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

    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
       explicit zz_pX(zz_p a); // promotion
       explicit zz_pX(long a); // promotion

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

       ~zz_pX(); // destructor

       zz_pX(INIT_MONO_TYPE, long i, zz_p c);
       zz_pX(INIT_MONO_TYPE, long i, long c);
       // initialize to c*X^i, invoke as zz_pX(INIT_MONO, i, c)

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

       typedef zz_p coeff_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_pX& a);  // return deg(a); deg(0) == -1.

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

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

    const 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

    zz_p& zz_pX::operator[](long i);
    const zz_p& zz_pX::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_pX::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_pX::normalize();
    // f.normalize() strips leading zeros from coefficient vector of f

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

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

                                      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

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

    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

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

    typedef Vec<zz_pX> vec_zz_pX; // 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_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

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

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

    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")

    zz_pX::zz_pX(long i, zz_p c);
    zz_pX::zz_pX(long i, long c);
     // initialize to c*X^i, provided for backward compatibility