~/ntl-11.4.2/doc/ZZ_pX.cpp.html

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

    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
       explicit ZZ_pX(const ZZ_p& a); // promotion
       explicit ZZ_pX(long a); // promotion

       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(); // destructor

       ZZ_pX(ZZ_pX&& a);
       // move constructor (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set

    #ifndef NTL_DISABLE_MOVE_ASSIGN
       ZZ_pX& operator=(ZZ_pX&& a);
       // move assignment (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set
    #endif

       ZZ_pX(INIT_MONO_TYPE, long i, const 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, long c);
       // initialize to X^i, invoke as ZZ_pX(INIT_MONO, i)

       // typedefs to aid in generic programming
       typedef zz_p coeff_type;
       typedef zz_pE residue_type;
       typedef zz_pXModulus modulus_type;
       typedef zz_pXMultiplier multiplier_type;
       typedef fftRep fft_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, 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

    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);

    // 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

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

    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
    // NOTE: thread boosting enabled only if x does not alias a

    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

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

    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 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 thread_local 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.

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

            Faster Composition and Projection with Pre-Conditioning

    A new, experimental version of composition with preconditioning.
    This interface was introduced in NTL v10.3.0, and it should be
    considered a preliminary interface and subject to change.

    The class ZZ_pXNewArgument is similar to ZZ_pXArgument, but with
    a different internal layout.  Copy constructor and assignment work.

    Note that all NTL modular composition and power projection routines,
    as well as other routines that use modular composition power projection
    internally, now use this new class.

    Note also that these routines do not pay any attention to the
    ZZ_pXArgBound variable.

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

    class ZZ_pXNewArgument {
     // ...
    };

    void build(ZZ_pXNewArgument& H, const ZZ_pX& h, const ZZ_pXModulus& F, long m);
    // same functionality as the corresponding ZZ_pXArgument-based routine

    void CompMod(ZZ_pX& x, const ZZ_pX& g, const ZZ_pXNewArgument& H,
                 const ZZ_pXModulus& F);
    // same functionality as the corresponding ZZ_pXArgument-based routine

    void ProjectPowers(vec_ZZ_p& x, const vec_ZZ_p& a, long k,
                       const ZZ_pXNewArgument& H, const ZZ_pXModulus& F);
    // same functionality as the corresponding ZZ_pXArgument-based routine

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

                                  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

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

    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& 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")

    void ZZ_pX::swap(ZZ_pX& x);
    // swap member function

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