/**************************************************************************\
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).
