randomsubproduct:=function(list)
local len, product, i, eps;
len:=Length(list);
product:=One(list[1]);
for i in [1..len] do
eps:=Random([0,1]);
product:=product*(list[i]^eps);
od;
return product;
end;
olustur:=function(q)
local g,gens,gensg,mat,c1,c2,i,x,tensorgens,g1,g2,y;
g:=GL(2,q);
gensg:=GeneratorsOfGroup(g);
gens:=[ ];
for i in [1..5] do
for x in gensg do
mat:=IdentityMat(10,GF(q));
mat[2*i-1][2*i-1]:=x[1][1];
mat[2*i-1][2*i]:=x[1][2];
mat[2*i][2*i-1]:=x[2][1];
mat[2*i][2*i]:=x[2][2];
Add(gens,mat);
od;
od;
c1:=IdentityMat(10,GF(q));
c1[1][1]:=0*Z(q);
c1[2][2]:=0*Z(q);
c1[3][3]:=0*Z(q);
c1[4][4]:=0*Z(q);
c1[1][3]:=Z(q)^0;
c1[2][4]:=Z(q)^0;
c1[3][1]:=Z(q)^0;
c1[4][2]:=Z(q)^0;
c2:=NullMat(10,10,GF(q));
for i in [1..8] do
c2[i][i+2]:=Z(q)^0;
od;
c2[9][1]:=Z(q)^0;
c2[10][2]:=Z(q)^0;
tensorgens:=[];
for x in gens do
Add(tensorgens,KroneckerProduct(x,IdentityMat(3,GF(q))));
od;
Add(tensorgens,KroneckerProduct(c1,IdentityMat(3,GF(q))));
Add(tensorgens,KroneckerProduct(c2,IdentityMat(3,GF(q))));
g2:=SL(3,q);
for y in GeneratorsOfGroup(g2) do
Add(tensorgens,KroneckerProduct(IdentityMat(10,GF(q)),y));
od;
g1:=Group(tensorgens);
return g1;
end;
create:=function(q)
local g,gens,gensg,mat,c1,c2,i,x,tensorgens,G;
g:=GL(2,q);
gensg:=GeneratorsOfGroup(g);
gens:=[ ];
for i in [1..5] do
for x in gensg do
mat:=IdentityMat(10,GF(q));
mat[2*i-1][2*i-1]:=x[1][1];
mat[2*i-1][2*i]:=x[1][2];
mat[2*i][2*i-1]:=x[2][1];
mat[2*i][2*i]:=x[2][2];
Add(gens,mat);
od;
od;
c1:=IdentityMat(10,GF(q));
c1[1][1]:=0*Z(q);
c1[2][2]:=0*Z(q);
c1[3][3]:=0*Z(q);
c1[4][4]:=0*Z(q);
c1[1][3]:=Z(q)^0;
c1[2][4]:=Z(q)^0;
c1[3][1]:=Z(q)^0;
c1[4][2]:=Z(q)^0;
c2:=NullMat(10,10,GF(q));
for i in [1..8] do
c2[i][i+2]:=Z(q)^0;
od;
c2[9][1]:=Z(q)^0;
c2[10][2]:=Z(q)^0;
tensorgens:=[];
for x in gens do
Add(tensorgens,KroneckerProduct(x,IdentityMat(10,GF(q))));
Add(tensorgens,KroneckerProduct(IdentityMat(10,GF(q)),x));
od;
#Add(tensorgens,KroneckerProduct(c1,c1));
Add(tensorgens,KroneckerProduct(c2,IdentityMat(10,GF(q))));
Add(tensorgens,KroneckerProduct(IdentityMat(10,GF(q)),c2));
G:=Group(tensorgens);
return G;
end;
separate:=function(g,N)
local r, ready, s, t, order, factor, f1, f2, len, cl, cl2, i, count, divisors, x;
ready:=false;
count:=0;
repeat
r:=PseudoRandom(g);
order:=ProjectiveOrder(r)[1];
factor:=Collected(Factors(order));
len:=Length(factor);
if len >1 then
f1:=(factor[len][1])^(factor[len][2]);
f2:=(factor[len-1][1])^(factor[len-1][2]);
s:=r^(order/f1);
t:=r^(order/f2);
cl:=[s];
for i in [1..10] do
Add(cl,s^PseudoRandom(g));
od;
if ForAll (cl,x->Comm(x,t)=One(x)) then
cl2:=[t];
for i in [1..10] do
Add(cl2,t^PseudoRandom(g));
od;
for i in [1..10] do
Add(cl,randomsubproduct(cl)^PseudoRandom(g));
Add(cl2,randomsubproduct(cl2)^PseudoRandom(g));
od;
if ForAll (cl,x->ForAll(cl2,y->Comm(x,y)=One(x))) then
ready:=true;
fi;
fi;
fi;
count:=count+1;
until ready or count>N;
if ready then
for i in [1..2] do
r:=PseudoRandom(g);
divisors:=DivisorsInt(ProjectiveOrder(r)[1]);
for x in divisors do
if CheckAnswer(r^x,cl) then
Add(cl2,r^x);
elif CheckAnswer(r^x,cl2) then
Add(cl,r^x);
fi;
od;
od;
for i in [1..10] do
Add(cl,randomsubproduct(cl)^PseudoRandom(g));
Add(cl2,randomsubproduct(cl2)^PseudoRandom(g));
od;
return [Group(cl),Group(cl2)];
else
return fail;
fi;
end;
NextStep := function ( g, a )
local b, c,g1;
b := ProjectiveOrder(g)[1]/a;
if b=1 then return fail; fi;
if Gcd( b, a ) = 1 then
c := b^-1 mod a;
Print("b=",b," c=",c,"\n");
return g^(b*c);
fi;
return fail;
end;
CheckAnswer := function( x, Bgens )
local y;
if x = fail then return false; fi;
if Length(Bgens) = 0 then return true; fi;
for y in Bgens do
if Comm( x, y ) <> Bgens[1]^0 then
return false;
fi;
od;
B := Group( Bgens );
a := ProjectiveOrder(g)[1];
Print( "aoriginal=",a,"\n");
hs := [];
for i in [ 1 .. N ] do
h := PseudoRandom( B );
order:=ProjectiveOrder(g*h)[1];
Add( hs, [h, order ] );
a := Gcd( a, order);
od;
Print( " a = ", a, "\n" );
i := 1;
repeat
h := PseudoRandom(B);
a := Gcd( a, ProjectiveOrder( g*h )[1] );
ans := NextStep ( g * h, a);
# if ans<>fail then Error(); fi;
if CheckAnswer( ans, Bgens ) then
Print( "i = ", i, "\n" );
if CheckAnswer( g*ans^(-1), Agens) then
return [ans, g*ans^(-1)];
fi;
fi;
i := i + 1;
until i > 1000 * N;
# h := First( hs, i -> Gcd( a, i[2]/a) = 1);
end;
writeeuclid:=function(g,r,Agens,Bgens)
local order,factor,euclid,i,a,parts,b,c,x;
order:=ProjectiveOrder(r)[1];
factor:=Collected(Factors(order));
euclid:=[];
parts:=[];
for i in [1..Length(factor)] do
euclid[i]:=order/(factor[i][1])^(factor[i][2]);
x:=r^euclid[i];
if CheckAnswer(x,Bgens) then
parts[i]:=[x,x^0];
elif CheckAnswer(x,Agens) then
parts[i]:=[x^0,x];
else
parts[i]:=TensorDecompose(g,x,Agens,Bgens,100);
fi;
od;
a:=GcdRepresentation(euclid);
b:=1;
c:=1;
for i in [1..Length(factor)] do
b:=b*((parts[i][1])^a[i]);
c:=c*((parts[i][2])^a[i]);
od;
if r=b*c then
return [b,c];
else
return fail;
fi;
end;
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