source: git/ntl/doc/lzz_pEX.txt @ 2cfffe

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

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/lzz_pE.h>
#include <NTL/vec_lzz_pE.h>

class zz_pEX {
public:

   zz_pEX(); // initial value 0

   zz_pEX(const zz_pEX& a); // copy

   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(long i, const zz_pE& c); // initilaize to X^i*c
   zz_pEX(long i, const zz_p& c); 
   zz_pEX(long i, long c); 

   
};






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

                                  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

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

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

const zz_pE& coeff(const zz_pEX& a, long i);
// returns a read-only reference to the coefficient of X^i, or zero if
// i not in range

const zz_pE& LeadCoeff(const zz_pEX& a);
// read-only reference to leading term of a, or zero if a == 0

const zz_pE& ConstTerm(const zz_pEX& a);
// read-only reference to 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

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/lzz_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

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

NTL_vector_decl(zz_pEX,vec_zz_pEX)
// vec_zz_pEX

NTL_eq_vector_decl(zz_pEX,vec_zz_pEX)
// == and !=

NTL_io_vector_decl(zz_pEX,vec_zz_pEX)
// I/O operators



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

                              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

A zz_pEX f is represented as a vec_zz_pE, which can be accessed as
f.rep.  The constant term is f.rep[0] and the leading coefficient is
f.rep[f.rep.length()-1], except if f is zero, in which case
f.rep.length() == 0.  Note that the leading coefficient is always
nonzero (unless f is zero).  One can freely access and modify f.rep,
but one should always ensure that the leading coefficient is nonzero,
which can be done by invoking f.normalize().


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


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

void zz_pEX::normalize();  
// f.normalize() strips leading zeros from f.rep.

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

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 swap(zz_pEX& x, zz_pEX& y); 
// swap x and y (via "pointer swapping")