zz_px.txt

数值算法库for Windows

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

MODULE: ZZ_pX

SUMMARY:

The class ZZ_pX implements polynomial arithmetic modulo p.

Polynomial arithmetic is implemented using the FFT, combined with the
Chinese Remainder Theorem.  A more detailed description of the
techniques used here can be found in [Shoup, J. Symbolic
Comp. 20:363-397, 1995].

Small degree polynomials are multiplied either with classical 
or Karatsuba algorithms.

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

#include <NTL/ZZ_p.h>
#include <NTL/vec_ZZ_p.h>

class ZZ_pX {
public:

   ZZ_pX(); // initialize to 0

   ZZ_pX(const ZZ_pX& a); // copy constructor

   ZZ_pX& operator=(const ZZ_pX& a); // assignment
   ZZ_pX& operator=(const ZZ_p& a); // assignment
   ZZ_pX& operator=(const long a); // assignment

   ZZ_pX(long i, const ZZ_p& c);  // initialize to X^i*c
   ZZ_pX(long i, long c);

   ~ZZ_pX(); // destructor
   
};






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

                                  Comparison

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


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

// PROMOTIONS: operators ==, != promote {long, ZZ_p} to ZZ_pX on (a, b).

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


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

                                   Addition

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


// operator notation:

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

ZZ_pX operator-(const ZZ_pX& a); // unary -

ZZ_pX& operator+=(ZZ_pX& x, const ZZ_pX& a);
ZZ_pX& operator+=(ZZ_pX& x, const ZZ_p& a);
ZZ_pX& operator+=(ZZ_pX& x, long a);

ZZ_pX& operator-=(ZZ_pX& x, const ZZ_pX& a);
ZZ_pX& operator-=(ZZ_pX& x, const ZZ_p& a);
ZZ_pX& operator-=(ZZ_pX& x, long a);

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

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

// procedural versions:


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


// PROMOTIONS: binary +, - and procedures add, sub promote
// {long, ZZ_p} to ZZ_pX on (a, b).


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

                               Multiplication

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

// operator notation:

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

ZZ_pX& operator*=(ZZ_pX& x, const ZZ_pX& a);
ZZ_pX& operator*=(ZZ_pX& x, const ZZ_p& a);
ZZ_pX& operator*=(ZZ_pX& x, long a);

// procedural versions:

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

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

// PROMOTIONS: operator * and procedure mul promote {long, ZZ_p} to ZZ_pX
// on (a, b).

void power(ZZ_pX& x, const ZZ_pX& a, long e);  // x = a^e (e >= 0)
ZZ_pX power(const ZZ_pX& 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_pX operator<<(const ZZ_pX& a, long n);
ZZ_pX operator>>(const ZZ_pX& a, long n);

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

// procedural versions:

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

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



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

                                  Division

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

// operator notation:

ZZ_pX operator/(const ZZ_pX& a, const ZZ_pX& b);
ZZ_pX operator/(const ZZ_pX& a, const ZZ_p& b);
ZZ_pX operator/(const ZZ_pX& a, long b);


ZZ_pX& operator/=(ZZ_pX& x, const ZZ_pX& b);
ZZ_pX& operator/=(ZZ_pX& x, const ZZ_p& b);
ZZ_pX& operator/=(ZZ_pX& x, long b);

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

ZZ_pX& operator%=(ZZ_pX& x, const ZZ_pX& b);


// procedural versions:


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

void div(ZZ_pX& q, const ZZ_pX& a, const ZZ_pX& b);
void div(ZZ_pX& q, const ZZ_pX& a, const ZZ_p& b);
void div(ZZ_pX& q, const ZZ_pX& a, long b);
// q = a/b

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

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

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


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

                                   GCD's

These routines are intended for use when p is prime.

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


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


void XGCD(ZZ_pX& d, ZZ_pX& s, ZZ_pX& t, const ZZ_pX& a, const ZZ_pX& b);
// d = gcd(a,b), a s + b t = d 


// NOTE: A classical algorithm is used, switching over to a
// "half-GCD" algorithm for large degree


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

                                  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 integers between 0 and p-1, and
a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
are arbitrary integers which are reduced modulo p, and leading zeros
stripped.

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

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


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

                              Some utility routines

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

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

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

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

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

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

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

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

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

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

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

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

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

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

void VectorCopy(vec_ZZ_p& x, const ZZ_pX& a, long n);
vec_ZZ_p VectorCopy(const ZZ_pX& 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_pX& x, long n);
ZZ_pX random_ZZ_pX(long n);
// generate a random polynomial of degree < n 



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

                    Polynomial Evaluation and related problems

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


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

void eval(ZZ_p& b, const ZZ_pX& f, const ZZ_p& a);
ZZ_p eval(const ZZ_pX& f, const ZZ_p& a);
// b = f(a)

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

void interpolate(ZZ_pX& f, const vec_ZZ_p& a, const vec_ZZ_p& b);
ZZ_pX interpolate(const vec_ZZ_p& a, const vec_ZZ_p& b);
// interpolates the polynomial f satisfying f(a[i]) = b[i].  p should
// be prime.

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

                       Arithmetic mod X^n

All routines require n >= 0, otherwise an error is raised.

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

void trunc(ZZ_pX& x, const ZZ_pX& a, long n); // x = a % X^n
ZZ_pX trunc(const ZZ_pX& a, long n); 

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

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

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

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

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

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

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

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

long InvModStatus(ZZ_pX& x, const ZZ_pX& a, const ZZ_pX& 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 a
ZZ_pXModulus F for f.  This pre-computes information about f that
speeds up subsequent computations.

It is required that deg(f) > 0 and that LeadCoeff(f) is invertible.

As an example, the following routine computes the product modulo f of a vector
of polynomials.

#include <NTL/ZZ_pX.h>

void product(ZZ_pX& x, const vec_ZZ_pX& v, const ZZ_pX& f)
{
   ZZ_pXModulus F(f);
   ZZ_pX res;
   res = 1;
   long i;