source: git/ntl/doc/lzz_pX.txt @ 26e030

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

MODULE: zz_pX

SUMMARY:

The class zz_pX implements polynomial arithmetic modulo p.

Polynomial arithmetic is implemented using a combination of classical
routines, Karatsuba, and FFT.

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

#include "zz_p.h"
#include "vec_zz_p.h"

class zz_pX {
public:

   zz_pX(); // initial value 0

   zz_pX(const zz_pX& a); // copy

   zz_pX& operator=(const zz_pX& a); // assignment
   zz_pX& operator=(zz_p a); 
   zz_pX& operator=(long a); 

   ~zz_pX(); // destructor

   zz_pX(long i, zz_p c); // initialize to X^i*c
   zz_pX(long i, long c); 
   
};





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

                                  Comparison

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


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

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

// PROMOTIONS: operators ==, != promote {long, zz_p} to zz_pX on (a, b)


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

                                   Addition

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

// operator notation:

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

zz_pX operator-(const zz_pX& a); // unary -

zz_pX& operator+=(zz_pX& x, const zz_pX& a);
zz_pX& operator+=(zz_pX& x, zz_p a);
zz_pX& operator+=(zz_pX& x, long a);

zz_pX& operator-=(zz_pX& x, const zz_pX& a);
zz_pX& operator-=(zz_pX& x, zz_p a);
zz_pX& operator-=(zz_pX& x, long a);

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

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

// procedural versions:


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

// PROMOTIONS: binary +, - and procedures add, sub promote {long, zz_p}
// to zz_pX on (a, b).


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

                               Multiplication

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

// operator notation:

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

zz_pX& operator*=(zz_pX& x, const zz_pX& a);
zz_pX& operator*=(zz_pX& x, zz_p a);
zz_pX& operator*=(zz_pX& x, long a);

// procedural versions:


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

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

// PROMOTIONS: operator * and procedure mul promote {long, zz_p} to zz_pX
// on (a, b).

void power(zz_pX& x, const zz_pX& a, long e);  // x = a^e (e >= 0)
zz_pX power(const zz_pX& 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_pX operator<<(const zz_pX& a, long n);
zz_pX operator>>(const zz_pX& a, long n);

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

// procedural versions:

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

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



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

                                  Division

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

// operator notation:

zz_pX operator/(const zz_pX& a, const zz_pX& b);
zz_pX operator%(const zz_pX& a, const zz_pX& b);

zz_pX& operator/=(zz_pX& x, const zz_pX& a);
zz_pX& operator/=(zz_pX& x, zz_p a);
zz_pX& operator/=(zz_pX& x, long a);

zz_pX& operator%=(zz_pX& x, const zz_pX& b);


// procedural versions:


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

void div(zz_pX& q, const zz_pX& a, const zz_pX& b);
// q = a/b

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

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

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

// PROMOTIONS: operator / and procedure div promote {long, zz_p} to zz_pX
// on (a, b).


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

                                   GCD's

These routines are intended for use when p is prime.

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


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


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


// NOTE: A classical algorithm is used, switching over to a
// "half-GCD" algorithm for large degree


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

                                  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 integers between 0 and p-1, amd
a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
are arbitrary integers which are reduced modulo p, and leading zeros
stripped.

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

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


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

                              Some utility routines

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

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

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

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

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

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

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

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

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

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


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

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

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

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

void VectorCopy(vec_zz_p& x, const zz_pX& a, long n);
vec_zz_p VectorCopy(const zz_pX& 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_pX& x, long n);
zz_pX random_zz_pX(long n);
// x = random polynomial of degree < n 


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

                    Polynomial Evaluation and related problems

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


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

void eval(zz_p& b, const zz_pX& f, zz_p a);
zz_p eval(const zz_pX& f, zz_p a);
// b = f(a)

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

void interpolate(zz_pX& f, const vec_zz_p& a, const vec_zz_p& b);
zz_pX interpolate(const vec_zz_p& a, const vec_zz_p& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i].  p should
// be prime.

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

                       Arithmetic mod X^n

It is required that n >= 0, otherwise an error is raised.

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

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

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

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

void InvTrunc(zz_pX& x, const zz_pX& a, long n);
zz_pX InvTrunc(const zz_pX& a, long n);
// computes x = a^{-1} % X^n.  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_pXModulus data structure and associated routines below for better
performance.

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

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

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

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

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

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


// for modular exponentiation, see below



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

                     Modular Arithmetic with Pre-Conditioning

If you need to do a lot of arithmetic modulo a fixed f, build
zz_pXModulus F for f.  This pre-computes information about f that
speeds up subsequent computations. Required: deg(f) > 0 and LeadCoeff(f)
invertible.

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

#include "zz_pX.h"

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


Note that automatic conversions are provided so that a zz_pX can
be used wherever a zz_pXModulus is required, and a zz_pXModulus
can be used wherever a zz_pX is required.



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

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

   zz_pXModulus(const zz_pXModulus&);  // copy

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

   zz_pXModulus(const zz_pX& f); // initialize with f, deg(f) > 0

   operator const zz_pX& () const; 
   // read-only access to f, implicit conversion operator

   const zz_pX& val() const; 
   // read-only access to f, explicit notation

};

void build(zz_pXModulus& F, const zz_pX& f);
// pre-computes information about f and stores it in F.
// Note that the declaration zz_pXModulus F(f) is equivalent to
// zz_pXModulus 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_pXModulus& F);  // return deg(f)

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

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

void PowerMod(zz_pX& x, const zz_pX& a, const ZZ& e, const zz_pXModulus& F);
zz_pX PowerMod(const zz_pX& a, const ZZ& e, const zz_pXModulus& F);

void PowerMod(zz_pX& x, const zz_pX& a, long e, const zz_pXModulus& F);
zz_pX PowerMod(const zz_pX& a, long e, const zz_pXModulus& F);

// x = a^e % f; deg(a) < n (e may be negative)

void PowerXMod(zz_pX& x, const ZZ& e, const zz_pXModulus& F);
zz_pX PowerXMod(const ZZ& e, const zz_pXModulus& F);

void PowerXMod(zz_pX& x, long e, const zz_pXModulus& F);
zz_pX PowerXMod(long e, const zz_pXModulus& F);

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

void PowerXPlusAMod(zz_pX& x, const zz_p& a, const ZZ& e, 
                    const zz_pXModulus& F);

zz_pX PowerXPlusAMod(const zz_p& a, const ZZ& e, 
                           const zz_pXModulus& F);

void PowerXPlusAMod(zz_pX& x, const zz_p& a, long e, 
                    const zz_pXModulus& F);

zz_pX PowerXPlusAMod(const zz_p& a, long e, 
                           const zz_pXModulus& F);

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


void rem(zz_pX& x, const zz_pX& a, const zz_pXModulus& F);
// x = a % f

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

void div(zz_pX& q, const zz_pX& a, const zz_pXModulus& F);
// q = a/f

// operator notation:

zz_pX operator/(const zz_pX& a, const zz_pXModulus& F);
zz_pX operator%(const zz_pX& a, const zz_pXModulus& F);

zz_pX& operator/=(zz_pX& x, const zz_pXModulus& F);
zz_pX& operator%=(zz_pX& x, const zz_pXModulus& F);




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


                        More Pre-Conditioning

If you need to compute a * b % f for a fixed b, but for many a's, it
is much more efficient to first build a zz_pXMultiplier B for b, and
then use the MulMod routine below.

Here is an example that multiplies each element of a vector by a fixed
polynomial modulo f.

#include "zz_pX.h"

void mul(vec_zz_pX& v, const zz_pX& b, const zz_pX& f)
{
   zz_pXModulus F(f);
   zz_pXMultiplier B(b, F);
   long i;
   for (i = 0; i < v.length(); i++)
      MulMod(v[i], v[i], B, F);
}

Note that a (trivial) conversion operator from zz_pXMultiplier to zz_pX
is provided, so that a zz_pXMultiplier can be used in a context
where a zz_pX is required.


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


class zz_pXMultiplier {
public:
   zz_pXMultiplier(); // initially zero

   zz_pXMultiplier(const zz_pX& b, const zz_pXModulus& F);
      // initializes with b mod F, where deg(b) < deg(F)

   zz_pXMultiplier(const zz_pXMultiplier&);
   zz_pXMultiplier& operator=(const zz_pXMultiplier&);

   ~zz_pXMultiplier();

   const zz_pX& val() const; // read-only access to b

};

void build(zz_pXMultiplier& B, const zz_pX& b, const zz_pXModulus& F);
// pre-computes information about b and stores it in B; deg(b) <
// deg(F)

void MulMod(zz_pX& x, const zz_pX& a, const zz_pXMultiplier& B,
                                      const zz_pXModulus& F);

zz_pX MulMod(const zz_pX& a, const zz_pXMultiplier& B, 
             const zz_pXModulus& F);

// x = (a * b) % F; deg(a) < deg(F)

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

                             vectors of zz_pX's

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

NTL_vector_decl(zz_pX,vec_zz_pX)
// vec_zz_pX

NTL_eq_vector_decl(zz_pX,vec_zz_pX)
// == and !=

NTL_io_vector_decl(zz_pX,vec_zz_pX)
// 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_pX& x, const zz_pX& g, const zz_pX& h, const zz_pXModulus& F);
zz_pX CompMod(const zz_pX& g, const zz_pX& h, const zz_pXModulus& F);
// x = g(h) mod f; deg(h) < n

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

void CompMod3(zz_pX& x1, zz_pX& x2, zz_pX& x3, 
              const zz_pX& g1, const zz_pX& g2, const zz_pX& g3,
              const zz_pX& h, const zz_pXModulus& 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_pXArgument 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_pXArgument {
   vec_zz_pX H;
};

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

void CompMod(zz_pX& x, const zz_pX& g, const zz_pXArgument& H, 
             const zz_pXModulus& F);

zz_pX CompMod(const zz_pX& g, const zz_pXArgument& H, 
             const zz_pXModulus& F);


extern long zz_pXArgBound;

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

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

                     power projection routines

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

void project(zz_p& x, const zz_pVector& a, const zz_pX& b);
zz_p project(const zz_pVector& a, const zz_pX& b);
// x = inner product of a with coefficient vector of b


void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k,
                   const zz_pX& h, const zz_pXModulus& F);

vec_zz_p ProjectPowers(const vec_zz_p& a, long k,
                   const zz_pX& h, const zz_pXModulus& 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.
// Input and output may have "high order" zeroes stripped.

void ProjectPowers(vec_zz_p& x, const vec_zz_p& a, long k,
                   const zz_pXArgument& H, const zz_pXModulus& F);

vec_zz_p ProjectPowers(const vec_zz_p& a, long k,
                   const zz_pXArgument& H, const zz_pXModulus& F);

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


void UpdateMap(vec_zz_p& x, const vec_zz_p& a,
               const zz_pXMultiplier& B, const zz_pXModulus& F);

vec_zz_p UpdateMap(const vec_zz_p& a,
               const zz_pXMultiplier& B, const zz_pXModulus& F);

// Computes the vector

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

// Restriction: a.length() <= deg(F).
// This is "transposed" MulMod by B.
// Input vector may have "high order" zeroes striped.
// The output will always have high order zeroes stripped.


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

                              Minimum Polynomials

These routines should be used with prime p.

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_pX& h, const vec_zz_p& 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_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m);

void ProbMinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX ProbMinPolyMod(const zz_pX& g, const zz_pXModulus& 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/p.

void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F, long m);

void MinPolyMod(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX MinPolyMod(const zz_pX& g, const zz_pXModulus& F);
// same as above, but guarantees that result is correct

void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F, long m);
zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& F, long m);

void IrredPoly(zz_pX& h, const zz_pX& g, const zz_pXModulus& F);
zz_pX IrredPoly(const zz_pX& g, const zz_pXModulus& 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).


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

                   Traces, norms, resultants

These routines should be used with prime p.

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


void TraceMod(zz_p& x, const zz_pX& a, const zz_pXModulus& F);
zz_p TraceMod(const zz_pX& a, const zz_pXModulus& F);

void TraceMod(zz_p& x, const zz_pX& a, const zz_pX& f);
zz_p TraceMod(const zz_pX& a, const zz_pXModulus& f);
// x = Trace(a mod f); deg(a) < deg(f)


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

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

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


void resultant(zz_p& x, const zz_pX& a, const zz_pX& b);
zz_pX resultant(zz_p& x, const zz_pX& a, const zz_pX& b);
// x = resultant(a, b)


void CharPolyMod(zz_pX& g, const zz_pX& a, const zz_pX& f);
zz_pX CharPolyMod(const zz_pX& a, const zz_pX& f);
// g = charcteristic polynomial of (a mod f); 0 < deg(f), deg(g) <
// deg(f).  This routine works for arbitrary f.  For irreducible f,
// is it faster to use IrredPolyMod, and then exponentiate as
// necessary, since in this case the characterstic polynomial
// is a power of the minimal polynomial.


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

                           Miscellany

A zz_pX f is represented as a vec_zz_p, 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_pX& x) // x = 0
void set(zz_pX& x); // x = 1

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

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

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

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

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

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