// constant term must be non-zero



inline GF2EX operator/(const GF2EX& a, const GF2EX& b)
   { GF2EX x; div(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX operator/(const GF2EX& a, const GF2E& b)
   { GF2EX x; div(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX operator/(const GF2EX& a, GF2 b)
   { GF2EX x; div(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX operator/(const GF2EX& a, long b)
   { GF2EX x; div(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX& operator/=(GF2EX& x, const GF2EX& b)
   { div(x, x, b); return x; }

inline GF2EX& operator/=(GF2EX& x, const GF2E& b)
   { div(x, x, b); return x; }

inline GF2EX& operator/=(GF2EX& x, GF2 b)
   { div(x, x, b); return x; }

inline GF2EX& operator/=(GF2EX& x, long b)
   { div(x, x, b); return x; }


inline GF2EX operator%(const GF2EX& a, const GF2EX& b)
   { GF2EX x; rem(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX& operator%=(GF2EX& x, const GF2EX& b)
   { rem(x, x, b); return x; }




/***********************************************************

                         GCD's

************************************************************/


void GCD(GF2EX& x, const GF2EX& a, const GF2EX& b);
inline GF2EX GCD(const GF2EX& a, const GF2EX& b)
   { GF2EX x; GCD(x, a, b); NTL_OPT_RETURN(GF2EX, x); }

// x = GCD(a, b),  x is always monic (or zero if a==b==0).

void XGCD(GF2EX& d, GF2EX& s, GF2EX& t, const GF2EX& a, const GF2EX& b);
// d = gcd(a,b), a s + b t = d 


/*************************************************************

             Modular Arithmetic without pre-conditioning

**************************************************************/

// arithmetic mod f.
// all inputs and outputs are polynomials of degree less than deg(f).
// ASSUMPTION: f is assumed monic, and deg(f) > 0.
// NOTE: if you want to do many computations with a fixed f,
//       use the GF2EXModulus data structure and associated routines below.



void MulMod(GF2EX& x, const GF2EX& a, const GF2EX& b, const GF2EX& f);
inline GF2EX MulMod(const GF2EX& a, const GF2EX& b, const GF2EX& f)
   { GF2EX x; MulMod(x, a, b, f); NTL_OPT_RETURN(GF2EX, x); }
// x = (a * b) % f

void SqrMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
inline GF2EX SqrMod(const GF2EX& a, const GF2EX& f)
   { GF2EX x; SqrMod(x, a, f); NTL_OPT_RETURN(GF2EX, x); }
// x = a^2 % f

void MulByXMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
inline GF2EX MulByXMod(const GF2EX& a, const GF2EX& f)
   { GF2EX x; MulByXMod(x, a, f); NTL_OPT_RETURN(GF2EX, x); }
// x = (a * X) mod f

void InvMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
inline GF2EX InvMod(const GF2EX& a, const GF2EX& f)
   { GF2EX x; InvMod(x, a, f); NTL_OPT_RETURN(GF2EX, x); }
// x = a^{-1} % f, error is a is not invertible

long InvModStatus(GF2EX& x, const GF2EX& a, const GF2EX& f);
// if (a, f) = 1, returns 0 and sets x = a^{-1} % f
// otherwise, returns 1 and sets x = (a, f)





/******************************************************************

        Modular Arithmetic with Pre-conditioning

*******************************************************************/


// If you need to do a lot of arithmetic modulo a fixed f,
// build GF2EXModulus F for f.  This pre-computes information about f
// that speeds up the computation a great deal.

class GF2EXModulus {
public:
   GF2EXModulus();
   ~GF2EXModulus() { }

   GF2EXModulus(const GF2EX& ff);

   GF2EX f;   // the modulus

   operator const GF2EX& () const { return f; }
   const GF2EX& val() const { return f; }

   long n; //  deg(f)

   long method; // GF2EX_MOD_PLAIN or GF2EX_MOD_MUL 

   GF2EX h0;
   GF2E hlc;
   GF2EX f0;

   vec_GF2E tracevec;

}; 


inline long deg(const GF2EXModulus& F) { return F.n; }

void build(GF2EXModulus& F, const GF2EX& f);



void rem(GF2EX& r, const GF2EX& a, const GF2EXModulus& F);
   
void DivRem(GF2EX& q, GF2EX& r, const GF2EX& a, const GF2EXModulus& F);

void div(GF2EX& q, const GF2EX& a, const GF2EXModulus& F);

void MulMod(GF2EX& c, const GF2EX& a, const GF2EX& b, const GF2EXModulus& F);
inline GF2EX MulMod(const GF2EX& a, const GF2EX& b, const GF2EXModulus& F)
   { GF2EX x; MulMod(x, a, b, F); NTL_OPT_RETURN(GF2EX, x); }

void SqrMod(GF2EX& c, const GF2EX& a, const GF2EXModulus& F);
inline GF2EX SqrMod(const GF2EX& a, const GF2EXModulus& F)
   { GF2EX x; SqrMod(x, a, F); NTL_OPT_RETURN(GF2EX, x); }


void PowerMod(GF2EX& h, const GF2EX& g, const ZZ& e, const GF2EXModulus& F);

inline void PowerMod(GF2EX& h, const GF2EX& g, long e, const GF2EXModulus& F)
   { PowerMod(h, g, ZZ_expo(e), F); }

inline GF2EX PowerMod(const GF2EX& g, const ZZ& e, const GF2EXModulus& F)
   { GF2EX x; PowerMod(x, g, e, F);  NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX PowerMod(const GF2EX& g, long e, const GF2EXModulus& F)
   { GF2EX x; PowerMod(x, g, e, F);  NTL_OPT_RETURN(GF2EX, x); }

void PowerXMod(GF2EX& hh, const ZZ& e, const GF2EXModulus& F);

inline void PowerXMod(GF2EX& h, long e, const GF2EXModulus& F)
   { PowerXMod(h, ZZ_expo(e), F); }


inline GF2EX PowerXMod(const ZZ& e, const GF2EXModulus& F)
   { GF2EX x; PowerXMod(x, e, F);  NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX PowerXMod(long e, const GF2EXModulus& F)
   { GF2EX x; PowerXMod(x, e, F);  NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX operator%(const GF2EX& a, const GF2EXModulus& F)
   { GF2EX x; rem(x, a, F); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX& operator%=(GF2EX& x, const GF2EXModulus& F)
   { rem(x, x, F); return x; }

inline GF2EX operator/(const GF2EX& a, const GF2EXModulus& F)
   { GF2EX x; div(x, a, F); NTL_OPT_RETURN(GF2EX, x); }

inline GF2EX& operator/=(GF2EX& x, const GF2EXModulus& F)
   { div(x, x, F); return x; }



/*****************************************************************

                       vectors of GF2EX's

*****************************************************************/



NTL_vector_decl(GF2EX,vec_GF2EX)

NTL_eq_vector_decl(GF2EX,vec_GF2EX)

NTL_io_vector_decl(GF2EX,vec_GF2EX)





/*******************************************************

              Evaluation and related problems

********************************************************/


void BuildFromRoots(GF2EX& x, const vec_GF2E& a);
inline GF2EX BuildFromRoots(const vec_GF2E& a)
   { GF2EX x; BuildFromRoots(x, a); NTL_OPT_RETURN(GF2EX, x); }
// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()


void eval(GF2E& b, const GF2EX& f, const GF2E& a);
inline GF2E eval(const GF2EX& f, const GF2E& a)
   { GF2E x; eval(x, f, a); NTL_OPT_RETURN(GF2E, x); }
// b = f(a)

void eval(vec_GF2E& b, const GF2EX& f, const vec_GF2E& a);
inline vec_GF2E eval(const GF2EX& f, const vec_GF2E& a)
   { vec_GF2E x; eval(x, f, a); NTL_OPT_RETURN(vec_GF2E, x); }
//  b[i] = f(a[i])

inline void eval(GF2E& b, const GF2X& f, const GF2E& a)
   { conv(b, CompMod(f, rep(a), GF2E::modulus())); }
   
inline GF2E eval(const GF2X& f, const GF2E& a)
   { GF2E x; eval(x, f, a); NTL_OPT_RETURN(GF2E, x); }
// b = f(a)


void interpolate(GF2EX& f, const vec_GF2E& a, const vec_GF2E& b);
inline GF2EX interpolate(const vec_GF2E& a, const vec_GF2E& b)
   { GF2EX x; interpolate(x, a, b); NTL_OPT_RETURN(GF2EX, x); }
// computes f such that f(a[i]) = b[i]




/**********************************************************

         Modular Composition and Minimal Polynomials

***********************************************************/


// algorithms for computing g(h) mod f




void CompMod(GF2EX& x, const GF2EX& g, const GF2EX& h, const GF2EXModulus& F);
inline GF2EX 
CompMod(const GF2EX& g, const GF2EX& h, const GF2EXModulus& F)
   { GF2EX x; CompMod(x, g, h, F); NTL_OPT_RETURN(GF2EX, x); }
// x = g(h) mod f

void Comp2Mod(GF2EX& x1, GF2EX& x2, const GF2EX& g1, const GF2EX& g2,
              const GF2EX& h, const GF2EXModulus& F);
// xi = gi(h) mod f (i=1,2)

void Comp3Mod(GF2EX& x1, GF2EX& x2, GF2EX& x3, 
              const GF2EX& g1, const GF2EX& g2, const GF2EX& g3,
              const GF2EX& h, const GF2EXModulus& F);
// xi = gi(h) mod f (i=1..3)



// The routine build (see below) which is implicitly called
// by the various compose and UpdateMap routines builds a table
// of polynomials.  
// If GF2EXArgBound > 0, then the table is limited in
// size to approximamtely that many KB.
// If GF2EXArgBound <= 0, then it is ignored, and space is allocated
// so as to maximize speed.
// Initially, GF2EXArgBound = 0.


// If a single h is going to be used with many g's
// then you should build a GF2EXArgument for h,
// and then use the compose routine below.
// build computes and stores h, h^2, ..., h^m mod f.
// After this pre-computation, composing a polynomial of degree 
// roughly n with h takes n/m multiplies mod f, plus n^2
// scalar multiplies.
// Thus, increasing m increases the space requirement and the pre-computation
// time, but reduces the composition time.
// If GF2EXArgBound > 0, a table of size less than m may be built.

struct GF2EXArgument {
   vec_GF2EX H;
};

extern long GF2EXArgBound;


void build(GF2EXArgument& H, const GF2EX& h, const GF2EXModulus& F, long m);

// m must be > 0, otherwise an error is raised

void CompMod(GF2EX& x, const GF2EX& g, const GF2EXArgument& H, 
             const GF2EXModulus& F);

inline GF2EX 
CompMod(const GF2EX& g, const GF2EXArgument& H, const GF2EXModulus& F)
   { GF2EX x; CompMod(x, g, H, F); NTL_OPT_RETURN(GF2EX, x); }
   



void MinPolySeq(GF2EX& h, const vec_GF2E& a, long m);
inline GF2EX MinPolySeq(const vec_GF2E& a, long m)
   { GF2EX x; MinPolySeq(x, a, m); NTL_OPT_RETURN(GF2EX, x); }


void MinPolyMod(GF2EX& hh, const GF2EX& g, const GF2EXModulus& F);
inline GF2EX MinPolyMod(const GF2EX& g, const GF2EXModulus& F)
   { GF2EX x; MinPolyMod(x, g, F); NTL_OPT_RETURN(GF2EX, x); }


void MinPolyMod(GF2EX& hh, const GF2EX& g, const GF2EXModulus& F, long m);
inline GF2EX MinPolyMod(const GF2EX& g, const GF2EXModulus& F, long m)
   { GF2EX x; MinPolyMod(x, g, F, m); NTL_OPT_RETURN(GF2EX, x); }

void ProbMinPolyMod(GF2EX& hh, const GF2EX& g, const GF2EXModulus& F);
inline GF2EX ProbMinPolyMod(const GF2EX& g, const GF2EXModulus& F)
   { GF2EX x; ProbMinPolyMod(x, g, F); NTL_OPT_RETURN(GF2EX, x); }

void ProbMinPolyMod(GF2EX& hh, const GF2EX& g, const GF2EXModulus& F, long m);
inline GF2EX ProbMinPolyMod(const GF2EX& g, const GF2EXModulus& F, long m)
   { GF2EX x; ProbMinPolyMod(x, g, F, m); NTL_OPT_RETURN(GF2EX, x); }

void IrredPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F);
inline GF2EX IrredPolyMod(const GF2EX& g, const GF2EXModulus& F)
   { GF2EX x; IrredPolyMod(x, g, F); NTL_OPT_RETURN(GF2EX, x); }

void IrredPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F, long m);
inline GF2EX IrredPolyMod(const GF2EX& g, const GF2EXModulus& F, long m)
   { GF2EX x; IrredPolyMod(x, g, F, m); NTL_OPT_RETURN(GF2EX, x); }


struct GF2EXTransMultiplier {
   GF2EX f0, fbi, b;
   long shamt, shamt_fbi, shamt_b;
};

void build(GF2EXTransMultiplier& B, const GF2EX& b, const GF2EXModulus& F);

void TransMulMod(GF2EX& x, const GF2EX& a, const GF2EXTransMultiplier& B,
               const GF2EXModulus& F);

void UpdateMap(vec_GF2E& x, const vec_GF2E& a, 
         const GF2EXTransMultiplier& B, const GF2EXModulus& F);

inline vec_GF2E UpdateMap(const vec_GF2E& a,
         const GF2EXTransMultiplier& B, const GF2EXModulus& F)
   { vec_GF2E x; UpdateMap(x, a, B, F); NTL_OPT_RETURN(vec_GF2E, x); }

void ProjectPowers(vec_GF2E& x, const vec_GF2E& a, long k, 
                   const GF2EXArgument& H, const GF2EXModulus& F);
inline vec_GF2E ProjectPowers(const vec_GF2E& a, long k, 
                   const GF2EXArgument& H, const GF2EXModulus& F)
   { vec_GF2E x; ProjectPowers(x, a, k, H, F); NTL_OPT_RETURN(vec_GF2E, x); }

void ProjectPowers(vec_GF2E& x, const vec_GF2E& a, long k, const GF2EX& h, 
                   const GF2EXModulus& F);
inline vec_GF2E ProjectPowers(const vec_GF2E& a, long k, 
                   const GF2EX& H, const GF2EXModulus& F)
   { vec_GF2E x; ProjectPowers(x, a, k, H, F); NTL_OPT_RETURN(vec_GF2E, x); }

inline void project(GF2E& x, const vec_GF2E& a, const GF2EX& b)
   { InnerProduct(x, a, b.rep); }

inline GF2E project(const vec_GF2E& a, const GF2EX& b)
   { GF2E x; InnerProduct(x, a, b.rep); NTL_OPT_RETURN(GF2E, x); }

/**********************************************************

         Modular Composition and Minimal Polynomials
                         in towers

***********************************************************/

// composition

void CompTower(GF2EX& x, const GF2X& g, const GF2EXArgument& A,
             const GF2EXModulus& F);

inline GF2EX CompTower(const GF2X& g, const GF2EXArgument& A,
             const GF2EXModulus& F)
   { GF2EX x; CompTower(x, g, A, F); NTL_OPT_RETURN(GF2EX, x); }

void CompTower(GF2EX& x, const GF2X& g, const GF2EX& h,
             const GF2EXModulus& F);

inline GF2EX CompTower(const GF2X& g, const GF2EX& h,
             const GF2EXModulus& F)
   { GF2EX x; CompTower(x, g, h, F); NTL_OPT_RETURN(GF2EX, x); }

// prob min poly

void ProbMinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F,
                      long m);

inline GF2X ProbMinPolyTower(const GF2EX& g, const GF2EXModulus& F,
                      long m)
   { GF2X x; ProbMinPolyTower(x, g, F, m); NTL_OPT_RETURN(GF2X, x); }

inline void ProbMinPolyTower(GF2X& h, const GF2EX& g, 
                             const GF2EXModulus& F)
   { ProbMinPolyTower(h, g, F, deg(F)*GF2E::degree()); }

inline GF2X ProbMinPolyTower(const GF2EX& g, const GF2EXModulus& F)
   { GF2X x; ProbMinPolyTower(x, g, F); NTL_OPT_RETURN(GF2X, x); }


// min poly


void MinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F,
                      long m);

inline GF2X MinPolyTower(const GF2EX& g, const GF2EXModulus& F,
                      long m)
   { GF2X x; MinPolyTower(x, g, F, m); NTL_OPT_RETURN(GF2X, x); }

inline void MinPolyTower(GF2X& h, const GF2EX& g, 
                             const GF2EXModulus& F)
   { MinPolyTower(h, g, F, deg(F)*GF2E::degree()); }

inline GF2X MinPolyTower(const GF2EX& g, const GF2EXModulus& F)
   { GF2X x; MinPolyTower(x, g, F); NTL_OPT_RETURN(GF2X, x); }

// irred poly


void IrredPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F,
                      long m);

inline GF2X IrredPolyTower(const GF2EX& g, const GF2EXModulus& F,
                      long m)
   { GF2X x; IrredPolyTower(x, g, F, m); NTL_OPT_RETURN(GF2X, x); }

inline void IrredPolyTower(GF2X& h, const GF2EX& g, 
                             const GF2EXModulus& F)
   { IrredPolyTower(h, g, F, deg(F)*GF2E::degree()); }

inline GF2X IrredPolyTower(const GF2EX& g, const GF2EXModulus& F)
   { GF2X x; IrredPolyTower(x, g, F); NTL_OPT_RETURN(GF2X, x); }



/*****************************************************************

                   Traces, norms, resultants

******************************************************************/

void TraceVec(vec_GF2E& S, const GF2EX& f);

inline vec_GF2E TraceVec(const GF2EX& f)
   { vec_GF2E x; TraceVec(x, f); NTL_OPT_RETURN(vec_GF2E, x); }


void TraceMod(GF2E& x, const GF2EX& a, const GF2EXModulus& F);

inline GF2E TraceMod(const GF2EX& a, const GF2EXModulus& F)
   { GF2E x; TraceMod(x, a, F); NTL_OPT_RETURN(GF2E, x); }

void TraceMod(GF2E& x, const GF2EX& a, const GF2EX& f);

inline GF2E TraceMod(const GF2EX& a, const GF2EX& f)
   { GF2E x; TraceMod(x, a, f); NTL_OPT_RETURN(GF2E, x); }





void NormMod(GF2E& x, const GF2EX& a, const GF2EX& f);

inline GF2E NormMod(const GF2EX& a, const GF2EX& f)
   { GF2E x; NormMod(x, a, f); NTL_OPT_RETURN(GF2E, x); }

void resultant(GF2E& rres, const GF2EX& a, const GF2EX& b);

inline GF2E resultant(const GF2EX& a, const GF2EX& b)
   { GF2E x; resultant(x, a, b); NTL_OPT_RETURN(GF2E, x); }


NTL_CLOSE_NNS 

#endif