poly2.cpp

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/*
 * C++ class to implement a polynomial type and to allow 
 * arithmetic on polynomials whose elements are from
 * the finite field GF(2^m)
 *
 * WARNING: This class has been cobbled together for a specific use with
 * the MIRACL library. It is not complete, and may not work in other 
 * applications
 *
 * See Knuth The Art of Computer Programming Vol.2, Chapter 4.6 
 */

#include "poly2.h"

Poly2::Poly2(const GF2m& c,int p)
{
    start=NULL;
    addterm(c,p);
}

Poly2::Poly2(Variable &x)
{
    start=NULL;
    addterm((GF2m)1,1);
}

Poly2 operator*(const GF2m& c,Variable x)
{
    Poly2 t(c,1);
    return t;
}

Poly2& Poly2::operator=(const GF2m& m)
{
    clear();
    if (!m.iszero()) addterm(m,0);
    return *this;
}

Poly2 pow2(Variable x,int n)
{
    Poly2 r((GF2m)1,n);
    return r;
}

Poly2::Poly2(const Poly2& p)
{
    term2 *ptr=p.start;
    term2 *pos=NULL;
    start=NULL;
    while (ptr!=NULL)
    {  
        pos=addterm(ptr->an,ptr->n,pos);
        ptr=ptr->next;
    }    
}

Poly2::~Poly2()
{
   term2 *nx;
   while (start!=NULL)
   {   
       nx=start->next;
       delete start;
       start=nx;
   }
}

GF2m Poly2::coeff(int power)  const
{
    GF2m c=0;
    term2 *ptr=start;
    while (ptr!=NULL)
    {
        if (ptr->n==power)
        {
            c=ptr->an;
            return c;
        }
        ptr=ptr->next;
    }
    return c;
}

GF2m Poly2::F(const GF2m& x) const
{
    GF2m f=0;
    int diff;
    term2 *ptr=start;

// Horner's rule

    if (ptr==NULL) return f;
    f=ptr->an;

    while (ptr->next!=NULL)
    {
        diff=ptr->n-ptr->next->n;
        if (diff==1) f=f*x+ptr->next->an;
        else         f=f*pow(x,diff)+ptr->next->an;    
        ptr=ptr->next;
    }
    f*=pow(x,ptr->n);

    return f;
}

GF2m Poly2:: min() const
{
    term2 *ptr=start;
    if (start==NULL) return (GF2m)0;
    
    while (ptr->next!=NULL) ptr=ptr->next;
    return (ptr->an);
}

Poly2 operator*(const Poly2& a,const Poly2& b)
{
    int i,d,dega,degb,deg;
    GF2m t;
    Poly2 prod;
    term2 *iptr,*pos;
    term2 *ptr=b.start;
    if (&a==&b)
    { // squaring - only diagonal terms count!
        pos=NULL;
        while (ptr!=NULL)
        { // diagonal terms
            pos=prod.addterm(ptr->an*ptr->an,ptr->n+ptr->n,pos);
            ptr=ptr->next;
        }
        return prod;
    }

    dega=degree(a);
    deg=dega;
    degb=degree(b);
    if (degb<dega) deg=degb;  // deg is smallest

    if (deg>=KARAT_BREAK_EVEN)
    { // use fast method 
        int len,m,inc;

        big *A,*B,*C,*T;
        deg=dega;
        if (dega<degb) deg=degb;   // deg is biggest
        m=deg; inc=1;
        while (m!=0) { m/=2; inc++; }

        len=2*(deg+inc);

        A=(big *)mr_alloc(deg+1,sizeof(big));
        B=(big *)mr_alloc(deg+1,sizeof(big));
        C=(big *)mr_alloc(len,sizeof(big));
        T=(big *)mr_alloc(len,sizeof(big));

        char *memc=(char *)memalloc(len);
        char *memt=(char *)memalloc(len);

        for (i=0;i<len;i++)
        {
            C[i]=mirvar_mem(memc,i);
            T[i]=mirvar_mem(memt,i);
        }

        ptr=a.start;
        while (ptr!=NULL)
        {
            A[ptr->n]=getbig(ptr->an);
            ptr=ptr->next;
        }
        ptr=b.start;
        while (ptr!=NULL)
        {
            B[ptr->n]=getbig(ptr->an);
            ptr=ptr->next;
        }

        karmul2_poly(deg+1,T,A,B,C);

        pos=NULL;
        for (d=dega+degb;d>=0;d--)
        {
            t=C[d];
            if (t.iszero()) continue;
            pos=prod.addterm(t,d,pos);
        }

        memkill(memc,len);
        memkill(memt,len);

        mr_free(T);
        mr_free(C);
        mr_free(B);
        mr_free(A);
        return prod;
    }

    while (ptr!=NULL)
    {
        pos=NULL;
        iptr=a.start;
        while (iptr!=NULL)
        {
            pos=prod.addterm(ptr->an*iptr->an,ptr->n+iptr->n,pos);
            iptr=iptr->next;
        }
        ptr=ptr->next;
    }

    return prod;
}

Poly2& Poly2::operator%=(const Poly2& v)
{
    GF2m m,pq;
    int power;
    term2 *rptr=start;
    term2 *vptr=v.start;
    term2 *ptr,*pos;
    if (degree(*this)<degree(v)) return *this;
    m=((GF2m)1/vptr->an);

    while (rptr!=NULL && rptr->n>=vptr->n)
    {
        pq=rptr->an*m;
        power=rptr->n-vptr->n;
        pos=NULL;
        ptr=v.start;
        while (ptr!=NULL)
        {
            pos=addterm(ptr->an*pq,ptr->n+power,pos);
            ptr=ptr->next;
        } 
        rptr=start;
    }
    return *this;
}

Poly2 operator%(const Poly2& u,const Poly2& v)
{
    Poly2 r=u;
    r%=v;
    return r;
}

Poly2 fulldiv(Poly2& r,const Poly2 &v)
{
    Poly2 q;
    GF2m pq,m;
    int power;
    term2 *rptr=r.start;
    term2 *vptr=v.start;
    term2 *ptr,*pos;
    m=(GF2m)1/vptr->an;

    while (rptr!=NULL && rptr->n>=vptr->n)
    {
        pq=rptr->an*m;
        power=rptr->n-vptr->n;
        q.addterm(pq,power);

        pos=NULL;
        ptr=v.start;
        while (ptr!=NULL)
        {
            pos=r.addterm(ptr->an*pq,ptr->n+power,pos);
            ptr=ptr->next;
        } 
        rptr=r.start;
    }
    return q;
}

Poly2 operator/(const Poly2& u,const Poly2 &v)
{
    Poly2 r=u;
    return fulldiv(r,v);
}

void swap(Poly2 &x,Poly2 &y)
{
    term2 *t;
    t=x.start;
    x.start=y.start;
    y.start=t;
}

void makemonic(Poly2& p)
{
    term2 *ptr = p.start;
    p.multerm((GF2m)1/ptr->an,0);
}

// A function to differentiate a polynomial
Poly2 differentiate(const Poly2& orig)
{
    Poly2 newpoly;
    term2 *ptr = orig.start;
    term2 *pos = NULL;
    int power;

    while (ptr!=NULL)
    {
        power = ptr->n;
        if(ptr->n > 0)
            pos = newpoly.addterm(ptr->an*(GF2m)(power % 2),ptr->n-1,pos);

        ptr = ptr->next;
    }

    return newpoly;
}

GF2m resultant(const Poly2& a,const Poly2 &b)
{ // borrowed from NTL
    GF2m lc,r=1;
    int d0,d1,d2;
    if (iszero(a) || iszero(b)) return (GF2m)0;
    if (degree(a)==0 && degree(b)==0) return (GF2m)1;

    Poly2 u=a;
    Poly2 v=b;
    term2 *ptr;

    forever
    {
        d0=degree(u);
        d1=degree(v);
        lc=v.coeff(d1);

        u%=v;

        ptr=v.start; v.start=u.start; u.start=ptr; // swap them

        d2=degree(v);
        if (!iszero(v))
        {
            lc=pow(lc,d0-d2);
            r*=lc;
            if (d0&d1&1) r=-r;
        }
        else
        {
            if (d1==0)
            {
                lc=pow(lc,d0);
                r*=lc;
            }
            else r=0;
            break;
        }
    }

    return r;
}


Poly2 operator-(const Poly2& a,const GF2m& z)
{
    Poly2 p=a;
    p.addterm((z),0);
    return p;
}

Poly2 operator-(const Poly2& a)
{
    Poly2 p=a;
    return p;
}

Poly2 operator-(const Poly2& a,const Poly2& b)
{
    Poly2 sum;
    sum=a;
    sum+=b;
    return sum;
}

Poly2 inverse(const Poly2& f,const Poly2& m)
{
    Poly2 r,s,q,p,x;
    term2 *ptr;
    x=f%m;
    r=1;
    s=0;
    p=m;
    while (!iszero(p))
    { // main euclidean loop */
        q=fulldiv(x,p);
        r+=s*q;
        swap(r,s);
        swap(x,p);
    }
    ptr=x.start;
    r.multerm((GF2m)1/ptr->an,0);
    return r;
}

Poly2 gcd(const Poly2& f,const Poly2& g)
{
    Poly2 a,b;
    a=f; b=g;
    term2 *ptr;

    forever
    {
        if (b.start==NULL)
        {
            ptr=a.start;
            a.multerm((GF2m)1/ptr->an,0);
            return a;
        }
        a%=b;
        if (a.start==NULL)
        {
            ptr=b.start;
            b.multerm((GF2m)1/ptr->an,0);
            return b;
        }
        b%=a;
    }
}

// The extended euclidean algorithm
// The result is returned in an array of Polys, with the gcd
// in first place, then the two coefficients
void egcd(Poly2 result[], const Poly2& u, const Poly2& v)
{
    Poly2 u1, u2, u3, v1, v2, v3, t1, t2, t3, zero, q;
	GF2m t;
	term2 *ptr;
	u1 = 1;
    u2 = 0;
    u3 = u;
    v1 = 0;
    v2 = 1;