~/ntl-11.4.2/doc/GF2EX.cpp.html

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

    MODULE: GF2EX

    SUMMARY:

    The class GF2EX represents polynomials over GF2E,
    and so can be used, for example, for arithmentic in GF(2^n)[X].
    However, except where mathematically necessary (e.g., GCD computations),
    GF2E need not be a field.

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

    #include <NTL/GF2E.h>
    #include <NTL/vec_GF2E.h>

    class GF2EX {
    public:

       GF2EX(); // initial value 0

       GF2EX(const GF2EX& a); // copy
       explicit GF2EX(const GF2E& a);  // promotion
       explicit GF2EX(GF2 a);
       explicit GF2EX(long a);

       GF2EX& operator=(const GF2EX& a); // assignment
       GF2EX& operator=(const GF2E& a);
       GF2EX& operator=(GF2 a);
       GF2EX& operator=(long a);

       ~GF2EX(); // destructor

       GF2EX(GF2EX&& a);
       // move constructor (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set

    #ifndef NTL_DISABLE_MOVE_ASSIGN
       GF2EX& operator=(GF2EX&& a);
       // move assignment (C++11 only)
       // declared noexcept unless NTL_EXCEPTIONS flag is set
    #endif

       GF2EX(INIT_MONO_TYPE, long i, const GF2E& c);
       GF2EX(INIT_MONO_TYPE, long i, GF2 c);
       GF2EX(INIT_MONO_TYPE, long i, long c);
       // initialize to c*X^i, invoke as GF2EX(INIT_MONO, i, c)

       GF2EX(INIT_MONO_TYPE, long i);
       // initialize to X^i, invoke as GF2EX(INIT_MONO, i)

       // typedefs to aid in generic programming

       typedef GF2E coeff_type;
       typedef GF2EXModulus 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: 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 GF2EX& a);  // return deg(a); deg(0) == -1.

    const GF2E& coeff(const GF2EX& a, long i);
    // returns the coefficient of X^i, or zero if i not in range

    const GF2E& LeadCoeff(const GF2EX& a);
    // returns leading term of a, or zero if a == 0

    const GF2E& ConstTerm(const GF2EX& a);
    // returns constant term of a, or zero if a == 0

    void SetCoeff(GF2EX& x, long i, const GF2E& a);
    void SetCoeff(GF2EX& x, long i, GF2 a);
    void SetCoeff(GF2EX& x, long i, long a);
    // makes coefficient of X^i equal to a; error is raised if i < 0

    void SetCoeff(GF2EX& x, long i);
    // makes coefficient of X^i equal to 1;  error is raised if i < 0

    void SetX(GF2EX& x); // x is set to the monomial X

    long IsX(const GF2EX& a); // test if x = X

    GF2E& GF2EX::operator[](long i);
    const GF2E& GF2EX::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 GF2EX::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 GF2EX::normalize();
    // f.normalize() strips leading zeros from coefficient vector of f

    void GF2EX::SetMaxLength(long n);
    // f.SetMaxLength(n) pre-allocate spaces for n coefficients.  The
    // polynomial that f represents is unchanged.

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

                                      Comparison

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

    long operator==(const GF2EX& a, const GF2EX& b);
    long operator!=(const GF2EX& a, const GF2EX& b);

    long IsZero(const GF2EX& a); // test for 0
    long IsOne(const GF2EX& a); // test for 1

    // PROMOTIONS: ==, != promote {long,GF2,GF2E} to GF2EX on (a, b).

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

                                       Addition

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

    // operator notation:

    GF2EX operator+(const GF2EX& a, const GF2EX& b);
    GF2EX operator-(const GF2EX& a, const GF2EX& b);
    GF2EX operator-(const GF2EX& a);

    GF2EX& operator+=(GF2EX& x, const GF2EX& a);
    GF2EX& operator+=(GF2EX& x, const GF2E& a);
    GF2EX& operator+=(GF2EX& x, GF2 a);
    GF2EX& operator+=(GF2EX& x, long a);

    GF2EX& operator++(GF2EX& x);  // prefix
    void operator++(GF2EX& x, int);  // postfix

    GF2EX& operator-=(GF2EX& x, const GF2EX& a);
    GF2EX& operator-=(GF2EX& x, const GF2E& a);
    GF2EX& operator-=(GF2EX& x, GF2 a);
    GF2EX& operator-=(GF2EX& x, long a);

    GF2EX& operator--(GF2EX& x);  // prefix
    void operator--(GF2EX& x, int);  // postfix

    // procedural versions:

    void add(GF2EX& x, const GF2EX& a, const GF2EX& b); // x = a + b
    void sub(GF2EX& x, const GF2EX& a, const GF2EX& b); // x = a - b
    void negate(GF2EX& x, const GF2EX& a); // x = - a

    // PROMOTIONS: +, -, add, sub promote {long,GF2,GF2E} to GF2EX on (a, b).

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

                                   Multiplication

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

    // operator notation:

    GF2EX operator*(const GF2EX& a, const GF2EX& b);

    GF2EX& operator*=(GF2EX& x, const GF2EX& a);
    GF2EX& operator*=(GF2EX& x, const GF2E& a);
    GF2EX& operator*=(GF2EX& x, GF2 a);
    GF2EX& operator*=(GF2EX& x, long a);

    // procedural versions:

    void mul(GF2EX& x, const GF2EX& a, const GF2EX& b); // x = a * b

    void sqr(GF2EX& x, const GF2EX& a); // x = a^2
    GF2EX sqr(const GF2EX& a);

    // PROMOTIONS: *, mul promote {long,GF2,GF2E} to GF2EX on (a, b).

    void power(GF2EX& x, const GF2EX& a, long e);  // x = a^e (e >= 0)
    GF2EX power(const GF2EX& 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:

    GF2EX operator<<(const GF2EX& a, long n);
    GF2EX operator>>(const GF2EX& a, long n);

    GF2EX& operator<<=(GF2EX& x, long n);
    GF2EX& operator>>=(GF2EX& x, long n);

    // procedural versions:

    void LeftShift(GF2EX& x, const GF2EX& a, long n);
    GF2EX LeftShift(const GF2EX& a, long n);

    void RightShift(GF2EX& x, const GF2EX& a, long n);
    GF2EX RightShift(const GF2EX& a, long n);

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

                                      Division

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

    // operator notation:

    GF2EX operator/(const GF2EX& a, const GF2EX& b);
    GF2EX operator/(const GF2EX& a, const GF2E& b);
    GF2EX operator/(const GF2EX& a, GF2 b);
    GF2EX operator/(const GF2EX& a, long b);

    GF2EX operator%(const GF2EX& a, const GF2EX& b);

    GF2EX& operator/=(GF2EX& x, const GF2EX& a);
    GF2EX& operator/=(GF2EX& x, const GF2E& a);
    GF2EX& operator/=(GF2EX& x, GF2 a);
    GF2EX& operator/=(GF2EX& x, long a);

    GF2EX& operator%=(GF2EX& x, const GF2EX& a);

    // procedural versions:

    void DivRem(GF2EX& q, GF2EX& r, const GF2EX& a, const GF2EX& b);
    // q = a/b, r = a%b

    void div(GF2EX& q, const GF2EX& a, const GF2EX& b);
    void div(GF2EX& q, const GF2EX& a, const GF2E& b);
    void div(GF2EX& q, const GF2EX& a, GF2 b);
    void div(GF2EX& q, const GF2EX& a, long b);
    // q = a/b

    void rem(GF2EX& r, const GF2EX& a, const GF2EX& b);
    // r = a%b

    long divide(GF2EX& q, const GF2EX& a, const GF2EX& b);
    // if b | a, sets q = a/b and returns 1; otherwise returns 0

    long divide(const GF2EX& a, const GF2EX& b);
    // if b | a, sets q = a/b and returns 1; otherwise returns 0

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

                                       GCD's

    These routines are intended for use when GF2E is a field.

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

    void GCD(GF2EX& x, const GF2EX& a, const GF2EX& b);
    GF2EX GCD(const GF2EX& a, const GF2EX& b);
    // x = GCD(a, b),  x is always monic (or zero if a==b==0).

    void XGCD(GF2EX& d, GF2EX& s, GF2EX& t, const GF2EX& a, const GF2EX& 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 polynomials of degree < GF2E::degree() and
    a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
    are arbitrary polynomials which are reduced modulo GF2E::modulus(), and leading
    zeros stripped.

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

    istream& operator>>(istream& s, GF2EX& x);
    ostream& operator<<(ostream& s, const GF2EX& a);

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

                                  Some utility routines

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

    void diff(GF2EX& x, const GF2EX& a); // x = derivative of a
    GF2EX diff(const GF2EX& a);

    void MakeMonic(GF2EX& x);
    // if x != 0 makes x into its monic associate; LeadCoeff(x) must be
    // invertible in this case

    void reverse(GF2EX& x, const GF2EX& a, long hi);
    GF2EX reverse(const GF2EX& a, long hi);

    void reverse(GF2EX& x, const GF2EX& a);
    GF2EX reverse(const GF2EX& a);

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

    void VectorCopy(vec_GF2E& x, const GF2EX& a, long n);
    vec_GF2E VectorCopy(const GF2EX& 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(GF2EX& x, long n);
    GF2EX random_GF2EX(long n);
    // x = random polynomial of degree < n

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

                        Polynomial Evaluation and related problems

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

    void BuildFromRoots(GF2EX& x, const vec_GF2E& a);
    GF2EX BuildFromRoots(const vec_GF2E& a);
    // computes the polynomial (X-a[0]) ... (X-a[n-1]), where n = a.length()

    void eval(GF2E& b, const GF2EX& f, const GF2E& a);
    GF2E eval(const GF2EX& f, const GF2E& a);
    // b = f(a)

    void eval(GF2E& b, const GF2X& f, const GF2E& a);
    GF2E eval(const GF2EX& f, const GF2E& a);
    // b = f(a); uses ModComp algorithm for GF2X

    void eval(vec_GF2E& b, const GF2EX& f, const vec_GF2E& a);
    vec_GF2E eval(const GF2EX& f, const vec_GF2E& a);
    //  b.SetLength(a.length()); b[i] = f(a[i]) for 0 <= i < a.length()

    void interpolate(GF2EX& f, const vec_GF2E& a, const vec_GF2E& b);
    GF2EX interpolate(const vec_GF2E& a, const vec_GF2E& b);
    // interpolates the polynomial f satisfying f(a[i]) = b[i].

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

                           Arithmetic mod X^n

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

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

    void trunc(GF2EX& x, const GF2EX& a, long n); // x = a % X^n
    GF2EX trunc(const GF2EX& a, long n);

    void MulTrunc(GF2EX& x, const GF2EX& a, const GF2EX& b, long n);
    GF2EX MulTrunc(const GF2EX& a, const GF2EX& b, long n);
    // x = a * b % X^n

    void SqrTrunc(GF2EX& x, const GF2EX& a, long n);
    GF2EX SqrTrunc(const GF2EX& a, long n);
    // x = a^2 % X^n

    void InvTrunc(GF2EX& x, const GF2EX& a, long n);
    GF2EX InvTrunc(GF2EX& x, const GF2EX& 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
    GF2EXModulus data structure and associated routines below for better
    performance.

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

    void MulMod(GF2EX& x, const GF2EX& a, const GF2EX& b, const GF2EX& f);
    GF2EX MulMod(const GF2EX& a, const GF2EX& b, const GF2EX& f);
    // x = (a * b) % f

    void SqrMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
    GF2EX SqrMod(const GF2EX& a, const GF2EX& f);
    // x = a^2 % f

    void MulByXMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
    GF2EX MulByXMod(const GF2EX& a, const GF2EX& f);
    // x = (a * X) mod f

    void InvMod(GF2EX& x, const GF2EX& a, const GF2EX& f);
    GF2EX InvMod(const GF2EX& a, const GF2EX& f);
    // x = a^{-1} % f, error is a is not invertible

    long InvModStatus(GF2EX& x, const GF2EX& a, const GF2EX& f);
    // if (a, f) = 1, returns 0 and sets x = a^{-1} % f; otherwise,
    // returns 1 and sets x = (a, f)

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

                         Modular Arithmetic with Pre-Conditioning

    If you need to do a lot of arithmetic modulo a fixed f, build
    GF2EXModulus F for f.  This pre-computes information about f that
    speeds up subsequent computations.

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

    #include <NTL/GF2EX.h>

    void product(GF2EX& x, const vec_GF2EX& v, const GF2EX& f)
    {
       GF2EXModulus F(f);
       GF2EX res;
       res = 1;
       long i;
       for (i = 0; i < v.length(); i++)
          MulMod(res, res, v[i], F);
       x = res;
    }

    NOTE: A GF2EX may be used wherever a GF2EXModulus is required,
    and a GF2EXModulus may be used wherever a GF2EX is required.

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

    class GF2EXModulus {
    public:
       GF2EXModulus(); // initially in an unusable state

       GF2EXModulus(const GF2EX& f); // initialize with f, deg(f) > 0

       GF2EXModulus(const GF2EXModulus&); // copy

       GF2EXModulus& operator=(const GF2EXModulus&); // assignment

       ~GF2EXModulus(); // destructor

       operator const GF2EX& () const; // implicit read-only access to f

       const GF2EX& val() const; // explicit read-only access to f
    };

    void build(GF2EXModulus& F, const GF2EX& f);
    // pre-computes information about f and stores it in F.  Must have
    // deg(f) > 0.  Note that the declaration GF2EXModulus F(f) is
    // equivalent to GF2EXModulus 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 GF2EXModulus& F);  // return n=deg(f)

    void MulMod(GF2EX& x, const GF2EX& a, const GF2EX& b, const GF2EXModulus& F);
    GF2EX MulMod(const GF2EX& a, const GF2EX& b, const GF2EXModulus& F);
    // x = (a * b) % f; deg(a), deg(b) < n

    void SqrMod(GF2EX& x, const GF2EX& a, const GF2EXModulus& F);
    GF2EX SqrMod(const GF2EX& a, const GF2EXModulus& F);
    // x = a^2 % f; deg(a) < n

    void PowerMod(GF2EX& x, const GF2EX& a, const ZZ& e, const GF2EXModulus& F);
    GF2EX PowerMod(const GF2EX& a, const ZZ& e, const GF2EXModulus& F);

    void PowerMod(GF2EX& x, const GF2EX& a, long e, const GF2EXModulus& F);
    GF2EX PowerMod(const GF2EX& a, long e, const GF2EXModulus& F);

    // x = a^e % f; e >= 0, deg(a) < n.  Uses a sliding window algorithm.
    // (e may be negative)

    void PowerXMod(GF2EX& x, const ZZ& e, const GF2EXModulus& F);
    GF2EX PowerXMod(const ZZ& e, const GF2EXModulus& F);

    void PowerXMod(GF2EX& x, long e, const GF2EXModulus& F);
    GF2EX PowerXMod(long e, const GF2EXModulus& F);

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

    void rem(GF2EX& x, const GF2EX& a, const GF2EXModulus& F);
    // x = a % f

    void DivRem(GF2EX& q, GF2EX& r, const GF2EX& a, const GF2EXModulus& F);
    // q = a/f, r = a%f

    void div(GF2EX& q, const GF2EX& a, const GF2EXModulus& F);
    // q = a/f

    // operator notation:

    GF2EX operator/(const GF2EX& a, const GF2EXModulus& F);
    GF2EX operator%(const GF2EX& a, const GF2EXModulus& F);

    GF2EX& operator/=(GF2EX& x, const GF2EXModulus& F);
    GF2EX& operator%=(GF2EX& x, const GF2EXModulus& F);

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

                                 vectors of GF2EX's

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

    typedef Vec<GF2EX> vec_GF2EX; // 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(GF2EX& x, const GF2EX& g, const GF2EX& h, const GF2EXModulus& F);
    GF2EX CompMod(const GF2EX& g, const GF2EX& h,
                        const GF2EXModulus& F);

    // x = g(h) mod f; deg(h) < n

    void Comp2Mod(GF2EX& x1, GF2EX& x2, const GF2EX& g1, const GF2EX& g2,
                  const GF2EX& h, const GF2EXModulus& F);
    // xi = gi(h) mod f (i=1,2); deg(h) < n.

    void Comp3Mod(GF2EX& x1, GF2EX& x2, GF2EX& x3,
                  const GF2EX& g1, const GF2EX& g2, const GF2EX& g3,
                  const GF2EX& h, const GF2EXModulus& 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 GF2EXArgument 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 GF2EXArgument {
       vec_GF2EX H;
    };

    void build(GF2EXArgument& H, const GF2EX& h, const GF2EXModulus& F, long m);
    // Pre-Computes information about h.  m > 0, deg(h) < n.

    void CompMod(GF2EX& x, const GF2EX& g, const GF2EXArgument& H,
                 const GF2EXModulus& F);

    GF2EX CompMod(const GF2EX& g, const GF2EXArgument& H,
                        const GF2EXModulus& F);

    extern thread_local long GF2EXArgBound;

    // Initially 0.  If this is set to a value greater than zero, then
    // composition routines will allocate a table of no than about
    // GF2EXArgBound KB.  Setting this value affects all compose routines
    // and the power projection and minimal polynomial routines below,
    // and indirectly affects many routines in GF2EXFactoring.

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

                         power projection routines

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

    void project(GF2E& x, const GF2EVector& a, const GF2EX& b);
    GF2E project(const GF2EVector& a, const GF2EX& b);
    // x = inner product of a with coefficient vector of b

    void ProjectPowers(vec_GF2E& x, const vec_GF2E& a, long k,
                       const GF2EX& h, const GF2EXModulus& F);

    vec_GF2E ProjectPowers(const vec_GF2E& a, long k,
                       const GF2EX& h, const GF2EXModulus& 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_GF2E& x, const vec_GF2E& a, long k,
                       const GF2EXArgument& H, const GF2EXModulus& F);

    vec_GF2E ProjectPowers(const vec_GF2E& a, long k,
                       const GF2EXArgument& H, const GF2EXModulus& F);

    // same as above, but uses a pre-computed GF2EXArgument

    class GF2EXTransMultiplier { /* ... */ };

    void build(GF2EXTransMultiplier& B, const GF2EX& b, const GF2EXModulus& F);

    void UpdateMap(vec_GF2E& x, const vec_GF2E& a,
                   const GF2EXMultiplier& B, const GF2EXModulus& F);

    vec_GF2E UpdateMap(const vec_GF2E& a,
                   const GF2EXMultiplier& B, const GF2EXModulus& F);

    // Computes the vector

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

    // Restriction: a.length() <= deg(F), deg(b) < deg(F).
    // This is "transposed" MulMod by B.
    // Input may have "high order" zeroes stripped.
    // Output always has high order zeroes stripped.

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

                                  Minimum Polynomials

    These routines should be used only when GF2E is a field.

    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(GF2EX& h, const vec_GF2E& a, long m);
    GF2EX MinPolySeq(const vec_GF2E& 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(GF2EX& h, const GF2EX& g, const GF2EXModulus& F, long m);
    GF2EX ProbMinPolyMod(const GF2EX& g, const GF2EXModulus& F, long m);

    void ProbMinPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F);
    GF2EX ProbMinPolyMod(const GF2EX& g, const GF2EXModulus& 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/2^{GF2E::degree()}.

    void MinPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F, long m);
    GF2EX MinPolyMod(const GF2EX& g, const GF2EXModulus& F, long m);

    void MinPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F);
    GF2EX MinPolyMod(const GF2EX& g, const GF2EXModulus& F);

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

    void IrredPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F, long m);
    GF2EX IrredPolyMod(const GF2EX& g, const GF2EXModulus& F, long m);

    void IrredPolyMod(GF2EX& h, const GF2EX& g, const GF2EXModulus& F);
    GF2EX IrredPolyMod(const GF2EX& g, const GF2EXModulus& 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).

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

               Composition and Minimal Polynomials in towers

    These are implementations of algorithms that will be described
    and analyzed in a forthcoming paper.

    GF2E need not be a field.

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

    void CompTower(GF2EX& x, const GF2X& g, const GF2EXArgument& h,
                 const GF2EXModulus& F);

    GF2EX CompTower(const GF2X& g, const GF2EXArgument& h,
                 const GF2EXModulus& F);

    void CompTower(GF2EX& x, const GF2X& g, const GF2EX& h,
                 const GF2EXModulus& F);

    GF2EX CompTower(const GF2X& g, const GF2EX& h,
                 const GF2EXModulus& F);

    // x = g(h) mod f

    void ProbMinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F,
                          long m);

    GF2X ProbMinPolyTower(const GF2EX& g, const GF2EXModulus& F, long m);

    void ProbMinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F);

    GF2X ProbMinPolyTower(const GF2EX& g, const GF2EXModulus& F);

    // Uses a probabilistic algorithm to compute the minimal
    // polynomial of (g mod f) over GF2.
    // The parameter m is a bound on the degree of the minimal polynomial
    // (default = deg(f)*GF2E::degree()).
    // In general, the result will be a divisor of the true minimimal
    // polynomial.  For correct results, use the MinPoly routines below.

    void MinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F, long m);

    GF2X MinPolyTower(const GF2EX& g, const GF2EXModulus& F, long m);

    void MinPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F);

    GF2X MinPolyTower(const GF2EX& g, const GF2EXModulus& F);

    // Same as above, but result is always correct.

    void IrredPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F, long m);

    GF2X IrredPolyTower(const GF2EX& g, const GF2EXModulus& F, long m);

    void IrredPolyTower(GF2X& h, const GF2EX& g, const GF2EXModulus& F);

    GF2X IrredPolyTower(const GF2EX& g, const GF2EXModulus& F);

    // Same as above, but assumes the minimal polynomial is
    // irreducible, and uses a slightly faster, deterministic algorithm.

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

                       Traces, norms, resultants

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

    void TraceMod(GF2E& x, const GF2EX& a, const GF2EXModulus& F);
    GF2E TraceMod(const GF2EX& a, const GF2EXModulus& F);

    void TraceMod(GF2E& x, const GF2EX& a, const GF2EX& f);
    GF2E TraceMod(const GF2EX& a, const GF2EXModulus& f);
    // x = Trace(a mod f); deg(a) < deg(f)

    void TraceVec(vec_GF2E& S, const GF2EX& f);
    vec_GF2E TraceVec(const GF2EX& 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(GF2E& x, const GF2EX& a, const GF2EX& f);
    GF2E NormMod(const GF2EX& a, const GF2EX& f);
    // x = Norm(a mod f); 0 < deg(f), deg(a) < deg(f)

    void resultant(GF2E& x, const GF2EX& a, const GF2EX& b);
    GF2E resultant(const GF2EX& a, const GF2EX& b);
    // x = resultant(a, b)

    // NormMod and resultant require that GF2E is a field.

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

                               Miscellany

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

    void clear(GF2EX& x) // x = 0
    void set(GF2EX& x); // x = 1

    void GF2EX::kill();
    // f.kill() sets f to 0 and frees all memory held by f.  Equivalent to
    // f.rep.kill().

    GF2EX::GF2EX(INIT_SIZE_TYPE, long n);
    // GF2EX(INIT_SIZE, n) initializes to zero, but space is pre-allocated
    // for n coefficients

    static const GF2EX& zero();
    // GF2EX::zero() is a read-only reference to 0

    void GF2EX::swap(GF2EX& x);
    void swap(GF2EX& x, GF2EX& y);
    // swap (via "pointer swapping")

    GF2EX::GF2EX(long i, const GF2E& c);
    GF2EX::GF2EX(long i, GF2 c);
    GF2EX::GF2EX(long i, long c);
    // initialize to X^i*c, provided for backward compatibility