/**************************************************************************\
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(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)
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 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