The class GF2E is used to represent polynomials in F_2[X] modulo a
polynomial P.  The modulus P may be any polynomial with deg(P) > 0,
not necessarily irreducible.  

Objects of the class GF2E are represented as a GF2X of degree < deg(P).

An executing program maintains a "current modulus", which is set to P using
GF2E::init(P).  The current modulus *must* be initialized before any operations
on GF2E's are performed.  The modulus may be changed, and a mechanism is provided
for saving and restoring a modulus (see classes GF2EPush and GF2EContext below).

NOTE: if P is a trinomial X^n + X^k + 1, or a pentanomial
X^n + X^k3 + X^k2 + X^k1 + 1, or of the form X^n + g, where
g has low degree, then performance will be somewhat improved.
Such polynomials are constructed by the routines
BuildSparseIrred and BuildIrred in GF2XFactoring.


#include <NTL/GF2X.h>
#include <NTL/SmartPtr.h>

class GF2E {

   GF2E(); // initial value 0

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

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

   ~GF2E(); // destructor

   GF2E(GF2E&& a);
   // move constructor (C++11 only)
   // declared noexcept unless NTL_EXCEPTIONS flag is set

   GF2E& operator=(GF2E&& a);
   // move assignment (C++11 only)
   // declared noexcept unless NTL_EXCEPTIONS flag is set

   static void init(const GF2X& P);
   // GF2E::init(P) initializes the current modulus to P;
   // required: deg(P) >= 1.

   static const GF2XModulus& modulus();
   // GF2E::modulus() yields read-only reference to the current modulus 

   static long degree();
   // GF2E::degree() returns deg(P)

   // typedefs to aid generic programming
   typedef GF2X rep_type;
   typedef GF2EContext context_type;
   typedef GF2EBak bak_type;
   typedef GF2EPush push_type;
   typedef GF2EX poly_type;


const GF2X& rep(const GF2E& a); // read-only access to representation of a




long operator==(const GF2E& a, const GF2E& b);
long operator!=(const GF2E& a, const GF2E& b);

long IsZero(const GF2E& a);  // test for 0
long IsOne(const GF2E& a);  // test for 1

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




// operator notation:

GF2E operator+(const GF2E& a, const GF2E& b);

GF2E operator-(const GF2E& a, const GF2E& b);
GF2E operator-(const GF2E& a);

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

GF2E& operator++(GF2E& x); // prefix
void operator++(GF2E& x, int); // postfix

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

GF2E& operator--(GF2E& x); // prefix
void operator--(GF2E& x, int); // postfix

// procedural versions:

void add(GF2E& x, const GF2E& a, const GF2E& b); // x = a + b
void sub(GF2E& x, const GF2E& a, const GF2E& b); // x = a - b = a + b
void negate(GF2E& x, const GF2E& a); // x = - a = a

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




// operator notation:

GF2E operator*(const GF2E& a, const GF2E& b);

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

// procedural versions:

void mul(GF2E& x, const GF2E& a, const GF2E& b); // x = a * b

void sqr(GF2E& x, const GF2E& a); // x = a^2
GF2E sqr(const GF2E& a);

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




// operator notation:

GF2E operator/(const GF2E& a, const GF2E& b);

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

// procedural versions:

void div(GF2E& x, const GF2E& a, const GF2E& b);
// x = a/b.  If b is not invertible, an error is raised.

void inv(GF2E& x, const GF2E& a);
GF2E inv(const GF2E& a);
// x = 1/a

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




void power(GF2E& x, const GF2E& a, const ZZ& e);
GF2E power(const GF2E& a, const ZZ& e);

void power(GF2E& x, const GF2E& a, long e);
GF2E power(const GF2E& a, long e);

// x = a^e (e may be negative)


                               Random Elements


void random(GF2E& x);
GF2E random_GF2E();
// x = random element in GF2E.




void trace(GF2& x, const GF2E& a);  // x = trace of a
GF2 trace(const GF2E& a);




ostream& operator<<(ostream& s, const GF2E& a);

istream& operator>>(istream& s, GF2E& x);
// a GF2X is read and reduced mod p


                       Modulus Switching 

A class GF2EPush is provided for "backing up" the current modulus
and installing a new one.

Here is what you do to save the current modulus, temporarily
set it to P, and automatically restore it:

      GF2EPush push(P); 



The constructor for push will save the current modulus, and install P as the
current modulus.  The destructor for push will restore the old modulus when the
scope enclosing it exits.  This is the so-called RAII (resource acquisition is
initialization) paradigm.

You could also do the following:

      GF2EPush push; // just backup current modulus


      GF2E::init(P1); // install P1 


      GF2E::init(P2); // install P2

      // reinstall original modulus as close of scope

The GF2EPush interface is good for implementing simple stack-like
modulus "context switching".  For more general context switching,
see GF2EContext below.  There is also an older GF2EBak class
that may also be useful.


It is critical that GF2E objects created under one GF2E modulus are not used in
any non-trivial way "out of context", i.e., under a different (or undefined)
GF2E modulus.  However, for ease-of-use, some operations may be safely
performed out of context.  These safe operations include: the default and copy
constructor, the destructor, and the assignment operator.  In addition is is
generally safe to read any GF2E object out of context (i.e., printing it out, or
fetching its underlying representive using the rep() function).

Any unsafe uses out of context are not in general checked, and may 
lead to unpredictable behavior.

NOTE: the implementation of Vec<GF2E> is specialized to manage memory more
efficiently than in the default implementation of Vec<T>.  Specifically,
contiguous elements in a Vec<GF2E> are allocated in a contiguous region of
memory.  This reduces the number of calls to the memory allocator, and --- more
significantly --- leads to greater locality of reference.  A consequence of
this implementation is that any calls to SetLength on a Vec<GF2E> object will
need to use information about the current modulus, and so such calls should
only be done "in context".  That said, it is still safe to construct a
Vec<GF2E> using the default or copy contructor, and to assign or append one
Vec<GF2E> to another "out of context".


// A convenient interface for common cases

class GF2EPush {

GF2EPush();  // backup current modulus
explicit GF2EPush(const GF2X& P);
explicit GF2EPush(const GF2EContext& context);
  // backup current modulus and install the given one

GF2EPush(const GF2EPush&); // disabled
void operator=(const GF2EPush&); // disabled


// more general context switching:
// A GF2EContext object has a modulus Q (possibly "null"),

class GF2EContext {


GF2EContext(); // Q = "null"
explicit GF2EContext(const GF2X& P); // Q = P

void save(); // Q = CurrentModulus
void restore() const; // CurrentModulus = Q

GF2EContext(const GF2EContext&);  // copy
GF2EContext& operator=(const GF2EContext&); // assignment
~GF2EContext(); // destructor


// An older interface:
// To describe this logic, think of a GF2EBak object
// of having two components: a modulus Q (possibly "null") and 
// an "auto-restore bit" b.

class GF2EBak {

   GF2EBak();  // Q = "null", b = 0

   ~GF2EBak();  // if (b) CurrentModulus = Q

   void save();  // Q = CurrentModulus, b = 1 
   void restore();  // CurrentModulus = Q, b = 0

   GF2EBak(const GF2EBak&);  // copy disabled
   void operator=(const GF2EBak&);  // assignment disabled




void clear(GF2E& x); // x = 0
void set(GF2E& x); // x = 1

static const GF2E& GF2E::zero();
// GF2E::zero() yields a read-only reference to zero

static long GF2X::WordLength();
// GF2E::size() returns # of words needed to store a polynomial of
// degree < GF2E::degree()

void GF2E::swap(GF2E& x);
void swap(GF2E& x, GF2E& y);
// swap (done by "pointer swapping", if possible).

static ZZ& GF2E::cardinality();
// yields the cardinality, i.e., 2^{GF2E::degree()}

// special constructor: invoke as GF2E x(INIT_NO_ALLOC);
// initializes x to 0, but allocates no space (this is now the default)

// special constructor: invoke as GF2E x(INIT_ALLOC);
// initializes x to 0, but allocates space 

// useful in conjunction with the INIT_NO_ALLLOC constructor:
// x.allocate() will pre-allocate space for x, using the
// current modulus