gf2x.txt

数值算法库for Windows

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

MODULE: GF2X

SUMMARY:

The class GF2X implements polynomial arithmetic modulo 2.

Polynomial arithmetic is implemented using a combination of classical
routines and Karatsuba.

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

#include <NTL/GF2.h>
#include <NTL/vec_GF2.h>

class GF2X {
public:

   GF2X(); // initial value 0

   GF2X(const GF2X& a); // copy

   GF2X& operator=(const GF2X& a); // assignment
   GF2X& operator=(GF2 a); 
   GF2X& operator=(long a); 

   ~GF2X(); // destructor

   GF2X(long i, GF2 c); // initialize to X^i*c
   GF2X(long i, long c); 
   
};


// SIZE INVARIANT: for any f in GF2X, def(f)+1 < 2^(NTL_BITS_PER_LONG-4).



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

                                  Comparison

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


long operator==(const GF2X& a, const GF2X& b);
long operator!=(const GF2X& a, const GF2X& b);

long IsZero(const GF2X& a); // test for 0
long IsOne(const GF2X& a); // test for 1

// PROMOTIONS: operators ==, != promote {long, GF2} to GF2X on (a, b)


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

                                   Addition

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

// operator notation:

GF2X operator+(const GF2X& a, const GF2X& b);
GF2X operator-(const GF2X& a, const GF2X& b);

GF2X operator-(const GF2X& a); // unary -

GF2X& operator+=(GF2X& x, const GF2X& a);
GF2X& operator+=(GF2X& x, GF2 a);
GF2X& operator+=(GF2X& x, long a);

GF2X& operator-=(GF2X& x, const GF2X& a);
GF2X& operator-=(GF2X& x, GF2 a);
GF2X& operator-=(GF2X& x, long a);

GF2X& operator++(GF2X& x);  // prefix
void operator++(GF2X& x, int);  // postfix

GF2X& operator--(GF2X& x);  // prefix
void operator--(GF2X& x, int);  // postfix

// procedural versions:


void add(GF2X& x, const GF2X& a, const GF2X& b); // x = a + b
void sub(GF2X& x, const GF2X& a, const GF2X& b); // x = a - b
void negate(GF2X& x, const GF2X& a); // x = -a

// PROMOTIONS: binary +, - and procedures add, sub promote {long, GF2}
// to GF2X on (a, b).


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

                               Multiplication

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

// operator notation:

GF2X operator*(const GF2X& a, const GF2X& b);

GF2X& operator*=(GF2X& x, const GF2X& a);
GF2X& operator*=(GF2X& x, GF2 a);
GF2X& operator*=(GF2X& x, long a);

// procedural versions:

void mul(GF2X& x, const GF2X& a, const GF2X& b); // x = a * b

void sqr(GF2X& x, const GF2X& a); // x = a^2
GF2X sqr(const GF2X& a);

// PROMOTIONS: operator * and procedure mul promote {long, GF2} to GF2X
// on (a, b).


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

                               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:

GF2X operator<<(const GF2X& a, long n);
GF2X operator>>(const GF2X& a, long n);

GF2X& operator<<=(GF2X& x, long n);
GF2X& operator>>=(GF2X& x, long n);

// procedural versions:

void LeftShift(GF2X& x, const GF2X& a, long n); 
GF2X LeftShift(const GF2X& a, long n);

void RightShift(GF2X& x, const GF2X& a, long n); 
GF2X RightShift(const GF2X& a, long n); 

void MulByX(GF2X& x, const GF2X& a); 
GF2X MulByX(const GF2X& a); 


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

                                  Division

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

// operator notation:

GF2X operator/(const GF2X& a, const GF2X& b);
GF2X operator%(const GF2X& a, const GF2X& b);

GF2X& operator/=(GF2X& x, const GF2X& a);
GF2X& operator/=(GF2X& x, GF2 a);
GF2X& operator/=(GF2X& x, long a);

GF2X& operator%=(GF2X& x, const GF2X& b);


// procedural versions:


void DivRem(GF2X& q, GF2X& r, const GF2X& a, const GF2X& b);
// q = a/b, r = a%b

void div(GF2X& q, const GF2X& a, const GF2X& b);
// q = a/b

void rem(GF2X& r, const GF2X& a, const GF2X& b);
// r = a%b

long divide(GF2X& q, const GF2X& a, const GF2X& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0

long divide(const GF2X& a, const GF2X& b);
// if b | a, sets q = a/b and returns 1; otherwise returns 0

// PROMOTIONS: operator / and procedure div promote {long, GF2} to GF2X
// on (a, b).


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

                                   GCD's

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


void GCD(GF2X& x, const GF2X& a, const GF2X& b);
GF2X GCD(const GF2X& a, const GF2X& b); 
// x = GCD(a, b) (zero if a==b==0).


void XGCD(GF2X& d, GF2X& s, GF2X& t, const GF2X& a, const GF2X& 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 0 or 1, and
a_n not zero (the zero polynomial is [ ]).  On input, the coefficients
may be arbitrary integers which are reduced modulo 2, and leading zeros
stripped.

There is also a more compact hex I/O format.  To output in this
format, set GF2X::HexOutput to a nonzero value.  On input, if the first
non-blank character read is 'x' or 'X', then a hex format is assumed.


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

istream& operator>>(istream& s, GF2X& x);
ostream& operator<<(ostream& s, const GF2X& a);


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

                              Some utility routines

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

long deg(const GF2X& a);  // return deg(a); deg(0) == -1.

GF2 coeff(const GF2X& a, long i);
// returns the coefficient of X^i, or zero if i not in range

GF2 LeadCoeff(const GF2X& a);
// returns leading term of a, or zero if a == 0

GF2 ConstTerm(const GF2X& a);
// returns constant term of a, or zero if a == 0

void SetCoeff(GF2X& x, long i, GF2 a);
void SetCoeff(GF2X& x, long i, long a);
// makes coefficient of X^i equal to a; error is raised if i < 0

void SetCoeff(GF2X& x, long i);
// makes coefficient of X^i equal to 1;  error is raised if i < 0

void SetX(GF2X& x); // x is set to the monomial X

long IsX(const GF2X& a); // test if x = X

void diff(GF2X& x, const GF2X& a);
GF2X diff(const GF2X& a); 
// x = derivative of a


void reverse(GF2X& x, const GF2X& a, long hi);
GF2X reverse(const GF2X& a, long hi);

void reverse(GF2X& x, const GF2X& a);
GF2X reverse(const GF2X& a);

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


void VectorCopy(vec_GF2& x, const GF2X& a, long n);
vec_GF2 VectorCopy(const GF2X& a, long n);
// x = copy of coefficient vector of a of length exactly n.
// input is truncated or padded with zeroes as appropriate.

// Note that there is also a conversion routine from GF2X to vec_GF2
// that makes the length of the vector match the number of coefficients
// of the polynomial.

long weight(const GF2X& a);
// returns the # of nonzero coefficients in a

void GF2XFromBytes(GF2X& x, const unsigned char *p, long n);
GF2X GF2XFromBytes(const unsigned char *p, long n);
// conversion from byte vector to polynomial.
// x = sum(p[i]*X^(8*i), i = 0..n-1), where the bits of p[i] are interpretted
// as a polynomial in the natural way (i.e., p[i] = 1 is interpretted as 1,
// p[i] = 2 is interpretted as X, p[i] = 3 is interpretted as X+1, etc.).
// In the unusual event that characters are wider than 8 bits,
// only the low-order 8 bits of p[i] are used.

void BytesFromGF2X(unsigned char *p, const GF2X& a, long n);
// conversion from polynomial to byte vector.
// p[0..n-1] are computed so that 
//     a = sum(p[i]*X^(8*i), i = 0..n-1) mod X^(8*n),
// where the values p[i] are interpretted as polynomials as in GF2XFromBytes
// above.

long NumBits(const GF2X& a);
// returns number of bits of a, i.e., deg(a) + 1.

long NumBytes(const GF2X& a);
// returns number of bytes of a, i.e., floor((NumBits(a)+7)/8)




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

                             Random Polynomials

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

void random(GF2X& x, long n);
GF2X random_GF2X(long n);
// x = random polynomial of degree < n 



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

                       Arithmetic mod X^n

Required: n >= 0; otherwise, an error is raised.

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

void trunc(GF2X& x, const GF2X& a, long n); // x = a % X^n
GF2X trunc(const GF2X& a, long n); 

void MulTrunc(GF2X& x, const GF2X& a, const GF2X& b, long n);
GF2X MulTrunc(const GF2X& a, const GF2X& b, long n);
// x = a * b % X^n

void SqrTrunc(GF2X& x, const GF2X& a, long n);
GF2X SqrTrunc(const GF2X& a, long n);
// x = a^2 % X^n

void InvTrunc(GF2X& x, const GF2X& a, long n);
GF2X InvTrunc(const GF2X& a, long n);
// computes x = a^{-1} % X^n.  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
GF2XModulus data structure and associated routines below for better
performance.

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

void MulMod(GF2X& x, const GF2X& a, const GF2X& b, const GF2X& f);
GF2X MulMod(const GF2X& a, const GF2X& b, const GF2X& f);
// x = (a * b) % f

void SqrMod(GF2X& x, const GF2X& a, const GF2X& f);
GF2X SqrMod(const GF2X& a, const GF2X& f);
// x = a^2 % f

void MulByXMod(GF2X& x, const GF2X& a, const GF2X& f);
GF2X MulByXMod(const GF2X& a, const GF2X& f);
// x = (a * X) mod f

void InvMod(GF2X& x, const GF2X& a, const GF2X& f);