat| (QREFELT |$| 24)) |FFIELDC-;conditionP;MU;5|) (COND ((OR (NULL |l|) (SPADCALL (ELT |$| 14) (|SPADfirst| |l|) (QREFELT |$| 27))) (EXIT (CONS 1 "failed")))) (EXIT (CONS 0 (SPADCALL (ELT |$| 28) (|SPADfirst| |l|) (QREFELT |$| 30)))))))) (DEFUN |FFIELDC-;charthRoot;2S;6| (|x| |$|) (SPADCALL |x| (QUOTIENT2 (SPADCALL (QREFELT |$| 35)) (SPADCALL (QREFELT |$| 36))) (QREFELT |$| 37))) (DEFUN |FFIELDC-;charthRoot;SU;7| (|x| |$|) (CONS 0 (SPADCALL |x| (QREFELT |$| 28)))) (DEFUN |FFIELDC-;createPrimitiveElement;S;8| (|$|) (PROG (|sm1| |start| |i| #1=#:G83175 |e| |found|) (RETURN (SEQ (LETT |sm1| (|-| (SPADCALL (QREFELT |$| 35)) 1) |FFIELDC-;createPrimitiveElement;S;8|) (LETT |start| (COND ((SPADCALL (SPADCALL (QREFELT |$| 42)) (CONS 1 "polynomial") (QREFELT |$| 43)) (SPADCALL (QREFELT |$| 36))) ((QUOTE T) 1)) |FFIELDC-;createPrimitiveElement;S;8|) (LETT |found| (QUOTE NIL) |FFIELDC-;createPrimitiveElement;S;8|) (SEQ (LETT |i| |start| |FFIELDC-;createPrimitiveElement;S;8|) G190 (COND ((NULL (COND (|found| (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |e| (SPADCALL (PROG1 (LETT #1# |i| |FFIELDC-;createPrimitiveElement;S;8|) (|check-subtype| (|>| #1# 0) (QUOTE (|PositiveInteger|)) #1#)) (QREFELT |$| 12)) |FFIELDC-;createPrimitiveElement;S;8|) (EXIT (LETT |found| (EQL (SPADCALL |e| (QREFELT |$| 17)) |sm1|) |FFIELDC-;createPrimitiveElement;S;8|))) (LETT |i| (|+| |i| 1) |FFIELDC-;createPrimitiveElement;S;8|) (GO G190) G191 (EXIT NIL)) (EXIT |e|))))) (DEFUN |FFIELDC-;primitive?;SB;9| (|a| |$|) (PROG (|explist| |q| |exp| #1=#:G83187 |equalone|) (RETURN (SEQ (COND ((SPADCALL |a| (QREFELT |$| 14)) (QUOTE NIL)) ((QUOTE T) (SEQ (LETT |explist| (SPADCALL (QREFELT |$| 47)) |FFIELDC-;primitive?;SB;9|) (LETT |q| (|-| (SPADCALL (QREFELT |$| 35)) 1) |FFIELDC-;primitive?;SB;9|) (LETT |equalone| (QUOTE NIL) |FFIELDC-;primitive?;SB;9|) (SEQ (LETT |exp| NIL |FFIELDC-;primitive?;SB;9|) (LETT #1# |explist| |FFIELDC-;primitive?;SB;9|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |exp| (CAR #1#) |FFIELDC-;primitive?;SB;9|) NIL) (NULL (COND (|equalone| (QUOTE NIL)) ((QUOTE T) (QUOTE T))))) (GO G191))) (SEQ (EXIT (LETT |equalone| (SPADCALL (SPADCALL |a| (QUOTIENT2 |q| (QCAR |exp|)) (QREFELT |$| 48)) (QREFELT |$| 49)) |FFIELDC-;primitive?;SB;9|))) (LETT #1# (CDR #1#) |FFIELDC-;primitive?;SB;9|) (GO G190) G191 (EXIT NIL)) (EXIT (COND (|equalone| (QUOTE NIL)) ((QUOTE T) (QUOTE T))))))))))) (DEFUN |FFIELDC-;order;SPi;10| (|e| |$|) (PROG (|lof| |rec| #1=#:G83195 |primeDivisor| |j| #2=#:G83196 |a| |goon| |ord|) (RETURN (SEQ (COND ((SPADCALL |e| (|spadConstant| |$| 7) (QREFELT |$| 51)) (|error| "order(0) is not defined ")) ((QUOTE T) (SEQ (LETT |ord| (|-| (SPADCALL (QREFELT |$| 35)) 1) |FFIELDC-;order;SPi;10|) (LETT |a| 0 |FFIELDC-;order;SPi;10|) (LETT |lof| (SPADCALL (QREFELT |$| 47)) |FFIELDC-;order;SPi;10|) (SEQ (LETT |rec| NIL |FFIELDC-;order;SPi;10|) (LETT #1# |lof| |FFIELDC-;order;SPi;10|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |rec| (CAR #1#) |FFIELDC-;order;SPi;10|) NIL)) (GO G191))) (SEQ (LETT |a| (QUOTIENT2 |ord| (LETT |primeDivisor| (QCAR |rec|) |FFIELDC-;order;SPi;10|)) |FFIELDC-;order;SPi;10|) (LETT |goon| (SPADCALL (SPADCALL |e| |a| (QREFELT |$| 48)) (QREFELT |$| 49)) |FFIELDC-;order;SPi;10|) (SEQ (LETT |j| 0 |FFIELDC-;order;SPi;10|) (LETT #2# (|-| (QCDR |rec|) 2) |FFIELDC-;order;SPi;10|) G190 (COND ((OR (QSGREATERP |j| #2#) (NULL |goon|)) (GO G191))) (SEQ (LETT |ord| |a| |FFIELDC-;order;SPi;10|) (LETT |a| (QUOTIENT2 |ord| |primeDivisor|) |FFIELDC-;order;SPi;10|) (EXIT (LETT |goon| (SPADCALL (SPADCALL |e| |a| (QREFELT |$| 48)) (QREFELT |$| 49)) |FFIELDC-;order;SPi;10|))) (LETT |j| (QSADD1 |j|) |FFIELDC-;order;SPi;10|) (GO G190) G191 (EXIT NIL)) (EXIT (COND (|goon| (LETT |ord| |a| |FFIELDC-;order;SPi;10|))))) (LETT #1# (CDR #1#) |FFIELDC-;order;SPi;10|) (GO G190) G191 (EXIT NIL)) (EXIT |ord|)))))))) (DEFUN |FFIELDC-;discreteLog;SNni;11| (|b| |$|) (PROG (|faclist| |gen| |groupord| |f| #1=#:G83216 |fac| |t| #2=#:G83217 |exp| |exptable| |n| |end| |i| |rho| |found| |disc1| |c| |mult| |disclog| |a|) (RETURN (SEQ (COND ((SPADCALL |b| (QREFELT |$| 14)) (|error| "discreteLog: logarithm of zero")) ((QUOTE T) (SEQ (LETT |faclist| (SPADCALL (QREFELT |$| 47)) |FFIELDC-;discreteLog;SNni;11|) (LETT |a| |b| |FFIELDC-;discreteLog;SNni;11|) (LETT |gen| (SPADCALL (QREFELT |$| 53)) |FFIELDC-;discreteLog;SNni;11|) (EXIT (COND ((SPADCALL |b| |gen| (QREFELT |$| 51)) 1) ((QUOTE T) (SEQ (LETT |disclog| 0 |FFIELDC-;discreteLog;SNni;11|) (LETT |mult| 1 |FFIELDC-;discreteLog;SNni;11|) (LETT |groupord| (|-| (SPADCALL (QREFELT |$| 35)) 1) |FFIELDC-;discreteLog;SNni;11|) (LETT |exp| |groupord| |FFIELDC-;discreteLog;SNni;11|) (SEQ (LETT |f| NIL |FFIELDC-;discreteLog;SNni;11|) (LETT #1# |faclist| |FFIELDC-;discreteLog;SNni;11|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |f| (CAR #1#) |FFIELDC-;discreteLog;SNni;11|) NIL)) (GO G191))) (SEQ (LETT |fac| (QCAR |f|) |FFIELDC-;discreteLog;SNni;11|) (EXIT (SEQ (LETT |t| 0 |FFIELDC-;discreteLog;SNni;11|) (LETT #2# (|-| (QCDR |f|) 1) |FFIELDC-;discreteLog;SNni;11|) G190 (COND ((QSGREATERP |t| #2#) (GO G191))) (SEQ (LETT |exp| (QUOTIENT2 |exp| |fac|) |FFIELDC-;discreteLog;SNni;11|) (LETT |exptable| (SPADCALL |fac| (QREFELT |$| 55)) |FFIELDC-;discreteLog;SNni;11|) (LETT |n| (SPADCALL |exptable| (QREFELT |$| 56)) |FFIELDC-;discreteLog;SNni;11|) (LETT |c| (SPADCALL |a| |exp| (QREFELT |$| 48)) |FFIELDC-;discreteLog;SNni;11|) (LETT |end| (QUOTIENT2 (|-| |fac| 1) |n|) |FFIELDC-;discreteLog;SNni;11|) (LETT |found| (QUOTE NIL) |FFIELDC-;discreteLog;SNni;11|) (LETT |disc1| 0 |FFIELDC-;discreteLog;SNni;11|) (SEQ (LETT |i| 0 |FFIELDC-;discreteLog;SNni;11|) G190 (COND ((OR (QSGREATERP |i| |end|) (NULL (COND (|found| (QUOTE NIL)) ((QUOTE T) (QUOTE T))))) (GO G191))) (SEQ (LETT |rho| (SPADCALL (SPADCALL |c| (QREFELT |$| 11)) |exptable| (QREFELT |$| 58)) |FFIELDC-;discreteLog;SNni;11|) (EXIT (COND ((QEQCAR |rho| 0) (SEQ (LETT |found| (QUOTE T) |FFIELDC-;discreteLog;SNni;11|) (EXIT (LETT |disc1| (|*| (|+| (|*| |n| |i|) (QCDR |rho|)) |mult|) |FFIELDC-;discreteLog;SNni;11|)))) ((QUOTE T) (LETT |c| (SPADCALL |c| (SPADCALL |gen| (|*| (QUOTIENT2 |groupord| |fac|) (|-| |n|)) (QREFELT |$| 48)) (QREFELT |$| 59)) |FFIELDC-;discreteLog;SNni;11|))))) (LETT |i| (QSADD1 |i|) |FFIELDC-;discreteLog;SNni;11|) (GO G190) G191 (EXIT NIL)) (EXIT (COND (|found| (SEQ (LETT |mult| (|*| |mult| |fac|) |FFIELDC-;discreteLog;SNni;11|) (LETT |disclog| (|+| |disclog| |disc1|) |FFIELDC-;discreteLog;SNni;11|) (EXIT (LETT |a| (SPADCALL |a| (SPADCALL |gen| (|-| |disc1|) (QREFELT |$| 48)) (QREFELT |$| 59)) |FFIELDC-;discreteLog;SNni;11|)))) ((QUOTE T) (|error| "discreteLog: ?? discrete logarithm"))))) (LETT |t| (QSADD1 |t|) |FFIELDC-;discreteLog;SNni;11|) (GO G190) G191 (EXIT NIL)))) (LETT #1# (CDR #1#) |FFIELDC-;discreteLog;SNni;11|) (GO G190) G191 (EXIT NIL)) (EXIT |disclog|)))))))))))) (DEFUN |FFIELDC-;discreteLog;2SU;12| (|logbase| |b| |$|) (PROG (|groupord| |faclist| |f| #1=#:G83235 |fac| |primroot| |t| #2=#:G83236 |exp| |rhoHelp| #3=#:G83234 |rho| |disclog| |mult| |a|) (RETURN (SEQ (EXIT (COND ((SPADCALL |b| (QREFELT |$| 14)) (SEQ (SPADCALL "discreteLog: logarithm of zero" (QREFELT |$| 64)) (EXIT (CONS 1 "failed")))) ((SPADCALL |logbase| (QREFELT |$| 14)) (SEQ (SPADCALL "discreteLog: logarithm to base zero" (QREFELT |$| 64)) (EXIT (CONS 1 "failed")))) ((SPADCALL |b| |logbase| (QREFELT |$| 51)) (CONS 0 1)) ((QUOTE T) (COND ((NULL (ZEROP (REMAINDER2 (LETT |groupord| (SPADCALL |logbase| (QREFELT |$| 17)) |FFIELDC-;discreteLog;2SU;12|) (SPADCALL |b| (QREFELT |$| 17))))) (SEQ (SPADCALL "discreteLog: second argument not in cyclic group generated by first argument" (QREFELT |$| 64)) (EXIT (CONS 1 "failed")))) ((QUOTE T) (SEQ (LETT |faclist| (SPADCALL (SPADCALL |groupord| (QREFELT |$| 66)) (QREFELT |$| 68)) |FFIELDC-;discreteLog;2SU;12|) (LETT |a| |b| |FFIELDC-;discreteLog;2SU;12|) (LETT |disclog| 0 |FFIELDC-;discreteLog;2SU;12|) (LETT |mult| 1 |FFIELDC-;discreteLog;2SU;12|) (LETT |exp| |groupord| |FFIELDC-;discreteLog;2SU;12|) (SEQ (LETT |f| NIL |FFIELDC-;discreteLog;2SU;12|) (LETT #1# |faclist| |FFIELDC-;discreteLog;2SU;12|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |f| (CAR #1#) |FFIELDC-;discreteLog;2SU;12|) NIL)) (GO G191))) (SEQ (LETT |fac| (QCAR |f|) |FFIELDC-;discreteLog;2SU;12|) (LETT |primroot| (SPADCALL |logbase| (QUOTIENT2 |groupord| |fac|) (QREFELT |$| 48)) |FFIELDC-;discreteLog;2SU;12|) (EXIT (SEQ (LETT |t| 0 |FFIELDC-;discreteLog;2SU;12|) (LETT #2# (|-| (QCDR |f|) 1) |FFIELDC-;discreteLog;2SU;12|) G190 (COND ((QSGREATERP |t| #2#) (GO G191))) (SEQ (LETT |exp| (QUOTIENT2 |exp| |fac|) |FFIELDC-;discreteLog;2SU;12|) (LETT |rhoHelp| (SPADCALL |primroot| (SPADCALL |a| |exp| (QREFELT |$| 48)) |fac| (QREFELT |$| 70)) |FFIELDC-;discreteLog;2SU;12|) (EXIT (COND ((QEQCAR |rhoHelp| 1) (PROGN (LETT #3# (CONS 1 "failed") |FFIELDC-;discreteLog;2SU;12|) (GO #3#))) ((QUOTE T) (SEQ (LETT |rho| (|*| (QCDR |rhoHelp|) |mult|) |FFIELDC-;discreteLog;2SU;12|) (LETT |disclog| (|+| |disclog| |rho|) |FFIELDC-;discreteLog;2SU;12|) (LETT |mult| (|*| |mult| |fac|) |FFIELDC-;discreteLog;2SU;12|) (EXIT (LETT |a| (SPADCALL |a| (SPADCALL |logbase| (|-| |rho|) (QREFELT |$| 48)) (QREFELT |$| 59)) |FFIELDC-;discreteLog;2SU;12|))))))) (LETT |t| (QSADD1 |t|) |FFIELDC-;discreteLog;2SU;12|) (GO G190) G191 (EXIT NIL)))) (LETT #1# (CDR #1#) |FFIELDC-;discreteLog;2SU;12|) (GO G190) G191 (EXIT NIL)) (EXIT (CONS 0 |disclog|)))))))) #3# (EXIT #3#))))) (DEFUN |FFIELDC-;squareFreePolynomial| (|f| |$|) (SPADCALL |f| (QREFELT |$| 75))) (DEFUN |FFIELDC-;factorPolynomial| (|f| |$|) (SPADCALL |f| (QREFELT |$| 77))) (DEFUN |FFIELDC-;factorSquareFreePolynomial| (|f| |$|) (PROG (|flist| |u| #1=#:G83248 #2=#:G83245 #3=#:G83243 #4=#:G83244) (RETURN (SEQ (COND ((SPADCALL |f| (|spadConstant| |$| 78) (QREFELT |$| 79)) (|spadConstant| |$| 80)) ((QUOTE T) (SEQ (LETT |flist| (SPADCALL |f| (QUOTE T) (QREFELT |$| 83)) |FFIELDC-;factorSquareFreePolynomial|) (EXIT (SPADCALL (SPADCALL (QCAR |flist|) (QREFELT |$| 84)) (PROGN (LETT #4# NIL |FFIELDC-;factorSquareFreePolynomial|) (SEQ (LETT |u| NIL |FFIELDC-;factorSquareFreePolynomial|) (LETT #1# (QCDR |flist|) |FFIELDC-;factorSquareFreePolynomial|) G190 (COND ((OR (ATOM #1#) (PROGN (LETT |u| (CAR #1#) |FFIELDC-;factorSquareFreePolynomial|) NIL)) (GO G191))) (SEQ (EXIT (PROGN (LETT #2# (SPADCALL (QCAR |u|) (QCDR |u|) (QREFELT |$| 85)) |FFIELDC-;factorSquareFreePolynomial|) (COND (#4# (LETT #3# (SPADCALL #3# #2# (QREFELT |$| 86)) |FFIELDC-;factorSquareFreePolynomial|)) ((QUOTE T) (PROGN (LETT #3# #2# |FFIELDC-;factorSquareFreePolynomial|) (LETT #4# (QUOTE T) |FFIELDC-;factorSquareFreePolynomial|))))))) (LETT #1# (CDR #1#) |FFIELDC-;factorSquareFreePolynomial|) (GO G190) G191 (EXIT NIL)) (COND (#4# #3#) ((QUOTE T) (|spadConstant| |$| 87)))) (QREFELT |$| 88)))))))))) (DEFUN |FFIELDC-;gcdPolynomial;3Sup;16| (|f| |g| |$|) (SPADCALL |f| |g| (QREFELT |$| 90))) (DEFUN |FiniteFieldCategory&| (|#1|) (PROG (|DV$1| |dv$| |$| |pv$|) (RETURN (PROGN (LETT |DV$1| (|devaluate| |#1|) . #1=(|FiniteFieldCategory&|)) (LETT |dv$| (LIST (QUOTE |FiniteFieldCategory&|) |DV$1|) . #1#) (LETT |$| (GETREFV 93) . #1#) (QSETREFV |$| 0 |dv$|) (QSETREFV |$| 3 (LETT |pv$| (|buildPredVector| 0 0 NIL) . #1#)) (|stuffDomainSlots| |$|) (QSETREFV |$| 6 |#1|) |$|)))) (MAKEPROP (QUOTE |FiniteFieldCategory&|) (QUOTE |infovec|) (LIST (QUOTE #(NIL NIL NIL NIL NIL NIL (|local| |#1|) (0 . |Zero|) |FFIELDC-;differentiate;2S;1| |FFIELDC-;init;S;2| (|PositiveInteger|) (4 . |lookup|) (9 . |index|) (|Boolean|) (14 . |zero?|) (|Union| |$| (QUOTE "failed")) |FFIELDC-;nextItem;SU;3| (19 . |order|) (|Integer|) (|OnePointCompletion| 10) (24 . |coerce|) |FFIELDC-;order;SOpc;4| (|List| 26) (|Matrix| 6) (29 . |nullSpace|) (|Mapping| 13 6) (|Vector| 6) (34 . |every?|) (40 . |charthRoot|) (|Mapping| 6 6) (45 . |map|) (|Union| (|Vector| |$|) (QUOTE "failed")) (|Matrix| |$|) |FFIELDC-;conditionP;MU;5| (|NonNegativeInteger|) (51 . |size|) (55 . |characteristic|) (59 . |**|) |FFIELDC-;charthRoot;2S;6| |FFIELDC-;charthRoot;SU;7| (65 . |One|) (|Union| (QUOTE "prime") (QUOTE "polynomial") (QUOTE "normal") (QUOTE "cyclic")) (69 . |representationType|) (73 . |=|) |FFIELDC-;createPrimitiveElement;S;8| (|Record| (|:| |factor| 18) (|:| |exponent| 18)) (|List| 45) (79 . |factorsOfCyclicGroupSize|) (83 . |**|) (89 . |one?|) |FFIELDC-;primitive?;SB;9| (94 . |=|) |FFIELDC-;order;SPi;10| (100 . |primitiveElement|) (|Table| 10 34) (104 . |tableForDiscreteLogarithm|) (109 . |#|) (|Union| 34 (QUOTE "failed")) (114 . |search|) (120 . |*|) |FFIELDC-;discreteLog;SNni;11| (|Void|) (|String|) (|OutputForm|) (126 . |messagePrint|) (|Factored| |$|) (131 . |factor|) (|Factored| 18) (136 . |factors|) (|DiscreteLogarithmPackage| 6) (141 . |shanksDiscLogAlgorithm|) |FFIELDC-;discreteLog;2SU;12| (|Factored| 73) (|SparseUnivariatePolynomial| 6) (|UnivariatePolynomialSquareFree| 6 73) (148 . |squareFree|) (|DistinctDegreeFactorize| 6 73) (153 . |factor|) (158 . |Zero|) (162 . |=|) (168 . |Zero|) (|Record| (|:| |irr| 73) (|:| |pow| 18)) (|Record| (|:| |cont| 6) (|:| |factors| (|List| 81))) (172 . |distdfact|) (178 . |coerce|) (183 . |primeFactor|) (189 . |*|) (195 . |One|) (199 . |*|) (|EuclideanDomain&| 73) (205 . |gcd|) (|SparseUnivariatePolynomial| |$|) |FFIELDC-;gcdPolynomial;3Sup;16|)) (QUOTE #(|primitive?| 211 |order| 216 |nextItem| 226 |init| 231 |gcdPolynomial| 235 |discreteLog| 241 |differentiate| 252 |createPrimitiveElement| 257 |conditionP| 261 |charthRoot| 266)) (QUOTE NIL) (CONS (|makeByteWordVec2| 1 (QUOTE NIL)) (CONS (QUOTE #()) (CONS (QUOTE #()) (|makeByteWordVec2| 92 (QUOTE (0 6 0 7 1 6 10 0 11 1 6 0 10 12 1 6 13 0 14 1 6 10 0 17 1 19 0 18 20 1 23 22 0 24 2 26 13 25 0 27 1 6 0 0 28 2 26 0 29 0 30 0 6 34 35 0 6 34 36 2 6 0 0 34 37 0 6 0 40 0 6 41 42 2 41 13 0 0 43 0 6 46 47 2 6 0 0 18 48 1 6 13 0 49 2 6 13 0 0 51 0 6 0 53 1 6 54 18 55 1 54 34 0 56 2 54 57 10 0 58 2 6 0 0 0 59 1 63 61 62 64 1 18 65 0 66 1 67 46 0 68 3 69 57 6 6 34 70 1 74 72 73 75 1 76 72 73 77 0 73 0 78 2 73 13 0 0 79 0 72 0 80 2 76 82 73 13 83 1 73 0 6 84 2 72 0 73 18 85 2 72 0 0 0 86 0 72 0 87 2 72 0 73 0 88 2 89 0 0 0 90 1 0 13 0 50 1 0 10 0 52 1 0 19 0 21 1 0 15 0 16 0 0 0 9 2 0 91 91 91 92 1 0 34 0 60 2 0 57 0 0 71 1 0 0 0 8 0 0 0 44 1 0 31 32 33 1 0 0 0 38 1 0 15 0 39)))))) (QUOTE |lookupComplete|))) @ \section{package FFSLPE FiniteFieldSolveLinearPolynomialEquation} <>= )abbrev package FFSLPE FiniteFieldSolveLinearPolynomialEquation ++ Author: Davenport ++ Date Created: 1991 ++ Date Last Updated: ++ Basic Functions: ++ Related Constructors: ++ Also See: ++ AMS Classifications: ++ Keywords: ++ References: ++ Description: ++ This package solves linear diophantine equations for Bivariate polynomials ++ over finite fields FiniteFieldSolveLinearPolynomialEquation(F:FiniteFieldCategory, FP:UnivariatePolynomialCategory F, FPP:UnivariatePolynomialCategory FP): with solveLinearPolynomialEquation: (List FPP, FPP) -> Union(List FPP,"failed") ++ solveLinearPolynomialEquation([f1, ..., fn], g) ++ (where the fi are relatively prime to each other) ++ returns a list of ai such that ++ \spad{g/prod fi = sum ai/fi} ++ or returns "failed" if no such list of ai's exists. == add oldlp:List FPP := [] slpePrime: FP := monomial(1,1) oldtable:Vector List FPP := [] lp: List FPP p: FPP import DistinctDegreeFactorize(F,FP) solveLinearPolynomialEquation(lp,p) == if (oldlp ^= lp) then -- we have to generate a new table deg:= +/[degree u for u in lp] ans:Union(Vector List FPP,"failed"):="failed" slpePrime:=monomial(1,1)+monomial(1,0) -- x+1: our starting guess while (ans case "failed") repeat ans:=tablePow(deg,slpePrime,lp)$GenExEuclid(FP,FPP) if (ans case "failed") then slpePrime:= nextItem(slpePrime)::FP while (degree slpePrime > 1) and not irreducible? slpePrime repeat slpePrime := nextItem(slpePrime)::FP oldtable:=(ans:: Vector List FPP) answer:=solveid(p,slpePrime,oldtable) answer @ \section{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. @ <<*>>= <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}