source: git/ntl/doc/ZZ_pX.txt @ 6ce030f

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

MODULE: ZZ_pX

SUMMARY:

The class ZZ_pX implements polynomial arithmetic modulo p.

Polynomial arithmetic is implemented using the FFT, combined with the
Chinese Remainder Theorem.  A more detailed description of the
techniques used here can be found in [Shoup, J. Symbolic
Comp. 20:363-397, 1995].

Small degree polynomials are multiplied either with classical 
or Karatsuba algorithms.

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

#include <NTL/ZZ_p.h>
#include <NTL/vec_ZZ_p.h>

class ZZ_pX {
public:

   ZZ_pX(); // initialize to 0

   ZZ_pX(const ZZ_pX& a); // copy constructor

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

   ZZ_pX(long i, const ZZ_p& c);  // initialize to X^i*c
   ZZ_pX(long i, long c);

   ~ZZ_pX(); // destructor
   
};






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

                                  Comparison

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


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

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

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


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

                                   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, const 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, const 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, const 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_p& b);
ZZ_pX operator/(const ZZ_pX& a, long b);


ZZ_pX& operator/=(ZZ_pX& x, const ZZ_pX& b);
ZZ_pX& operator/=(ZZ_pX& x, const ZZ_p& b);
ZZ_pX& operator/=(ZZ_pX& x, long b);

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

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);
void div(ZZ_pX& q, const ZZ_pX& a, const ZZ_p& b);
void div(ZZ_pX& q, const ZZ_pX& a, long 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


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

                                   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, and
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.

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

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

const ZZ_p& ConstTerm(const ZZ_pX& a);
// read-only reference to constant term of a, or zero if a == 0

void SetCoeff(ZZ_pX& x, long i, const 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); // x = derivative of a
ZZ_pX diff(const ZZ_pX& 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);
// generate a 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, const ZZ_p& a);
ZZ_p eval(const ZZ_pX& f, const 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

All routines require 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^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_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 a
ZZ_pXModulus F for f.  This pre-computes information about f that
speeds up subsequent computations.

It is required that deg(f) > 0 and that LeadCoeff(f) is invertible.

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

#include <NTL/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(const ZZ_pXModulus&);  // copy

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

   ~ZZ_pXModulus();

   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 n=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 <NTL/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);
}

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


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&);  // copy

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

   ~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);

// 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 Comp3Mod(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.

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: must have a.length() <= deg(F).
// This is "transposed" MulMod by B.
// Input may have "high order" zeroes stripped.
// 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);
ZZ_pX MinPolySeq(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 IrredPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F, long m);
ZZ_pX IrredPolyMod(const ZZ_pX& g, const ZZ_pXModulus& F, long m);

void IrredPolyMod(ZZ_pX& h, const ZZ_pX& g, const ZZ_pXModulus& F);
ZZ_pX IrredPolyMod(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 trace 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_p resultant(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;  if f is irreducible,
// it is faster to use the IrredPolyMod routine, and then exponentiate
// if necessary (since in this case the CharPoly is just a power of
// the IrredPoly).


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

                           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& 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")