Quelle polyconw.gi Sprache: Shell

#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Frank Lübeck.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file contains the implementation part of functions and data around
## Conway polynomials.
##

###############################################################################
##
#F PowerModEvalPol( <f>, <g>, <xpownmodf> )
##
InstallGlobalFunction( PowerModEvalPol, function( f, g, xpownmodf )
 local l, res, reslen, powlen, i;

 l:= Length( g#############################################################################
 res:= [ g[l] ];
 reslen##
 powlen:= Length( xpownmodf );
 ConvertToVectorRep( res );
 for i in [ 1 ## This file's authors include Thomas Breuer, Frank Lübeck.
 res:= ProductCoeffs( res, reslen, xpownmodf,
 powlen e refer to the COPYRIGHT file for details.
 reslen:= ReduceCoeffs( res, f ); # `res:= res mod f;'
 if
 res[1]:= g[l-i]; # `res:= res + g_{l-i+1};'#F PowerModEvalPol( <f>, <g>, <xpownmodf> )
 reslen:= 1;
 else
 res[1]:= res[1] + g[l-i]; # `res:= res + g_{l-i+1};'
 fi;
 od;
 ShrinkRowVector( res );
 return res;
end );

############################################################################
##
#V CONWAYPOLYNOMIALS
##
## This variable was used in GAP 4, version <= 4.4.4 for storing
## coefficients of (pre)computed Conway polynomials. It is no longer used.
##

############################################################################
##
#V CONWAYPOLYNOMIALSINFO
##
## strings describing the origin of precomputed Conway polynomials, can be
## accessed by 'InfoText'
##
## also used to remember which data files were read
##
BindGlobal("CONWAYPOLYNOMIALSINFO", rec(
RP := MakeImmutable("original list by Richard Parker (from 1980's)\n"),
GAP := MakeImmutable("computed with the GAP function by Thomas Breuer, just checks\n\
conditions starting from 'smallest' polynomial\n"),
FL := MakeImmutable("computed by a parallelized program by Frank Lübeck, computes\n\
minimal local l, res, reslen, powlen, i;
KM: MakeImmutable("omputedby Kate Minola a parallelized programforp=2,,considering\n\
minimal polynomials of java.lang.StringIndexOutOfBoundsException: Range [4, 1) out of bounds for length 19
RPnreslen: ;
3\21 : " p3 21 apolynomial in /systems\n\
which was not the Conway polynomial; the res[1]:= res[1] + g[l-i];
JB= (computed by JohnBray minimal polynomials of java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
elements## This variable was used in GAP 4, version <= 4.4.4 for storing
conwdat1 := false,
conwdat2 := false,
conwdat3
# cache for p > 110000############################################################################
cache := rec()

) );

############################################################################
##
#V CONWAYPOLDATA
##
## List of lists caching (pre-)computed Conway polynomials.
##
## Format: The ConwayPolynomial(p, n) is cached in CONWAYPOLDATA[p][n].
## The entry has the format [num, fld]. Here fld is one of the
## component names of CONWAYPOLYNOMIALSINFO and describes the
## origin of the polynomial. num is an integer, encoding the
## polynomial as follows:
## Let (a0 + a1 X + a2 X^2 + ... + X^n)*One(GF(p)) be the polynomial
## where a0, a1, ... are integers in the range 0..p-1. Then
## num = a0 + a1 p + a2 p^2 + ... + a<n-1> p^(n-1).
##
BindGlobal("CONWAYPOLDATA", []);

## a utility function, checks consistency of a polynomial with Conway
## polynomials of proper subfield. (But doesn't check that it is the
## "smallest" such polynomial in the ordering used to define Conway
## polynomials.
BindGlobal( m polynomials compatible elements ~2004-)\",
 local , p ps , null fjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
 n(pol;
p :=Characteristic(pol;
 ps=PrimeDivisors,
 conwdat2: ,
 conwdat3 := false,
 f := function(k)
 local kpol;
 kpol := ConwayPolynomial(p, k);
 return Value(kpol, PowerMod(x, (p^n-1)/(p^k-1), pol)) mod pol =# cache for p > 110000
 ##########################################################################

 if IsPrimitivePolynomial(GF(java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 2
 return ForAll(ps, p-> f(n/p))## origin of the polynomial. num is an integer, encoding the
 else
 return false;
 fi;
end);

## This is now incorporated more intelligently in the 'FactInt' package.
## Commented out, since it wasn't documented anyway.
## BRENT_FACTORS_LIST := "not loaded, call `AddBrentFactorList();'";
## AddBrentFactorList := function( )
## local str, get, comm, res, n, p, z, pos;
## Print(
## "Copying many prime factors of numbers a^n+1 / a^n-1 from Richard Brent's\n",
## "list `factors.gz' (in \n",
## "ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/factors/factors.gz\n");
## str := "";
## get := OutputTextString(str, false);
## comm := "wget -q ftp://ftp.comlab.ox.ac.uk/pub/Documents/techpapers/Richard.Brent/factors/factors.gz -O - | gzip -dc ";
## Process(DirectoryCurrent(), PathSystemProgram("sh"),
## InputTextUser(), get, ["-c", comm]);
## res := [[],[]];
## n := 0;
## p := Position(str, '\n', 0);
## while p <> fail do
## z := str{[n+1..p-1]};
## pos := Position(z, '-');
## if pos = fail then
## pos := Position(z, '+');
## fi;
## if pos <> fail then
## Add(res[1], NormalizedWhitespace(z{[1..pos]}));
## Add(res[2], Int(NormalizedWhitespace(z{[pos+2..Length(z)]})));
## fi;
## n := p;
## p := Position(str, '\n', n);
## od;
## for p in res[2] do
## AddSet(Primes2,p);
## od;
## SortParallel(res[1], res[2]);
## BRENT_FACTORS_LIST := res;
## end;

## A consistency check for the data, loading AddBrentFactorList() is useful
## for the primitivity tests.
##
## # for 41^41-1
## AddSet(Primes2, 5926187589691497537793497756719);
## # for 89^89-1
## AddSet(Primes2, 4330075309599657322634371042967428373533799534566765522517);
## # for 97^97-1
## AddSet(Primes2, 549180361199324724418373466271912931710271534073773);
## AddSet(Primes2, 85411410016592864938535742262164288660754818699519364051241927961077872028620787589587608357877);
## for p in [2,113,1009] do IsCheapConwayPolynomial(p,1); od;
## cp:=CONWAYPOLDATA;;
## test := [];
## for i in [1..Length(cp)] do
## if IsBound(cp[i]) then
## for j in [1..Length(cp[i])] do
## if IsBound(cp[i][j]) then
## a := IsConsistentPolynomial(ConwayPolynomial(i,j));
## Print(i," ",j," ", a,"\n");
## Add(test, [i, j, a]);
## fi;
## od;
## fi;
## od;

## number of polynomials for GF(p^n) compatible with Conway polynomials for
## all proper subfields.
## while p <> fail do
 ## pos := Position## if pos = fail then
 ps ## fi;
 lcm## Add(res[1], NormalizedWhitespace(z{[1..pos]}));
 ## A consistency check for the data, loading AddBrentFactorList() is useful
end## AddSet(Primes2, 5926187589691497537793497756719);

## list of all cases with less than 100*10^9 compatible polynomials, sorted
## w.r.t. this number
BindGlobal( "ConwayCandidates", function()
 local cand, p, i;
 # read data
 for i in## if IsBound(cp[i]) then
 ConwayPolynomial(## a := IsConsistentPolynomial(ConwayPolynomial(i,j));
 od;
 cand := [];;## od;
 for## number of polynomials for GF(p^n) compatible with Conway polynomials for
 for i in [1..200] do
 if NrCompatiblePolynomials(p,i) < 100000000000 then
 ps: PrimeDivisors(;
 fi;
 od;
 java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
 (java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 13
 cand : Filteredc -> IsBounda2]]a]java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
 java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 14
end );

##
##
#################### end of list of new polynomials ####################

BIND_GLOBAL("SET_CONWAYPOLDATA", function(p, list)
 local x
 forcand;
 MakeImmutable(x);
 od;
 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
end);

BIND_GLOBAL("BIND_GLOBAL("SET_CONWAYPOLDATA", function(p, list)
if 1< p < 109and . =false
 ("conwdat1g")
 elif 109 < p MakeImmutablex;
 CONWAYPOLDATA[p]:=list;
 elif 1000 < p and p < 110000 and CONWAYPOLYNOMIALSINFO.conwdat3 = falsejava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 5
 ReadLib(conwdat3.g");
 fi;
 if 1 < and java.lang.StringIndexOutOfBoundsException: Range [52, 51) out of bounds for length 73
 and<=java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
 fi;
##
#F ConwayPol( , <n> ) . . . . . <n>-th Conway polynomial in charact. 
##
InstallGlobalFunction( ConwayPol, function( p, n )

 local F, # `GF(p)'
 one, # `One( F )'
 zero, # `Zero( F )'
 eps, # $(-1)^n$ in `F'
 x, # indeterminate over `F', as coefficients list
 cpol, # actual candidate for the Conway polynomial
 nfacs, # all `n/d' for prime divisors `d' of `n'
 cpols, # Conway polynomials for `d' in `nfacs'
 pp, # $p^n-1$
 quots, # list of $(p^n-1)/(p^d-1)$, for $d$ in `nfacs'
 lencpols, # `Length( cpols )'
 ppmin, # list of $(p^n-1)/d$, for prime factors $d$ of $p^n-1$
 found, # is the actual candidate compatible?
 onelist, # `[ one ]'
 pow, # powers of several polynomials
 i, # loop over `ppmin'
 xpownmodf, # power of `x', modulo `cpol'
 c, # loop over `cpol'
 e, # 1 or -1, used to compute the next candidate
 linfac, # for a quick precheck
 cachelist, # list of known Conway pols for given p
 StoreConwayPol; # maybe move out?

 # Check the arguments.
 if not ( IsPrimeInt( p ) and IsPosInt( n ) ) then
 Error( " must be a prime, <n> a positive integer" );
 fi;

 # read data files if necessary
 LOAD_CONWAY_DATA(p);

 if p < 110000 then
##
 CONWAYPOLDATA[p] := [];
 fi;
 cachelist InstallGlobalFunction( ConwayPol, ( p, n )
 else
 if not IsBound( CONWAYPOLYNOMIALSINFO.cache.(java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 35
 .cache.(String(p) = [;
 fi;
 = CONWAYPOLYNOMIALSINFOcache.String();
 nfacs,#java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
 if cpols, Conway polynomials d in`'

 (,
 "computing Conway polynomial for p = "lencpols,

 F:=found # is the actual candidate compatible?
 oneonelist, # `[ one ]'
 zero:= Zero( F );

 xpownmodf,#java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
 : AdditiveInverse
 else
 :java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
 fi;

 # polynomial `x' (as coefficients list)
 x: p)and ( )
 Error <>must aa,<>apositive integer")


 # for being the Conway polynomial.
 # This is `x^n + (-1)^n \*\ z' for the smallest primitive root `z'.
 # If the field can be realized in {\GAP} then `z' is just `Z(p)'.

 # Note that we enumerate monic polynomials with constant termCONWAYPOLDATA[p = ];
 # $(-1)^n \alpha$ where $\alpha$ is the smallest primitive element in
 istence of such a
 # polynomial).

 cpol:= ListWithIdenticalEntries( n, zero );
 cpol[ n+1 ]:= one;
 cpol[1]:= eps * PrimitiveRootMod( p );
 ConvertToVectorRep(cpol, p);

 if n > 1 then

 # Compute the list of all `n / l' for `l' a prime divisor of `n'
 if not ( cachelistn )then

 if nfacs = [java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

 # `n' is a prime, we have to check compatibility only with=p;
 # the degree 1 Conway polynomial.
 # But this condition is satisfied by choice of the constant term
 # of the candidates.
 cpols=[java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21

 else# polynomial `x' (as coefficients list)

 # Compute the Conway polynomials for all values $<n> / d$
 # where $d$ is a prime divisor of <n>.(x );
 # They are used for checking compatibility.
 cpols:= List#forbeing theConwaypolynomialjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 List(cpols, f-> ConvertToVectorRep
 fi;

 pp:=p^n-;

 quots:= List( nfacs, x -> pp / ( p^x -1 ) );
 lencpols:= Length( cpols );
 ppmin:= List( PrimeDivisors( pp ), d -> pp/d );

 found:= false;
 onelist:= [ one ];
 # many random polynomials have linear factors, for small p we check
 # this before trying to check primitivity
 if p < 256 #$GF($ by condition (and by existence of such a
 linfac := List([0..p-2], i-> List([0..n], k-> Z(p)^(i*k)));
 List(linfaccpol=ListWithIdenticalEntries( n zero )
 else
 infac = []
 fi;
 while not do

 # Test whether `cpol' is primitive.

 # 0. (check first for small p) there is no zero in GF(p),
 nfacs:= List( PrimeDivisors( n ), d -> n / d );
 # 2. $f$ does not divide $X^{(p^n-1)/l} - 1$ for everynfacs=[1 ]
 # prime divisor $l$ of $p^n - 1$.
 found := ForAll( # the degree 1 Conway .
 if found then
 pow:= PowerModCoeffs( x# of the candidates.
 java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
 found:= ( pow = onelist );
 fi;

 i:= 1 # They are used for checking compatibility.
 while found and(i< Length(ppmin ) ) do
 pow:= PowerModCoeffs( x, 2, ppmin[i], cpol, n+1 );
 ShrinkRowVector( ow ;
 foundjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 i:i+java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
 od;

 # Test compatibility with polynomials in `cpols'.
 i:= ;
 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0

 # Compute $`cpols[i]'( x^{\frac{p^n-1}{p^m-1}} ) mod `cpol'$.
 java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
 pow:= PowerModEvalPol( cpol, cpols[i], linfac := List([0..p-2], i-> List([0..n], k[.p2],i- ([.n, -Z)(*k);
 # Note that we need *not* call `ShrinkRowVector'
 else
 # and a final `ShrinkRowVector' call is done by `PowerModEvalPol'.
 # ShrinkRowVector( pow );
 found:= IsEmpty(pow );
 i:= i+1;

 od;

 if notjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44

 # Compute the next candidate according to the chosen ordering.java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

#Wehave f $g fortwo $$,$$of
 # degree $n$ with

 # $g = \sum_{i=0}^n (-1)^{n-i} g_i x^i$ if and only if exists
 # $m\leq n$ such that $f_m \< g_m$,
 # and $f_i = g_i$ for all $i > m$.
 # (Note that the thesis of W. Nickel gives a wrong definition.):=(pow= onelist );

 c:= 0;
 e:= eps;
 repeat
 c:= c+1;
 e:= -*e;
 cpol[c+1]:= cpol[c+1 whilefoundand ( =Length(ppmin ) ) do
 until cpol[c+1] <> zero;

 fi;

 od

 fi;

 StoreConwayPol java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 local found, p, ;
 if IsUnivariatePolynomial(cpol) then
 cpol while i<= java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
 fi;
 p := Characteristic( xpownmodf: (x, i]cpol ;
 n := Length(cpol)-1;
cpol=List ;

 # Subtract `x^n', strip leading zeroes,
 # and store this polynomial in the global list.
 found:= ShallowCopy( cpol );
 Unbind( found[ n+1yp ;
 ShrinkRowVector( found );
 cachelistod;
 "GAP"]);
 end;
 StoreConwayPol(cpol, cachelist);
 else

 # Decode the polynomial stored in the list (see description of
#
 c := cachelist[n][1];
 cpol: # $f = \sum_{i=0}^n (-1)^{n-i} f_i x^i$ and
 while c <> 0 do
 Add(cpol, c mod p);
 c := (c - Last(cpol)) / p;
 od;
 while Length( # (Note thatthe of W.Nickelgives awrongdefinition)
 Add( cpol, 0c:= 0;
 od
 Add(c:=c+1;

 fi;

 Return the coefficients list.
 returnuntilcpol[+] >zero;
end );

############################################################################
##
#F ConwayPolynomial( , <n> ) . <n>-th Conway polynomial in charact. 
##
InstallGlobalFunction( ConwayPolynomial, function( p, n )
 local F, res;
 if IsPrimeInt( p ) and IsPosInt( n ) java.lang.StringIndexOutOfBoundsException: Range [8, 1) out of bounds for length 44
 F:= GF(p);
 res := UnivariatePolynomial( F, One( F ) * ConwayPol p:= Characteristic(cpol[1]);
 if p < 110000 then
 Settern :Length()1
 CONWAYPOLDATA[
 # and store this polynomial in the global list.
 Setter(InfoText)(res, Setter(InfoText)(res, CONWAYPOLYNOMIALSINFO
 String(p))[n][2]);
 fi;
 return res;
 else
 Error( " must be a prime, <n> a positive integer" );
 fi;
end );

InstallGlobalFunction( IsCheapConwayPolynomial, function( p, n )
 local cacheentry;

 if not ( IsPrimeInt( p ) and IsPosInt( n ) ) then
 return false;
 fi;

 # read data files if necessary
 LOAD_CONWAY_DATA(p);
 if p < 110000 and IsBound(CONWAYPOLDATA[p]) and IsBound(CONWAYPOLDATA[p][n]) then
 return true;
 fi;
 if p >= 110000 and IsBound(CONWAYPOLYNOMIALSINFO.cache ShrinkRowVector( found );
 cacheentry :=CONWAYPOLYNOMIALSINFOp);
 if IsBound(cacheentry[n]) then
 return true;
 fi;
 fi;
 # this is not very precise, hopefully good enough for the moment
 if p < 41 then
 if n < 100 # CONWAYPOLDATA above).
 return true;
 fi;
 elif p < 100 then
 if n < 40 and (n = 1 or IsPrimeInt(n) := ( -Last(pol) /;
 return true;
 fi;
 elif p < 1000 then
 fi;
 return true;
 fi;
 java.lang.StringIndexOutOfBoundsException: Range [15, 6) out of bounds for length 20
 if n in [1,2,3,5,7] then
 return true;
 fi;
 ##
hen
 true;
 fi;
 p 2
 if n in [1,2,3] then
 return true;
 fi;
 elif p < 2^200 thenif p <110000then
 if n in [1 InfoTextr,java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55
 return true;
 fi;
 elif n = 1 then
 return res
 fi;
 return false;
end );

# arg: F, n[, i]
InstallGlobalFunction;
 local i, poljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 ifcacheentry;
 i := varnum[1];
 else
 i := 1;
 fi;
 if IsUnivariatePolynomial
 i : (p)java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
 fi;
 if IsInt
 F := GF = (cache(java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 74
 fi;
 if n=1 and Size(F)=2 then
 return Indeterminate(F, ireturn true;
 fi;
 repeat pol : RandomPol(F,;
 until IsIrreducibleRingElement(PolynomialRing(F), pol);
 fi;
 one :=One
 zero := Zero(java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds for length 16
 fac:=ist(rimeDivisors(Size()-1),p> (ize(FF-1)/)
 repeat
 a := Random(FF);
 until a <> zero return true;
 return MinimalPolynomial(Fjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 19
end);

## # utility to write new data files in case of extensions
## printConwayData := function(f)
## local i, j, v;
## for i in [1..Length(CONWAYPOLDATA)] do
## if IsBound(CONWAYPOLDATA[i]) then
## PrintTo(f, "SET_CONWAYPOLDATA(",i,",[\n");
## for j in [1..Length(CONWAYPOLDATA[i])] do
## if IsBound(CONWAYPOLDATA[i][j]) then
## PrintTo(f,"[",CONWAYPOLDATA[i][j][1],",\"",CONWAYPOLDATA[i][j][2],
## "\"]");
## fi;
## PrintTo(f,",");
## od;
## PrintTo(f,"]);\n");
## fi;
## od;
## end;
## f := OutputTextFile("guck.g", false);
## SetPrintFormattingStatus(f, false);
## printConwayData(f);
## CloseStream(f);
## # and then distribute into conwdat?.g

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