ENHANCE: RootFFE as 3-argument function with field first.

lib/ffe.gd

@@ -659,18 +659,19 @@ DeclareAttribute( "AsInternalFFE", IsFFE);
 ##
 ##  <#GAPDoc Label="RootFFE">
 ##  <ManSection>
-##  <Oper Name="RootFFE" Arg='z, k'/>
+##  <Oper Name="RootFFE" Arg='F, z, k'/>
 ##
 ##  <Description>
 ##  <Ref Func="RootFFE"/> returns a finite field element 
-##  <A>r</A> whose <A>k</A>-th power is <A>z</A>. If no such element exists
+##  <A>r</A> from <F> whose <A>k</A>-th power is <A>z</A>.
+##  If no such element exists
 ##  then
 ##  <K>fail</K> is returned.
 ##  </Description>
 ##  </ManSection>
 ##  <#/GAPDoc>
 ##
-DeclareOperation( "RootFFE", [ IsFFE, IsObject ] );
+DeclareOperation( "RootFFE", [ IsObject, IsFFE, IsObject ] );
 
 
 #############################################################################

lib/ffe.gi

@@ -1149,17 +1149,33 @@ end);
 ##
 #M  RootFFE( <z>, <k> ) 
 ##
-InstallMethod( RootFFE, "use LogFFE",true,[IsFFE,IsPosInt],
-function(z,k)
-local q,e;
-  q:=Characteristic(z)^DegreeFFE(z);
+InstallMethod(RootFFE,"use field order",IsCollsElmsX,
+  [IsField,IsFFE,IsPosInt],0,
+function(F,z,k)
+  return RootFFE(Size(F),z,k);
+end);
+
+InstallOtherMethod(RootFFE,"use LogFFE",true,[IsPosInt,IsFFE,IsPosInt],
+function(q,z,k)
+local e,m,p,a;
+  if IsZero(z) or IsOne(z) then return z;fi;
+  m:=q-1;
   e:=LogFFE(z,Z(q));
-  e:=e/k;
-  if Gcd(DenominatorRat(e),q-1)=1 then
-    return Z(q)^(e mod (q-1));
-  else
-    return fail;
+  p:=GcdInt(m,e); #power for subgroup it is in
+  k:=k mod m;
+  a:=GcdInt(m,k); # power for subgroup we want
+  if p mod a<>0 then
+    return fail; # element does not lie in subgroup of a-th powers
   fi;
+  # k/a is coprime to order of elts in subgroup and elt is an a-th power
+  a:=(e*(a/k mod (m/p))/a mod m);
+  return Z(q)^a;
+end);
+
+InstallOtherMethod(RootFFE,"without field",true,
+  [IsFFE,IsPosInt],0,
+function(z,k)
+  return RootFFE(Characteristic(z)^DegreeFFE(z),z,k);
 end);
 
 

lib/zmodnz.gi

@@ -660,16 +660,28 @@ InstallMethod( LogFFE,
 ##
 #M  RootFFE( <z>, <k> )  . . . . . . . . . . . . . . . . . . for `IsZmodpZObj'
 ##
-InstallMethod(RootFFE,"for modulus rep, using RootMod",true,
-  [IsZmodpZObj and IsModulusRep,IsPosInt],
-function( z, k )
+InstallOtherMethod(RootFFE,"for modulus rep, using RootMod",true,
+  [IsPosInt,IsZmodpZObj and IsModulusRep,IsPosInt],
+function( A, z, k )
 local r,fam;
   fam:=FamilyObj(z);
-  r:=RootMod(z![1],k,fam!.Characteristic);
+  if A<>fam!.Characteristic then
+    TryNextMethod();
+  fi;
+  if k=1 or z![1]=0 or z![1]=1 then return z;fi;
+  r:=RootMod(z![1],k,A);
   if r=fail then return r;fi;
   return ZmodnZObj(fam,r);
 end );
 
+InstallOtherMethod(RootFFE,"for modulus rep",true,
+  [IsZmodpZObj and IsModulusRep,IsPosInt],
+function(z,k)
+  return RootFFE(FamilyObj(z)!.Characteristic,z,k);
+end);
+
+
+
 
 #############################################################################
 ##

tst/testinstall/ffe.tst

@@ -119,6 +119,23 @@ gap> Subfields( GF(81) );
 [ GF(3), GF(3^2), GF(3^4) ]
 gap> Subfields( GF(2^6) );
 [ GF(2), GF(2^2), GF(2^3), GF(2^6) ]
+
+# RootFFE
+gap> Rochambeau:=function(F)
+> local e,i,p,a,r;
+>  e:=Elements(F);
+>  for i in [1..2*Size(F)] do
+>    p:=Set(List(e,x->x^i));
+>    for a in e do
+>      r:=RootFFE(F,a,i);
+>      if a in p and r=fail then Error("-1"); return -1;fi;
+>      if r<>fail and a<>r^i then Error("1");return 1;fi;
+>    od;
+>  od;
+>  return 0;
+> end;;
+gap> ForAll(Filtered([1..256],IsPrimePowerInt),x->Rochambeau(GF(x))=0); 
+true
 gap> STOP_TEST( "ffe.tst", 470000);
 
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