ZZ_pEX.txt

/**************************************************************************\

MODULE: ZZ_pEX

SUMMARY:

The class ZZ_pEX represents polynomials over ZZ_pE,
and so can be used, for example, for arithmentic in GF(p^n)[X].
However, except where mathematically necessary (e.g., GCD computations),
ZZ_pE need not be a field.

\**************************************************************************/

#include <NTL/ZZ_pE.h>
#include <NTL/vec_ZZ_pE.h>

class ZZ_pEX {
public:

   ZZ_pEX(); // initial value 0

   ZZ_pEX(const ZZ_pEX& a); // copy

   explicit ZZ_pEX(const ZZ_pE& a); // promotion
   explicit ZZ_pEX(const ZZ_p& a); 
   explicit ZZ_pEX(long a); 

   ZZ_pEX& operator=(const ZZ_pEX& a); // assignment
   ZZ_pEX& operator=(const ZZ_pE& a);
   ZZ_pEX& operator=(const ZZ_p& a);
   ZZ_pEX& operator=(long a);

   ~ZZ_pEX(); // destructor

   ZZ_pEX(INIT_MONO_TYPE, long i, const ZZ_pE& c); 
   ZZ_pEX(INIT_MONO_TYPE, long i, const ZZ_p& c); 
   ZZ_pEX(INIT_MONO_TYPE, long i, long c); 
   // initialize to c*X^i, invoke as ZZ_pEX(INIT_MONO, i, c)

   ZZ_pEX(INIT_MONO_TYPE, long i); 
   // initialize to X^i, invoke as ZZ_pEX(INIT_MONO, i)

   // typedefs to aid in generic programming
   typedef ZZ_pE coeff_type;
   typedef ZZ_pEXModulus modulus_type;

   // ...

};




/**************************************************************************\

                              Accessing coefficients

The degree of a polynomial f is obtained as deg(f),
where the zero polynomial, by definition, has degree -1.

A polynomial f is represented as a coefficient vector.
Coefficients may be accesses in one of two ways.

The safe, high-level method is to call the function
coeff(f, i) to get the coefficient of X^i in the polynomial f,
and to call the function SetCoeff(f, i, a) to set the coefficient
of X^i in f to the scalar a.

One can also access the coefficients more directly via a lower level 
interface.  The coefficient of X^i in f may be accessed using 
subscript notation f[i].  In addition, one may write f.SetLength(n)
to set the length of the underlying coefficient vector to n,
and f.SetMaxLength(n) to allocate space for n coefficients,
without changing the coefficient vector itself.

After setting coefficients using this low-level interface,
one must ensure that leading zeros in the coefficient vector
are stripped afterwards by calling the function f.normalize().

NOTE: the coefficient vector of f may also be accessed directly
as f.rep; however, this is not recommended. Also, for a properly
normalized polynomial f, we have f.rep.length() == deg(f)+1,
and deg(f) >= 0  =>  f.rep[deg(f)] != 0.

\**************************************************************************/



long deg(const ZZ_pEX& a);  // return deg(a); deg(0) == -1.

const ZZ_pE& coeff(const ZZ_pEX& a, long i);
// returns the coefficient of X^i, or zero if i not in range

const ZZ_pE& LeadCoeff(const ZZ_pEX& a);
// returns leading term of a, or zero if a == 0

const ZZ_pE& ConstTerm(const ZZ_pEX& a);
// returns constant term of a, or zero if a == 0

void SetCoeff(ZZ_pEX& x, long i, const ZZ_pE& a);
void SetCoeff(ZZ_pEX& x, long i, const ZZ_p& a);
void SetCoeff(ZZ_pEX& x, long i, long a);
// makes coefficient of X^i equal to a; error is raised if i < 0

void SetCoeff(ZZ_pEX& x, long i);
// makes coefficient of X^i equal to 1;  error is raised if i < 0

void SetX(ZZ_pEX& x); // x is set to the monomial X

long IsX(const ZZ_pEX& a); // test if x = X




ZZ_pE& ZZ_pEX::operator[](long i); 
const ZZ_pE& ZZ_pEX::operator[](long i) const;
// indexing operators: f[i] is the coefficient of X^i ---
// i should satsify i >= 0 and i <= deg(f).
// No range checking (unless NTL_RANGE_CHECK is defined).

void ZZ_pEX::SetLength(long n);
// f.SetLength(n) sets the length of the inderlying coefficient
// vector to n --- after this call, indexing f[i] for i = 0..n-1
// is valid.

void ZZ_pEX::normalize();  
// f.normalize() strips leading zeros from coefficient vector of f

void ZZ_pEX::SetMaxLength(long n);
// f.SetMaxLength(n) pre-allocate spaces for n coefficients.  The
// polynomial that f represents is unchanged.










/**************************************************************************\

                                  Comparison

\**************************************************************************/


long operator==(const ZZ_pEX& a, const ZZ_pEX& b);
long operator!=(const ZZ_pEX& a, const ZZ_pEX& b);

long IsZero(const ZZ_pEX& a); // test for 0
long IsOne(const ZZ_pEX& a); // test for 1

// PROMOTIONS: ==, != promote {long,ZZ_p,ZZ_pE} to ZZ_pEX on (a, b).

/**************************************************************************\

                                   Addition

\**************************************************************************/

// operator notation:

ZZ_pEX operator+(const ZZ_pEX& a, const ZZ_pEX& b);
ZZ_pEX operator-(const ZZ_pEX& a, const ZZ_pEX& b);
ZZ_pEX operator-(const ZZ_pEX& a);

ZZ_pEX& operator+=(ZZ_pEX& x, const ZZ_pEX& a);
ZZ_pEX& operator+=(ZZ_pEX& x, const ZZ_pE& a);
ZZ_pEX& operator+=(ZZ_pEX& x, const ZZ_p& a);
ZZ_pEX& operator+=(ZZ_pEX& x, long a);


ZZ_pEX& operator++(ZZ_pEX& x);  // prefix
void operator++(ZZ_pEX& x, int);  // postfix

ZZ_pEX& operator-=(ZZ_pEX& x, const ZZ_pEX& a);
ZZ_pEX& operator-=(ZZ_pEX& x, const ZZ_pE& a);
ZZ_pEX& operator-=(ZZ_pEX& x, const ZZ_p& a);
ZZ_pEX& operator-=(ZZ_pEX& x, long a);

ZZ_pEX& operator--(ZZ_pEX& x);  // prefix
void operator--(ZZ_pEX& x, int);  // postfix

// procedural versions:

void add(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b); // x = a + b
void sub(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b); // x = a - b 
void negate(ZZ_pEX& x, const ZZ_pEX& a); // x = - a 

// PROMOTIONS: +, -, add, sub promote {long,ZZ_p,ZZ_pE} to ZZ_pEX on (a, b).



/**************************************************************************\

                               Multiplication

\**************************************************************************/

// operator notation:

ZZ_pEX operator*(const ZZ_pEX& a, const ZZ_pEX& b);

ZZ_pEX& operator*=(ZZ_pEX& x, const ZZ_pEX& a);
ZZ_pEX& operator*=(ZZ_pEX& x, const ZZ_pE& a);
ZZ_pEX& operator*=(ZZ_pEX& x, const ZZ_p& a);
ZZ_pEX& operator*=(ZZ_pEX& x, long a);


// procedural versions:


void mul(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b); // x = a * b

void sqr(ZZ_pEX& x, const ZZ_pEX& a); // x = a^2
ZZ_pEX sqr(const ZZ_pEX& a); 

// PROMOTIONS: *, mul promote {long,ZZ_p,ZZ_pE} to ZZ_pEX on (a, b).

void power(ZZ_pEX& x, const ZZ_pEX& a, long e);  // x = a^e (e >= 0)
ZZ_pEX power(const ZZ_pEX& a, long e);


/**************************************************************************\

                               Shift Operations

LeftShift by n means multiplication by X^n
RightShift by n means division by X^n

A negative shift amount reverses the direction of the shift.

\**************************************************************************/

// operator notation:

ZZ_pEX operator<<(const ZZ_pEX& a, long n);
ZZ_pEX operator>>(const ZZ_pEX& a, long n);

ZZ_pEX& operator<<=(ZZ_pEX& x, long n);
ZZ_pEX& operator>>=(ZZ_pEX& x, long n);

// procedural versions:

void LeftShift(ZZ_pEX& x, const ZZ_pEX& a, long n); 
ZZ_pEX LeftShift(const ZZ_pEX& a, long n);

void RightShift(ZZ_pEX& x, const ZZ_pEX& a, long n); 
ZZ_pEX RightShift(const ZZ_pEX& a, long n); 



/**************************************************************************\

                                  Division

\**************************************************************************/

// operator notation:

ZZ_pEX operator/(const ZZ_pEX& a, const ZZ_pEX& b);
ZZ_pEX operator/(const ZZ_pEX& a, const ZZ_pE& b);
ZZ_pEX operator/(const ZZ_pEX& a, const ZZ_p& b);
ZZ_pEX operator/(const ZZ_pEX& a, long b);

ZZ_pEX operator%(const ZZ_pEX& a, const ZZ_pEX& b);

ZZ_pEX& operator/=(ZZ_pEX& x, const ZZ_pEX& a);
ZZ_pEX& operator/=(ZZ_pEX& x, const ZZ_pE& a);
ZZ_pEX& operator/=(ZZ_pEX& x, const ZZ_p& a);
ZZ_pEX& operator/=(ZZ_pEX& x, long a);

ZZ_pEX& operator%=(ZZ_pEX& x, const ZZ_pEX& a);

// procedural versions:


void DivRem(ZZ_pEX& q, ZZ_pEX& r, const ZZ_pEX& a, const ZZ_pEX& b);
// q = a/b, r = a%b

void div(ZZ_pEX& q, const ZZ_pEX& a, const ZZ_pEX& b);
void div(ZZ_pEX& q, const ZZ_pEX& a, const ZZ_pE& b);
void div(ZZ_pEX& q, const ZZ_pEX& a, const ZZ_p& b);
void div(ZZ_pEX& q, const ZZ_pEX& a, long b);
// q = a/b

void rem(ZZ_pEX& r, const ZZ_pEX& a, const ZZ_pEX& b);
// r = a%b

long divide(ZZ_pEX& q, const ZZ_pEX& a, const ZZ_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0

long divide(const ZZ_pEX& a, const ZZ_pEX& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0


/**************************************************************************\

                                   GCD's

These routines are intended for use when ZZ_pE is a field.

\**************************************************************************/


void GCD(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b);
ZZ_pEX GCD(const ZZ_pEX& a, const ZZ_pEX& b); 
// x = GCD(a, b),  x is always monic (or zero if a==b==0).


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


/**************************************************************************\

                                  Input/Output

I/O format:

   [a_0 a_1 ... a_n],

represents the polynomial a_0 + a_1*X + ... + a_n*X^n.

On output, all coefficients will be polynomials of degree < ZZ_pE::degree() and
a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
are arbitrary polynomials which are reduced modulo ZZ_pE::modulus(), 
and leading zeros stripped.

\**************************************************************************/

istream& operator>>(istream& s, ZZ_pEX& x);
ostream& operator<<(ostream& s, const ZZ_pEX& a);


/**************************************************************************\

                              Some utility routines

\**************************************************************************/


void diff(ZZ_pEX& x, const ZZ_pEX& a); // x = derivative of a
ZZ_pEX diff(const ZZ_pEX& a); 

void MakeMonic(ZZ_pEX& x); 
// if x != 0 makes x into its monic associate; LeadCoeff(x) must be
// invertible in this case

void reverse(ZZ_pEX& x, const ZZ_pEX& a, long hi);
ZZ_pEX reverse(const ZZ_pEX& a, long hi);

void reverse(ZZ_pEX& x, const ZZ_pEX& a);
ZZ_pEX reverse(const ZZ_pEX& a);

// x = reverse of a[0]..a[hi] (hi >= -1);
// hi defaults to deg(a) in second version

void VectorCopy(vec_ZZ_pE& x, const ZZ_pEX& a, long n);
vec_ZZ_pE VectorCopy(const ZZ_pEX& a, long n);
// x = copy of coefficient vector of a of length exactly n.
// input is truncated or padded with zeroes as appropriate.




/**************************************************************************\

                             Random Polynomials

\**************************************************************************/

void random(ZZ_pEX& x, long n);
ZZ_pEX random_ZZ_pEX(long n);
// x = random polynomial of degree < n 


/**************************************************************************\

                    Polynomial Evaluation and related problems

\**************************************************************************/


void BuildFromRoots(ZZ_pEX& x, const vec_ZZ_pE& a);
ZZ_pEX BuildFromRoots(const vec_ZZ_pE& a);
// computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()

void eval(ZZ_pE& b, const ZZ_pEX& f, const ZZ_pE& a);
ZZ_pE eval(const ZZ_pEX& f, const ZZ_pE& a);
// b = f(a)

void eval(ZZ_pE& b, const ZZ_pX& f, const ZZ_pE& a);
ZZ_pE eval(const ZZ_pEX& f, const ZZ_pE& a);
// b = f(a); uses ModComp algorithm for ZZ_pX

void eval(vec_ZZ_pE& b, const ZZ_pEX& f, const vec_ZZ_pE& a);
vec_ZZ_pE eval(const ZZ_pEX& f, const vec_ZZ_pE& a);
//  b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()

void interpolate(ZZ_pEX& f, const vec_ZZ_pE& a, const vec_ZZ_pE& b);
ZZ_pEX interpolate(const vec_ZZ_pE& a, const vec_ZZ_pE& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i].  

/**************************************************************************\

                       Arithmetic mod X^n

Required: n >= 0; otherwise, an error is raised.

\**************************************************************************/

void trunc(ZZ_pEX& x, const ZZ_pEX& a, long n); // x = a % X^n
ZZ_pEX trunc(const ZZ_pEX& a, long n); 

void MulTrunc(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b, long n);
ZZ_pEX MulTrunc(const ZZ_pEX& a, const ZZ_pEX& b, long n);
// x = a * b % X^n

void SqrTrunc(ZZ_pEX& x, const ZZ_pEX& a, long n);
ZZ_pEX SqrTrunc(const ZZ_pEX& a, long n);
// x = a^2 % X^n

void InvTrunc(ZZ_pEX& x, const ZZ_pEX& a, long n);
ZZ_pEX InvTrunc(ZZ_pEX& x, const ZZ_pEX& a, long n);
// computes x = a^{-1} % X^m.  Must have ConstTerm(a) invertible.

/**************************************************************************\

                Modular Arithmetic (without pre-conditioning)

Arithmetic mod f.

All inputs and outputs are polynomials of degree less than deg(f), and
deg(f) > 0.


NOTE: if you want to do many computations with a fixed f, use the
ZZ_pEXModulus data structure and associated routines below for better
performance.

\**************************************************************************/

void MulMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b, const ZZ_pEX& f);
ZZ_pEX MulMod(const ZZ_pEX& a, const ZZ_pEX& b, const ZZ_pEX& f);
// x = (a * b) % f

void SqrMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& f);
ZZ_pEX SqrMod(const ZZ_pEX& a, const ZZ_pEX& f);
// x = a^2 % f

void MulByXMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& f);
ZZ_pEX MulByXMod(const ZZ_pEX& a, const ZZ_pEX& f);
// x = (a * X) mod f

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

long InvModStatus(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& 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
ZZ_pEXModulus F for f.  This pre-computes information about f that
speeds up subsequent computations.

As an example, the following routine the product modulo f of a vector
of polynomials.

#include <NTL/ZZ_pEX.h>

void product(ZZ_pEX& x, const vec_ZZ_pEX& v, const ZZ_pEX& f)
{
   ZZ_pEXModulus F(f);
   ZZ_pEX res;
   res = 1;
   long i;
   for (i = 0; i < v.length(); i++)
      MulMod(res, res, v[i], F); 
   x = res;
}

NOTE: A ZZ_pEX may be used wherever a ZZ_pEXModulus is required,
and a ZZ_pEXModulus may be used wherever a ZZ_pEX is required.


\**************************************************************************/

class ZZ_pEXModulus {
public:
   ZZ_pEXModulus(); // initially in an unusable state

   ZZ_pEXModulus(const ZZ_pEX& f); // initialize with f, deg(f) > 0

   ZZ_pEXModulus(const ZZ_pEXModulus&); // copy

   ZZ_pEXModulus& operator=(const ZZ_pEXModulus&); // assignment

   ~ZZ_pEXModulus(); // destructor

   operator const ZZ_pEX& () const; // implicit read-only access to f

   const ZZ_pEX& val() const; // explicit read-only access to f
};

void build(ZZ_pEXModulus& F, const ZZ_pEX& f);
// pre-computes information about f and stores it in F.  Must have
// deg(f) > 0.  Note that the declaration ZZ_pEXModulus F(f) is
// equivalent to ZZ_pEXModulus F; build(F, f).

// In the following, f refers to the polynomial f supplied to the
// build routine, and n = deg(f).


long deg(const ZZ_pEXModulus& F);  // return n=deg(f)

void MulMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEX& b, 
            const ZZ_pEXModulus& F);
ZZ_pEX MulMod(const ZZ_pEX& a, const ZZ_pEX& b, const ZZ_pEXModulus& F);
// x = (a * b) % f; deg(a), deg(b) < n

void SqrMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEXModulus& F);
ZZ_pEX SqrMod(const ZZ_pEX& a, const ZZ_pEXModulus& F);
// x = a^2 % f; deg(a) < n

void PowerMod(ZZ_pEX& x, const ZZ_pEX& a, const ZZ& e, const ZZ_pEXModulus& F);
ZZ_pEX PowerMod(const ZZ_pEX& a, const ZZ& e, const ZZ_pEXModulus& F);

void PowerMod(ZZ_pEX& x, const ZZ_pEX& a, long e, const ZZ_pEXModulus& F);
ZZ_pEX PowerMod(const ZZ_pEX& a, long e, const ZZ_pEXModulus& F);

// x = a^e % f; e >= 0, deg(a) < n.  Uses a sliding window algorithm.
// (e may be negative)

void PowerXMod(ZZ_pEX& x, const ZZ& e, const ZZ_pEXModulus& F);
ZZ_pEX PowerXMod(const ZZ& e, const ZZ_pEXModulus& F);

void PowerXMod(ZZ_pEX& x, long e, const ZZ_pEXModulus& F);
ZZ_pEX PowerXMod(long e, const ZZ_pEXModulus& F);

// x = X^e % f (e may be negative)

void rem(ZZ_pEX& x, const ZZ_pEX& a, const ZZ_pEXModulus& F);
// x = a % f

void DivRem(ZZ_pEX& q, ZZ_pEX& r, const ZZ_pEX& a, const ZZ_pEXModulus& F);
// q = a/f, r = a%f

void div(ZZ_pEX& q, const ZZ_pEX& a, const ZZ_pEXModulus& F);
// q = a/f

// operator notation:

ZZ_pEX operator/(const ZZ_pEX& a, const ZZ_pEXModulus& F);
ZZ_pEX operator%(const ZZ_pEX& a, const ZZ_pEXModulus& F);

ZZ_pEX& operator/=(ZZ_pEX& x, const ZZ_pEXModulus& F);
ZZ_pEX& operator%=(ZZ_pEX& x, const ZZ_pEXModulus& F);



/**************************************************************************\

                             vectors of ZZ_pEX's

\**************************************************************************/


typedef Vec<ZZ_pEX> vec_ZZ_pEX; // backward compatibility



/**************************************************************************\

                              Modular Composition

Modular composition is the problem of computing g(h) mod f for
polynomials f, g, and h.

The algorithm employed is that of Brent & Kung (Fast algorithms for
manipulating formal power series, JACM 25:581-595, 1978), which uses
O(n^{1/2}) modular polynomial multiplications, and O(n^2) scalar
operations.


\**************************************************************************/

void CompMod(ZZ_pEX& x, const ZZ_pEX& g, const ZZ_pEX& h, 
             const ZZ_pEXModulus& F);
ZZ_pEX CompMod(const ZZ_pEX& g, const ZZ_pEX& h, 
                    const ZZ_pEXModulus& F);

// x = g(h) mod f; deg(h) < n

void Comp2Mod(ZZ_pEX& x1, ZZ_pEX& x2, const ZZ_pEX& g1, const ZZ_pEX& g2,
              const ZZ_pEX& h, const ZZ_pEXModulus& F);
// xi = gi(h) mod f (i=1,2); deg(h) < n.


void Comp3Mod(ZZ_pEX& x1, ZZ_pEX& x2, ZZ_pEX& x3, 
              const ZZ_pEX& g1, const ZZ_pEX& g2, const ZZ_pEX& g3,
              const ZZ_pEX& h, const ZZ_pEXModulus& F);
// xi = gi(h) mod f (i=1..3); deg(h) < n.



/**************************************************************************\

                     Composition with Pre-Conditioning

If a single h is going to be used with many g's then you should build
a ZZ_pEXArgument for h, and then use the compose routine below.  The
routine 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.

\**************************************************************************/


struct ZZ_pEXArgument {
   vec_ZZ_pEX H;
};

void build(ZZ_pEXArgument& H, const ZZ_pEX& h, const ZZ_pEXModulus& F, long m);
// Pre-Computes information about h.  m > 0, deg(h) < n.

void CompMod(ZZ_pEX& x, const ZZ_pEX& g, const ZZ_pEXArgument& H, 
             const ZZ_pEXModulus& F);

ZZ_pEX CompMod(const ZZ_pEX& g, const ZZ_pEXArgument& H, 
                    const ZZ_pEXModulus& F);

extern long ZZ_pEXArgBound;

// Initially 0.  If this is set to a value greater than zero, then
// composition routines will allocate a table of no than about
// ZZ_pEXArgBound KB.  Setting this value affects all compose routines
// and the power projection and minimal polynomial routines below, 
// and indirectly affects many routines in ZZ_pEXFactoring.

/**************************************************************************\

                     power projection routines

\**************************************************************************/

void project(ZZ_pE& x, const ZZ_pEVector& a, const ZZ_pEX& b);
ZZ_pE project(const ZZ_pEVector& a, const ZZ_pEX& b);
// x = inner product of a with coefficient vector of b


void ProjectPowers(vec_ZZ_pE& x, const vec_ZZ_pE& a, long k,
                   const ZZ_pEX& h, const ZZ_pEXModulus& F);

vec_ZZ_pE ProjectPowers(const vec_ZZ_pE& a, long k,
                   const ZZ_pEX& h, const ZZ_pEXModulus& F);

// Computes the vector

//    project(a, 1), project(a, h), ..., project(a, h^{k-1} % f).  

// This operation is the "transpose" of the modular composition operation.

void ProjectPowers(vec_ZZ_pE& x, const vec_ZZ_pE& a, long k,
                   const ZZ_pEXArgument& H, const ZZ_pEXModulus& F);

vec_ZZ_pE ProjectPowers(const vec_ZZ_pE& a, long k,
                   const ZZ_pEXArgument& H, const ZZ_pEXModulus& F);

// same as above, but uses a pre-computed ZZ_pEXArgument


class ZZ_pEXTransMultiplier { /* ... */ };

void build(ZZ_pEXTransMultiplier& B, const ZZ_pEX& b, const ZZ_pEXModulus& F);

void UpdateMap(vec_ZZ_pE& x, const vec_ZZ_pE& a,
               const ZZ_pEXMultiplier& B, const ZZ_pEXModulus& F);

vec_ZZ_pE UpdateMap(const vec_ZZ_pE& a,
               const ZZ_pEXMultiplier& B, const ZZ_pEXModulus& F);

// Computes the vector

//    project(a, b), project(a, (b*X)%f), ..., project(a, (b*X^{n-1})%f)

// Required: a.length() <= deg(F), deg(b) < deg(F).
// This is "transposed" MulMod by B.
// Input may have "high order" zeroes stripped.
// Output always has high order zeroes stripped.


/**************************************************************************\

                              Minimum Polynomials

These routines should be used only when ZZ_pE is a field.

All of these routines implement the algorithm from [Shoup, J. Symbolic
Comp. 17:371-391, 1994] and [Shoup, J. Symbolic Comp. 20:363-397,
1995], based on transposed modular composition and the
Berlekamp/Massey algorithm.

\**************************************************************************/


void MinPolySeq(ZZ_pEX& h, const vec_ZZ_pE& a, long m);
ZZ_pEX MinPolySeq(const vec_ZZ_pE& a, long m);
// computes the minimum polynomial of a linealy generated sequence; m
// is a bound on the degree of the polynomial; required: a.length() >=
// 2*m


void ProbMinPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);
ZZ_pEX ProbMinPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void ProbMinPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);
ZZ_pEX ProbMinPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// computes the monic minimal polynomial if (g mod f).  m = a bound on
// the degree of the minimal polynomial; in the second version, this
// argument defaults to n.  The algorithm is probabilistic, always
// returns a divisor of the minimal polynomial, and returns a proper
// divisor with probability at most m/2^{ZZ_pE::degree()}.

void MinPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);
ZZ_pEX MinPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void MinPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);
ZZ_pEX MinPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// same as above, but guarantees that result is correct

void IrredPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);
ZZ_pEX IrredPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void IrredPolyMod(ZZ_pEX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);
ZZ_pEX IrredPolyMod(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// same as above, but assumes that f is irreducible, or at least that
// the minimal poly of g is itself irreducible.  The algorithm is
// deterministic (and is always correct).

/**************************************************************************\

           Composition and Minimal Polynomials in towers

These are implementations of algorithms that will be described
and analyzed in a forthcoming paper.

The routines require that p is prime, but ZZ_pE need not be a field.

\**************************************************************************/


void CompTower(ZZ_pEX& x, const ZZ_pX& g, const ZZ_pEXArgument& h,
             const ZZ_pEXModulus& F);

ZZ_pEX CompTower(const ZZ_pX& g, const ZZ_pEXArgument& h,
             const ZZ_pEXModulus& F);

void CompTower(ZZ_pEX& x, const ZZ_pX& g, const ZZ_pEX& h,
             const ZZ_pEXModulus& F);

ZZ_pEX CompTower(const ZZ_pX& g, const ZZ_pEX& h,
             const ZZ_pEXModulus& F);


// x = g(h) mod f


void ProbMinPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F,
                      long m);

ZZ_pX ProbMinPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void ProbMinPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);

ZZ_pX ProbMinPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// Uses a probabilistic algorithm to compute the minimal
// polynomial of (g mod f) over ZZ_p.
// The parameter m is a bound on the degree of the minimal polynomial
// (default = deg(f)*ZZ_pE::degree()).
// In general, the result will be a divisor of the true minimimal
// polynomial.  For correct results, use the MinPoly routines below.



void MinPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

ZZ_pX MinPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void MinPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);

ZZ_pX MinPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// Same as above, but result is always correct.


void IrredPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

ZZ_pX IrredPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F, long m);

void IrredPolyTower(ZZ_pX& h, const ZZ_pEX& g, const ZZ_pEXModulus& F);

ZZ_pX IrredPolyTower(const ZZ_pEX& g, const ZZ_pEXModulus& F);

// Same as above, but assumes the minimal polynomial is
// irreducible, and uses a slightly faster, deterministic algorithm.


/**************************************************************************\

                   Traces, norms, resultants

\**************************************************************************/


void TraceMod(ZZ_pE& x, const ZZ_pEX& a, const ZZ_pEXModulus& F);
ZZ_pE TraceMod(const ZZ_pEX& a, const ZZ_pEXModulus& F);

void TraceMod(ZZ_pE& x, const ZZ_pEX& a, const ZZ_pEX& f);
ZZ_pE TraceMod(const ZZ_pEX& a, const ZZ_pEXModulus& f);
// x = Trace(a mod f); deg(a) < deg(f)


void TraceVec(vec_ZZ_pE& S, const ZZ_pEX& f);
vec_ZZ_pE TraceVec(const ZZ_pEX& f);
// S[i] = Trace(X^i mod f), i = 0..deg(f)-1; 0 < deg(f)

// The above trace routines implement the asymptotically fast trace
// algorithm from [von zur Gathen and Shoup, Computational Complexity,
// 1992].

void NormMod(ZZ_pE& x, const ZZ_pEX& a, const ZZ_pEX& f);
ZZ_pE NormMod(const ZZ_pEX& a, const ZZ_pEX& f);
// x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)

void resultant(ZZ_pE& x, const ZZ_pEX& a, const ZZ_pEX& b);
ZZ_pE resultant(const ZZ_pEX& a, const ZZ_pEX& b);
// x = resultant(a, b)

// NormMod and resultant require that ZZ_pE is a field.




/**************************************************************************\

                           Miscellany


\**************************************************************************/


void clear(ZZ_pEX& x) // x = 0
void set(ZZ_pEX& x); // x = 1

void ZZ_pEX::kill();
// f.kill() sets f to 0 and frees all memory held by f.  Equivalent to
// f.rep.kill().

ZZ_pEX::ZZ_pEX(INIT_SIZE_TYPE, long n);
// ZZ_pEX(INIT_SIZE, n) initializes to zero, but space is pre-allocated
// for n coefficients

static const ZZ_pEX& zero();
// ZZ_pEX::zero() is a read-only reference to 0

void ZZ_pEX::swap(ZZ_pEX& x);
void swap(ZZ_pEX& x, ZZ_pEX& y); 
// swap (via "pointer swapping")


ZZ_pEX::ZZ_pEX(long i, const ZZ_pE& c); 
ZZ_pEX::ZZ_pEX(long i, const ZZ_p& c); 
ZZ_pEX::ZZ_pEX(long i, long c); 
// initialize to c*X^i, provided for backward compatibility