zz_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/ZZ_pE.h>
#include <NTL/vec_ZZ_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/ZZ_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;
}