~/ntl-11.4.2/doc/GF2X.cpp.html

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

    MODULE: GF2X

    SUMMARY:

    The class GF2X implements polynomial arithmetic modulo 2.

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

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

    #include <NTL/GF2.h>
    #include <NTL/vec_GF2.h>

    class GF2X {
    public:

       GF2X(); // initial value 0

       GF2X(const GF2X& a); // copy
       explicit GF2X(long a); // promotion
       explicit GF2X(GF2 a); // promotion

       GF2X& operator=(const GF2X& a); // assignment
       GF2X& operator=(GF2 a);
       GF2X& operator=(long a);

       ~GF2X(); // destructor

       GF2X(GF2X&& a);
       // move constructor (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set

    #ifndef NTL_DISABLE_MOVE_ASSIGN
       GF2X& operator=(GF2X&& a);
       // move assignment (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set
    #endif

       GF2X(INIT_MONO_TYPE, long i, GF2 c);
       GF2X(INIT_MONO_TYPE, long i, long c);
       // initialize to c*X^i, invoke as GF2X(INIT_MONO, i, c)

       GF2X(INIT_MONO_TYPE, long i);
       // initialize to c*X^i, invoke as GF2X(INIT_MONO, i)

       // typedefs to aid in generic programming
       typedef GF2 coeff_type;
       typedef GF2E residue_type;
       typedef GF2XModulus modulus_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: unlike other polynomial classes, the coefficient vector
    for GF2X has a special representation, packing coefficients into
    words.  This has two consequences.  First, when using the indexing
    notation on a non-const polynomial f, the return type is ref_GF2,
    rather than GF2&.  For the most part, a ref_GF2 may be used like
    a GF2& --- see GF2.txt for more details.  Second, when applying
    f.SetLength(n) to a polynomial f, this essentially has the effect
    of zeroing out the coefficients of X^i for i >= n.

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

    long deg(const GF2X& a);  // return deg(a); deg(0) == -1.

    const GF2 coeff(const GF2X& a, long i);
    // returns the coefficient of X^i, or zero if i not in range

    const GF2 LeadCoeff(const GF2X& a);
    // returns leading term of a, or zero if a == 0

    const GF2 ConstTerm(const GF2X& a);
    // returns constant term of a, or zero if a == 0

    void SetCoeff(GF2X& x, long i, GF2 a);
    void SetCoeff(GF2X& x, long i, long a);
    // makes coefficient of X^i equal to a; error is raised if i < 0

    void SetCoeff(GF2X& x, long i);
    // makes coefficient of X^i equal to 1;  error is raised if i < 0

    void SetX(GF2X& x); // x is set to the monomial X

    long IsX(const GF2X& a); // test if x = X

    ref_GF2 GF2X::operator[](long i);
    const GF2 GF2X::operator[](long i) const;
    // indexing operators: f[i] is the coefficient of X^i ---
    // i should satsify i >= 0 and i <= deg(f)

    void GF2X::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 GF2X::normalize();
    // f.normalize() strips leading zeros from coefficient vector of f

    void GF2X::SetMaxLength(long n);
    // f.SetMaxLength(n) pre-allocate spaces for n coefficients.  The
    // polynomial that f represents is unchanged.

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

                                      Comparison

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

    long operator==(const GF2X& a, const GF2X& b);
    long operator!=(const GF2X& a, const GF2X& b);

    long IsZero(const GF2X& a); // test for 0
    long IsOne(const GF2X& a); // test for 1

    // PROMOTIONS: operators ==, != promote {long, GF2} to GF2X on (a, b)

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

                                       Addition

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

    // operator notation:

    GF2X operator+(const GF2X& a, const GF2X& b);
    GF2X operator-(const GF2X& a, const GF2X& b);

    GF2X operator-(const GF2X& a); // unary -

    GF2X& operator+=(GF2X& x, const GF2X& a);
    GF2X& operator+=(GF2X& x, GF2 a);
    GF2X& operator+=(GF2X& x, long a);

    GF2X& operator-=(GF2X& x, const GF2X& a);
    GF2X& operator-=(GF2X& x, GF2 a);
    GF2X& operator-=(GF2X& x, long a);

    GF2X& operator++(GF2X& x);  // prefix
    void operator++(GF2X& x, int);  // postfix

    GF2X& operator--(GF2X& x);  // prefix
    void operator--(GF2X& x, int);  // postfix

    // procedural versions:

    void add(GF2X& x, const GF2X& a, const GF2X& b); // x = a + b
    void sub(GF2X& x, const GF2X& a, const GF2X& b); // x = a - b
    void negate(GF2X& x, const GF2X& a); // x = -a

    // PROMOTIONS: binary +, - and procedures add, sub promote {long, GF2}
    // to GF2X on (a, b).

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

                                   Multiplication

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

    // operator notation:

    GF2X operator*(const GF2X& a, const GF2X& b);

    GF2X& operator*=(GF2X& x, const GF2X& a);
    GF2X& operator*=(GF2X& x, GF2 a);
    GF2X& operator*=(GF2X& x, long a);

    // procedural versions:

    void mul(GF2X& x, const GF2X& a, const GF2X& b); // x = a * b

    void sqr(GF2X& x, const GF2X& a); // x = a^2
    GF2X sqr(const GF2X& a);

    // PROMOTIONS: operator * and procedure mul promote {long, GF2} to GF2X
    // on (a, b).

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

                                   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:

    GF2X operator<<(const GF2X& a, long n);
    GF2X operator>>(const GF2X& a, long n);

    GF2X& operator<<=(GF2X& x, long n);
    GF2X& operator>>=(GF2X& x, long n);

    // procedural versions:

    void LeftShift(GF2X& x, const GF2X& a, long n);
    GF2X LeftShift(const GF2X& a, long n);

    void RightShift(GF2X& x, const GF2X& a, long n);
    GF2X RightShift(const GF2X& a, long n);

    void MulByX(GF2X& x, const GF2X& a);
    GF2X MulByX(const GF2X& a);

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

                                      Division

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

    // operator notation:

    GF2X operator/(const GF2X& a, const GF2X& b);
    GF2X operator%(const GF2X& a, const GF2X& b);

    GF2X& operator/=(GF2X& x, const GF2X& a);
    GF2X& operator/=(GF2X& x, GF2 a);
    GF2X& operator/=(GF2X& x, long a);

    GF2X& operator%=(GF2X& x, const GF2X& b);

    // procedural versions:

    void DivRem(GF2X& q, GF2X& r, const GF2X& a, const GF2X& b);
    // q = a/b, r = a%b

    void div(GF2X& q, const GF2X& a, const GF2X& b);
    // q = a/b

    void rem(GF2X& r, const GF2X& a, const GF2X& b);
    // r = a%b

    long divide(GF2X& q, const GF2X& a, const GF2X& b);
    // if b | a, sets q = a/b and returns 1; otherwise returns 0

    long divide(const GF2X& a, const GF2X& b);
    // if b | a, sets q = a/b and returns 1; otherwise returns 0

    // PROMOTIONS: operator / and procedure div promote {long, GF2} to GF2X
    // on (a, b).

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

                                       GCD's

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

    void GCD(GF2X& x, const GF2X& a, const GF2X& b);
    GF2X GCD(const GF2X& a, const GF2X& b);
    // x = GCD(a, b) (zero if a==b==0).

    void XGCD(GF2X& d, GF2X& s, GF2X& t, const GF2X& a, const GF2X& b);
    // d = gcd(a,b), a s + b t = d

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

                                      Input/Output

    I/O format:

       [a_0 a_1 ... a_n],

    represents the polynomial a_0 + a_1*X + ... + a_n*X^n.

    On output, all coefficients will be 0 or 1, and
    a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
    may be arbitrary integers which are reduced modulo 2, and leading zeros
    stripped.

    There is also a more compact hex I/O format.  To output in this
    format, set GF2X::HexOutput to a nonzero value.  On input, if the first
    non-blank character read is 'x' or 'X', then a hex format is assumed.

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

    istream& operator>>(istream& s, GF2X& x);
    ostream& operator<<(ostream& s, const GF2X& a);

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

                                  Some utility routines

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

    void diff(GF2X& x, const GF2X& a);
    GF2X diff(const GF2X& a);
    // x = derivative of a

    void reverse(GF2X& x, const GF2X& a, long hi);
    GF2X reverse(const GF2X& a, long hi);

    void reverse(GF2X& x, const GF2X& a);
    GF2X reverse(const GF2X& a);

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

    void VectorCopy(vec_GF2& x, const GF2X& a, long n);
    vec_GF2 VectorCopy(const GF2X& a, long n);
    // x = copy of coefficient vector of a of length exactly n.
    // input is truncated or padded with zeroes as appropriate.

    // Note that there is also a conversion routine from GF2X to vec_GF2
    // that makes the length of the vector match the number of coefficients
    // of the polynomial.

    long weight(const GF2X& a);
    // returns the # of nonzero coefficients in a

    void GF2XFromBytes(GF2X& x, const unsigned char *p, long n);
    GF2X GF2XFromBytes(const unsigned char *p, long n);
    // conversion from byte vector to polynomial.
    // x = sum(p[i]*X^(8*i), i = 0..n-1), where the bits of p[i] are interpretted
    // as a polynomial in the natural way (i.e., p[i] = 1 is interpretted as 1,
    // p[i] = 2 is interpretted as X, p[i] = 3 is interpretted as X+1, etc.).
    // In the unusual event that characters are wider than 8 bits,
    // only the low-order 8 bits of p[i] are used.

    void BytesFromGF2X(unsigned char *p, const GF2X& a, long n);
    // conversion from polynomial to byte vector.
    // p[0..n-1] are computed so that
    //     a = sum(p[i]*X^(8*i), i = 0..n-1) mod X^(8*n),
    // where the values p[i] are interpretted as polynomials as in GF2XFromBytes
    // above.

    long NumBits(const GF2X& a);
    // returns number of bits of a, i.e., deg(a) + 1.

    long NumBytes(const GF2X& a);
    // returns number of bytes of a, i.e., floor((NumBits(a)+7)/8)

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

                                 Random Polynomials

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

    void random(GF2X& x, long n);
    GF2X random_GF2X(long n);
    // x = random polynomial of degree < n

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

                           Arithmetic mod X^n

    Required: n >= 0; otherwise, an error is raised.

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

    void trunc(GF2X& x, const GF2X& a, long n); // x = a % X^n
    GF2X trunc(const GF2X& a, long n);

    void MulTrunc(GF2X& x, const GF2X& a, const GF2X& b, long n);
    GF2X MulTrunc(const GF2X& a, const GF2X& b, long n);
    // x = a * b % X^n

    void SqrTrunc(GF2X& x, const GF2X& a, long n);
    GF2X SqrTrunc(const GF2X& a, long n);
    // x = a^2 % X^n

    void InvTrunc(GF2X& x, const GF2X& a, long n);
    GF2X InvTrunc(const GF2X& 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
    GF2XModulus data structure and associated routines below for better
    performance.

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

    void MulMod(GF2X& x, const GF2X& a, const GF2X& b, const GF2X& f);
    GF2X MulMod(const GF2X& a, const GF2X& b, const GF2X& f);
    // x = (a * b) % f

    void SqrMod(GF2X& x, const GF2X& a, const GF2X& f);
    GF2X SqrMod(const GF2X& a, const GF2X& f);
    // x = a^2 % f

    void MulByXMod(GF2X& x, const GF2X& a, const GF2X& f);
    GF2X MulByXMod(const GF2X& a, const GF2X& f);
    // x = (a * X) mod f

    void InvMod(GF2X& x, const GF2X& a, const GF2X& f);
    GF2X InvMod(const GF2X& a, const GF2X& f);
    // x = a^{-1} % f, error is a is not invertible

    long InvModStatus(GF2X& x, const GF2X& a, const GF2X& 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
    GF2XModulus F for f.  This pre-computes information about f that
    speeds up subsequent computations.

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

    #include <NTL/GF2X.h>

    void product(GF2X& x, const vec_GF2X& v, const GF2X& f)
    {
       GF2XModulus F(f);
       GF2X 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 GF2X can
    be used wherever a GF2XModulus is required, and a GF2XModulus
    can be used wherever a GF2X is required.

    The GF2XModulus routines optimize several important special cases:

      - f = X^n + X^k + 1, where k <= min((n+1)/2, n-NTL_BITS_PER_LONG)

      - f = X^n + X^{k_3} + X^{k_2} + X^{k_1} + 1,
          where k_3 <= min((n+1)/2, n-NTL_BITS_PER_LONG)

      - f = X^n + g, where deg(g) is small

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

    class GF2XModulus {
    public:
       GF2XModulus(); // initially in an unusable state
       ~GF2XModulus();

       GF2XModulus(const GF2XModulus&);  // copy

       GF2XModulus& operator=(const GF2XModulus&);   // assignment

       GF2XModulus(const GF2X& f); // initialize with f, deg(f) > 0

       operator const GF2X& () const;
       // read-only access to f, implicit conversion operator

       const GF2X& val() const;
       // read-only access to f, explicit notation

       long WordLength() const;
       // returns word-length of resisues
    };

    void build(GF2XModulus& F, const GF2X& f);
    // pre-computes information about f and stores it in F; deg(f) > 0.
    // Note that the declaration GF2XModulus F(f) is equivalent to
    // GF2XModulus 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 GF2XModulus& F);  // return deg(f)

    void MulMod(GF2X& x, const GF2X& a, const GF2X& b, const GF2XModulus& F);
    GF2X MulMod(const GF2X& a, const GF2X& b, const GF2XModulus& F);
    // x = (a * b) % f; deg(a), deg(b) < n

    void SqrMod(GF2X& x, const GF2X& a, const GF2XModulus& F);
    GF2X SqrMod(const GF2X& a, const GF2XModulus& F);
    // x = a^2 % f; deg(a) < n

    void MulByXMod(GF2X& x, const GF2X& a, const GF2XModulus& F);
    GF2X MulByXMod(const GF2X& a, const GF2XModulus& F);
    // x = (a * X) mod F

    void PowerMod(GF2X& x, const GF2X& a, const ZZ& e, const GF2XModulus& F);
    GF2X PowerMod(const GF2X& a, const ZZ& e, const GF2XModulus& F);

    void PowerMod(GF2X& x, const GF2X& a, long e, const GF2XModulus& F);
    GF2X PowerMod(const GF2X& a, long e, const GF2XModulus& F);

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

    void PowerXMod(GF2X& x, const ZZ& e, const GF2XModulus& F);
    GF2X PowerXMod(const ZZ& e, const GF2XModulus& F);

    void PowerXMod(GF2X& x, long e, const GF2XModulus& F);
    GF2X PowerXMod(long e, const GF2XModulus& F);

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

    void rem(GF2X& x, const GF2X& a, const GF2XModulus& F);
    // x = a % f

    void DivRem(GF2X& q, GF2X& r, const GF2X& a, const GF2XModulus& F);
    // q = a/f, r = a%f

    void div(GF2X& q, const GF2X& a, const GF2XModulus& F);
    // q = a/f

    // operator notation:

    GF2X operator/(const GF2X& a, const GF2XModulus& F);
    GF2X operator%(const GF2X& a, const GF2XModulus& F);

    GF2X& operator/=(GF2X& x, const GF2XModulus& F);
    GF2X& operator%=(GF2X& x, const GF2XModulus& F);

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

                                 vectors of GF2X's

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

    typedef Vec<GF2X> vec_GF2X; // 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(GF2X& x, const GF2X& g, const GF2X& h, const GF2XModulus& F);
    GF2X CompMod(const GF2X& g, const GF2X& h, const GF2XModulus& F);
    // x = g(h) mod f; deg(h) < n

    void Comp2Mod(GF2X& x1, GF2X& x2, const GF2X& g1, const GF2X& g2,
                  const GF2X& h, const GF2XModulus& F);
    // xi = gi(h) mod f (i=1,2), deg(h) < n.

    void CompMod3(GF2X& x1, GF2X& x2, GF2X& x3,
                  const GF2X& g1, const GF2X& g2, const GF2X& g3,
                  const GF2X& h, const GF2XModulus& 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 GF2XArgument 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 GF2XArgument {
       vec_GF2X H;
    };

    void build(GF2XArgument& H, const GF2X& h, const GF2XModulus& F, long m);
    // Pre-Computes information about h.  m > 0, deg(h) < n

    void CompMod(GF2X& x, const GF2X& g, const GF2XArgument& H,
                 const GF2XModulus& F);

    GF2X CompMod(const GF2X& g, const GF2XArgument& H,
                 const GF2XModulus& F);

    extern thread_local long GF2XArgBound;

    // Initially 0.  If this is set to a value greater than zero, then
    // composition routines will allocate a table of no than about
    // GF2XArgBound KB.  Setting this value affects all compose routines
    // and the power projection and minimal polynomial routines below,
    // and indirectly affects many routines in GF2XFactoring.

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

                         Power Projection routines

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

    void project(GF2& x, const vec_GF2& a, const GF2X& b);
    GF2 project(const vec_GF2& a, const GF2X& b);
    // x = inner product of a with coefficient vector of b

    void ProjectPowers(vec_GF2& x, const vec_GF2& a, long k,
                       const GF2X& h, const GF2XModulus& F);

    vec_GF2 ProjectPowers(const vec_GF2& a, long k,
                       const GF2X& h, const GF2XModulus& F);

    // Computes the vector

    //   (project(a, 1), project(a, h), ..., project(a, h^{k-1} % f).

    // Restriction: must have a.length <= deg(F) and deg(h) < deg(F).
    // This operation is really the "transpose" of the modular composition
    // operation.

    void ProjectPowers(vec_GF2& x, const vec_GF2& a, long k,
                       const GF2XArgument& H, const GF2XModulus& F);

    vec_GF2 ProjectPowers(const vec_GF2& a, long k,
                       const GF2XArgument& H, const GF2XModulus& F);

    // same as above, but uses a pre-computed GF2XArgument

    // lower-level routines for transposed modular multiplication:

    class GF2XTransMultiplier { /* ... */ };

    void build(GF2XTransMultiplier& B, const GF2X& b, const GF2XModulus& F);

    // build a GF2XTransMultiplier to use in the following routine:

    void UpdateMap(vec_GF2& x, const vec_GF2& a, const GF2XTransMultiplier& B,
             const GF2XModulus& F);

    vec_GF2 UpdateMap(const vec_GF2& a, const GF2XTransMultiplier& B,
             const GF2XModulus& 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) and deg(b) < deg(F).
    // This is really the transpose of modular multiplication.
    // Input may have "high order" zeroes stripped.
    // Output always has high order zeroes stripped.

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

                                  Minimum Polynomials

    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(GF2X& h, const vec_GF2& 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(GF2X& h, const GF2X& g, const GF2XModulus& F, long m);
    GF2X ProbMinPolyMod(const GF2X& g, const GF2XModulus& F, long m);

    void ProbMinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F);
    GF2X ProbMinPolyMod(const GF2X& g, const GF2XModulus& 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; it always
    // returns a divisor of the minimal polynomial, possibly a proper divisor.

    void MinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F, long m);
    GF2X MinPolyMod(const GF2X& g, const GF2XModulus& F, long m);

    void MinPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F);
    GF2X MinPolyMod(const GF2X& g, const GF2XModulus& F);

    // same as above, but guarantees that result is correct

    void IrredPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F, long m);
    GF2X IrredPolyMod(const GF2X& g, const GF2XModulus& F, long m);

    void IrredPolyMod(GF2X& h, const GF2X& g, const GF2XModulus& F);
    GF2X IrredPolyMod(const GF2X& g, const GF2XModulus& 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

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

    void TraceMod(GF2& x, const GF2X& a, const GF2XModulus& F);
    GF2 TraceMod(const GF2X& a, const GF2XModulus& F);

    void TraceMod(GF2& x, const GF2X& a, const GF2X& f);
    GF2 TraceMod(const GF2X& a, const GF2X& f);
    // x = Trace(a mod f); deg(a) < deg(f)

    void TraceVec(vec_GF2& S, const GF2X& f);
    vec_GF2 TraceVec(const GF2X& 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].

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

                               Miscellany

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

    void clear(GF2X& x) // x = 0
    void set(GF2X& x); // x = 1

    void GF2X::kill();
    // f.kill() sets f to 0 and frees all memory held by f.

    GF2X::GF2X(INIT_SIZE_TYPE, long n);
    // GF2X(INIT_SIZE, n) initializes to zero, but space is pre-allocated
    // for n coefficients

    static const GF2X& zero();
    // GF2X::zero() is a read-only reference to 0

    void GF2X::swap(GF2X& x);
    void swap(GF2X& x, GF2X& y);
    // swap (via "pointer swapping" -- if possible)

    GF2X::GF2X(long i, GF2 c);
    GF2X::GF2X(long i, long c);
    // initialize to c*X^i, provided for backward compatibility

    // SIZE INVARIANT: for any f in GF2X, deg(f)+1 < 2^(NTL_BITS_PER_LONG-4).