lzz_pex.txt

数值算法库for Windows

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

MODULE: zz_pEX

SUMMARY:

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

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

#include <NTL/lzz_pE.h>
#include <NTL/vec_lzz_pE.h>

class zz_pEX {
public:

   zz_pEX(); // initial value 0

   zz_pEX(const zz_pEX& a); // copy

   zz_pEX& operator=(const zz_pEX& a); // assignment
   zz_pEX& operator=(const zz_pE& a);
   zz_pEX& operator=(const zz_p& a);
   zz_pEX& operator=(long a);

   ~zz_pEX(); // destructor

   zz_pEX(long i, const zz_pE& c); // initilaize to X^i*c
   zz_pEX(long i, const zz_p& c); 
   zz_pEX(long i, long c); 

   
};






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

                                  Comparison

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


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

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

// PROMOTIONS: ==, != promote {long,zz_p,zz_pE} to zz_pEX on (a, b).

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

                                   Addition

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

// operator notation:

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

zz_pEX& operator+=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator+=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator+=(zz_pEX& x, const zz_p& a);
zz_pEX& operator+=(zz_pEX& x, long a);


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

zz_pEX& operator-=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator-=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator-=(zz_pEX& x, const zz_p& a);
zz_pEX& operator-=(zz_pEX& x, long a);

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

// procedural versions:

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

// PROMOTIONS: +, -, add, sub promote {long,zz_p,zz_pE} to zz_pEX on (a, b).



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

                               Multiplication

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

// operator notation:

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

zz_pEX& operator*=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator*=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator*=(zz_pEX& x, const zz_p& a);
zz_pEX& operator*=(zz_pEX& x, long a);


// procedural versions:


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

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

// PROMOTIONS: *, mul promote {long,zz_p,zz_pE} to zz_pEX on (a, b).

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

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

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

// procedural versions:

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

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



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

                                  Division

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

// operator notation:

zz_pEX operator/(const zz_pEX& a, const zz_pEX& b);
zz_pEX operator/(const zz_pEX& a, const zz_pE& b);
zz_pEX operator/(const zz_pEX& a, const zz_p& b);
zz_pEX operator/(const zz_pEX& a, long b);

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

zz_pEX& operator/=(zz_pEX& x, const zz_pEX& a);
zz_pEX& operator/=(zz_pEX& x, const zz_pE& a);
zz_pEX& operator/=(zz_pEX& x, const zz_p& a);
zz_pEX& operator/=(zz_pEX& x, long a);

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

// procedural versions:


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

void div(zz_pEX& q, const zz_pEX& a, const zz_pEX& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_pE& b);
void div(zz_pEX& q, const zz_pEX& a, const zz_p& b);
void div(zz_pEX& q, const zz_pEX& a, long b);
// q = a/b

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

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

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


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

                                   GCD's

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

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


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


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

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

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


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

                              Some utility routines

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

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

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

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

const zz_pE& ConstTerm(const zz_pEX& a);
// read-only reference to constant term of a, or zero if a == 0

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

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

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

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

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

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

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

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

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

void VectorCopy(vec_zz_pE& x, const zz_pEX& a, long n);
vec_zz_pE VectorCopy(const zz_pEX& 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(zz_pEX& x, long n);
zz_pEX random_zz_pEX(long n);
// x = random polynomial of degree < n 


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

                    Polynomial Evaluation and related problems

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


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

void eval(zz_pE& b, const zz_pEX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a)

void eval(zz_pE& b, const zz_pX& f, const zz_pE& a);
zz_pE eval(const zz_pEX& f, const zz_pE& a);
// b = f(a); uses ModComp algorithm for zz_pX

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

void interpolate(zz_pEX& f, const vec_zz_pE& a, const vec_zz_pE& b);
zz_pEX interpolate(const vec_zz_pE& a, const vec_zz_pE& 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(zz_pEX& x, const zz_pEX& a, long n); // x = a % X^n
zz_pEX trunc(const zz_pEX& a, long n); 

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

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

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

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

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

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

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

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

long InvModStatus(zz_pEX& x, const zz_pEX& a, const zz_pEX& 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
zz_pEXModulus 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/lzz_pEX.h>

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