source: git/ntl/doc/GF2EX.txt @ 2cfffe

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

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

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

   ~GF2EX(); // destructor

   GF2EX(long i, const GF2E& c); // initilaize to X^i*c
   GF2EX(long i, GF2 c); 
   GF2EX(long i, long c); 

   
};






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

                                  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

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

long deg(const GF2EX& a);  // return deg(a); deg(0) == -1.

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

const GF2E& LeadCoeff(const GF2EX& a);
// read-only reference to leading term of a, or zero if a == 0

const GF2E& ConstTerm(const GF2EX& a);
// read-only reference to 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

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

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

NTL_vector_decl(GF2EX,vec_GF2EX)
// vec_GF2EX

NTL_eq_vector_decl(GF2EX,vec_GF2EX)
// == and !=

NTL_io_vector_decl(GF2EX,vec_GF2EX)
// I/O operators



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

                              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

A GF2EX f is represented as a vec_GF2E, which can be accessed as
f.rep.  The constant term is f.rep[0] and the leading coefficient is
f.rep[f.rep.length()-1], except if f is zero, in which case
f.rep.length() == 0.  Note that the leading coefficient is always
nonzero (unless f is zero).  One can freely access and modify f.rep,
but one should always ensure that the leading coefficient is nonzero,
which can be done by invoking f.normalize().


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


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

void GF2EX::normalize();  
// f.normalize() strips leading zeros from f.rep.

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

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