zz_px1.cpp

数值算法库for Windows

#include <NTL/ZZ_pX.h>

#include <NTL/new.h>

NTL_START_IMPL




long divide(ZZ_pX& q, const ZZ_pX& a, const ZZ_pX& b)
{
   if (IsZero(b)) {
      if (IsZero(a)) {
         clear(q);
         return 1;
      }
      else
         return 0;
   }

   ZZ_pX lq, r;
   DivRem(lq, r, a, b);
   if (!IsZero(r)) return 0; 
   q = lq;
   return 1;
}

long divide(const ZZ_pX& a, const ZZ_pX& b)
{
   if (IsZero(b)) return IsZero(a);
   ZZ_pX lq, r;
   DivRem(lq, r, a, b);
   if (!IsZero(r)) return 0; 
   return 1;
}




void ZZ_pXMatrix::operator=(const ZZ_pXMatrix& M)
{
   elts[0][0] = M.elts[0][0];
   elts[0][1] = M.elts[0][1];
   elts[1][0] = M.elts[1][0];
   elts[1][1] = M.elts[1][1];
}


void RightShift(ZZ_pX& x, const ZZ_pX& a, long n)
{
   if (n < 0) {
      if (n < -NTL_MAX_LONG) Error("overflow in RightShift");
      LeftShift(x, a, -n);
      return;
   }

   long da = deg(a);
   long i;
 
   if (da < n) {
      clear(x);
      return;
   }

   if (&x != &a)
      x.rep.SetLength(da-n+1);

   for (i = 0; i <= da-n; i++)
      x.rep[i] = a.rep[i+n];

   if (&x == &a)
      x.rep.SetLength(da-n+1);

   x.normalize();
}

void LeftShift(ZZ_pX& x, const ZZ_pX& a, long n)
{
   if (n < 0) {
      if (n < -NTL_MAX_LONG) Error("overflow in LeftShift");
      RightShift(x, a, -n);
      return;
   }

   if (n >= (1L << (NTL_BITS_PER_LONG-4)))
      Error("overflow in LeftShift");

   if (IsZero(a)) {
      clear(x);
      return;
   }

   long m = a.rep.length();

   x.rep.SetLength(m+n);

   long i;
   for (i = m-1; i >= 0; i--)
      x.rep[i+n] = a.rep[i];

   for (i = 0; i < n; i++)
      clear(x.rep[i]);
}


void ShiftAdd(ZZ_pX& U, const ZZ_pX& V, long n)
// assumes input does not alias output
{
   if (IsZero(V))
      return;

   long du = deg(U);
   long dv = deg(V);

   long d = max(du, n+dv);

   U.rep.SetLength(d+1);
   long i;

   for (i = du+1; i <= d; i++)
      clear(U.rep[i]);

   for (i = 0; i <= dv; i++)
      add(U.rep[i+n], U.rep[i+n], V.rep[i]);

   U.normalize();
}

void ShiftSub(ZZ_pX& U, const ZZ_pX& V, long n)
// assumes input does not alias output
{
   if (IsZero(V))
      return;

   long du = deg(U);
   long dv = deg(V);

   long d = max(du, n+dv);

   U.rep.SetLength(d+1);
   long i;

   for (i = du+1; i <= d; i++)
      clear(U.rep[i]);

   for (i = 0; i <= dv; i++)
      sub(U.rep[i+n], U.rep[i+n], V.rep[i]);

   U.normalize();
}


void mul(ZZ_pX& U, ZZ_pX& V, const ZZ_pXMatrix& M)
// (U, V)^T = M*(U, V)^T
{
   long d = deg(U) - deg(M(1,1));
   long k = NextPowerOfTwo(d - 1);

   // When the GCD algorithm is run on polynomials of degree n, n-1, 
   // where n is a power of two, then d-1 is likely to be a power of two.
   // It would be more natural to set k = NextPowerOfTwo(d+1), but this
   // would be much less efficient in this case.

   // We optimize this case, as it does sometimes arise naturally
   // in some situations.

   long n = (1L << k);
   long xx;
   ZZ_p a0, a1, b0, b1, c0, d0, u0, u1, v0, v1, nu0, nu1, nv0;
   static ZZ t1, t2;

   if (n == d-1)
      xx = 1;
   else if (n == d)
      xx = 2;
   else 
      xx = 3;

   switch (xx) {
   case 1:
      GetCoeff(a0, M(0,0), 0);
      GetCoeff(a1, M(0,0), 1);
      GetCoeff(b0, M(0,1), 0);
      GetCoeff(b1, M(0,1), 1);
      GetCoeff(c0, M(1,0), 0);
      GetCoeff(d0, M(1,1), 0);

      GetCoeff(u0, U, 0);
      GetCoeff(u1, U, 1);
      GetCoeff(v0, V, 0);
      GetCoeff(v1, V, 1);

      mul(t1, rep(a0), rep(u0));
      mul(t2, rep(b0), rep(v0));
      add(t1, t1, t2); 
      conv(nu0, t1);

      mul(t1, rep(a1), rep(u0));
      mul(t2, rep(a0), rep(u1));
      add(t1, t1, t2);
      mul(t2, rep(b1), rep(v0));
      add(t1, t1, t2);
      mul(t2, rep(b0), rep(v1));
      add(t1, t1, t2);
      conv(nu1, t1);

      mul(t1, rep(c0), rep(u0));
      mul(t2, rep(d0), rep(v0));
      add (t1, t1, t2);
      conv(nv0, t1);
   
      break;

   case 2:
      GetCoeff(a0, M(0,0), 0);
      GetCoeff(b0, M(0,1), 0);

      GetCoeff(u0, U, 0);
      GetCoeff(v0, V, 0);

      mul(t1, rep(a0), rep(u0));
      mul(t2, rep(b0), rep(v0));
      add(t1, t1, t2); 
      conv(nu0, t1);

      break;

   case 3:
      break;

   }

   FFTRep RU(INIT_SIZE, k), RV(INIT_SIZE, k), R1(INIT_SIZE, k), 
          R2(INIT_SIZE, k);

   ToFFTRep(RU, U, k);  
   ToFFTRep(RV, V, k);  

   ToFFTRep(R1, M(0,0), k);
   mul(R1, R1, RU);
   ToFFTRep(R2, M(0,1), k);
   mul(R2, R2, RV);
   add(R1, R1, R2);
   FromFFTRep(U, R1, 0, d);

   ToFFTRep(R1, M(1,0), k);
   mul(R1, R1, RU);
   ToFFTRep(R2, M(1,1), k);
   mul(R2, R2, RV);
   add(R1, R1, R2);
   FromFFTRep(V, R1, 0, d-1);

   // now fix-up results

   switch (xx) {
   case 1:
      GetCoeff(u0, U, 0);
      sub(u0, u0, nu0);
      SetCoeff(U, d-1, u0);
      SetCoeff(U, 0, nu0);

      GetCoeff(u1, U, 1);
      sub(u1, u1, nu1);
      SetCoeff(U, d, u1);
      SetCoeff(U, 1, nu1);

      GetCoeff(v0, V, 0);
      sub(v0, v0, nv0);
      SetCoeff(V, d-1, v0);
      SetCoeff(V, 0, nv0);

      break;
      

   case 2:
      GetCoeff(u0, U, 0);
      sub(u0, u0, nu0);
      SetCoeff(U, d, u0);
      SetCoeff(U, 0, nu0);

      break;

   }
}


void mul(ZZ_pXMatrix& A, ZZ_pXMatrix& B, ZZ_pXMatrix& C)
// A = B*C, B and C are destroyed
{
   long db = deg(B(1,1));
   long dc = deg(C(1,1));
   long da = db + dc;

   long k = NextPowerOfTwo(da+1);

   FFTRep B00, B01, B10, B11, C0, C1, T1, T2;
   
   ToFFTRep(B00, B(0,0), k); B(0,0).kill();
   ToFFTRep(B01, B(0,1), k); B(0,1).kill();
   ToFFTRep(B10, B(1,0), k); B(1,0).kill();
   ToFFTRep(B11, B(1,1), k); B(1,1).kill();

   ToFFTRep(C0, C(0,0), k);  C(0,0).kill();
   ToFFTRep(C1, C(1,0), k);  C(1,0).kill();

   mul(T1, B00, C0);
   mul(T2, B01, C1);
   add(T1, T1, T2);
   FromFFTRep(A(0,0), T1, 0, da);

   mul(T1, B10, C0);
   mul(T2, B11, C1);
   add(T1, T1, T2);
   FromFFTRep(A(1,0), T1, 0, da);

   ToFFTRep(C0, C(0,1), k);  C(0,1).kill();
   ToFFTRep(C1, C(1,1), k);  C(1,1).kill();

   mul(T1, B00, C0);
   mul(T2, B01, C1);
   add(T1, T1, T2);
   FromFFTRep(A(0,1), T1, 0, da);

   mul(T1, B10, C0);
   mul(T2, B11, C1);
   add(T1, T1, T2);
   FromFFTRep(A(1,1), T1, 0, da);
}

void IterHalfGCD(ZZ_pXMatrix& M_out, ZZ_pX& U, ZZ_pX& V, long d_red)
{
   M_out(0,0).SetMaxLength(d_red);
   M_out(0,1).SetMaxLength(d_red);
   M_out(1,0).SetMaxLength(d_red);
   M_out(1,1).SetMaxLength(d_red);

   set(M_out(0,0));   clear(M_out(0,1));
   clear(M_out(1,0)); set(M_out(1,1));

   long goal = deg(U) - d_red;

   if (deg(V) <= goal)
      return;

   ZZVec tmp(deg(U)+1, ZZ_pInfo->ExtendedModulusSize);
   ZZ_pX Q, t(INIT_SIZE, d_red);

   while (deg(V) > goal) {
      PlainDivRem(Q, U, U, V, tmp);
      swap(U, V);

      mul(t, Q, M_out(1,0));
      sub(t, M_out(0,0), t);
      M_out(0,0) = M_out(1,0);
      M_out(1,0) = t;

      mul(t, Q, M_out(1,1));
      sub(t, M_out(0,1), t);
      M_out(0,1) = M_out(1,1);
      M_out(1,1) = t;
   }
}
   


void HalfGCD(ZZ_pXMatrix& M_out, const ZZ_pX& U, const ZZ_pX& V, long d_red)
{
   if (IsZero(V) || deg(V) <= deg(U) - d_red) {
      set(M_out(0,0));   clear(M_out(0,1));
      clear(M_out(1,0)); set(M_out(1,1));
 
      return;
   }


   long n = deg(U) - 2*d_red + 2;
   if (n < 0) n = 0;

   ZZ_pX U1, V1;

   RightShift(U1, U, n);
   RightShift(V1, V, n);

   if (d_red <= NTL_ZZ_pX_HalfGCD_CROSSOVER) {
      IterHalfGCD(M_out, U1, V1, d_red);
      return;
   }

   long d1 = (d_red + 1)/2;
   if (d1 < 1) d1 = 1;
   if (d1 >= d_red) d1 = d_red - 1;

   ZZ_pXMatrix M1;

   HalfGCD(M1, U1, V1, d1);
   mul(U1, V1, M1);

   long d2 = deg(V1) - deg(U) + n + d_red;

   if (IsZero(V1) || d2 <= 0) {
      M_out = M1;
      return;
   }


   ZZ_pX Q;
   ZZ_pXMatrix M2;

   DivRem(Q, U1, U1, V1);
   swap(U1, V1);

   HalfGCD(M2, U1, V1, d2);

   ZZ_pX t(INIT_SIZE, deg(M1(1,1))+deg(Q)+1);

   mul(t, Q, M1(1,0));
   sub(t, M1(0,0), t);
   swap(M1(0,0), M1(1,0));
   swap(M1(1,0), t);

   t.kill();

   t.SetMaxLength(deg(M1(1,1))+deg(Q)+1);

   mul(t, Q, M1(1,1));
   sub(t, M1(0,1), t);
   swap(M1(0,1), M1(1,1));
   swap(M1(1,1), t);

   t.kill();

   mul(M_out, M2, M1); 
}




void XHalfGCD(ZZ_pXMatrix& M_out, ZZ_pX& U, ZZ_pX& V, long d_red)
{
   if (IsZero(V) || deg(V) <= deg(U) - d_red) {
      set(M_out(0,0));   clear(M_out(0,1));
      clear(M_out(1,0)); set(M_out(1,1));
 
      return;
   }

   long du = deg(U);

   if (d_red <= NTL_ZZ_pX_HalfGCD_CROSSOVER) {
      IterHalfGCD(M_out, U, V, d_red);
      return;
   }

   long d1 = (d_red + 1)/2;
   if (d1 < 1) d1 = 1;
   if (d1 >= d_red) d1 = d_red - 1;

   ZZ_pXMatrix M1;

   HalfGCD(M1, U, V, d1);
   mul(U, V, M1);

   long d2 = deg(V) - du + d_red;

   if (IsZero(V) || d2 <= 0) {
      M_out = M1;
      return;
   }


   ZZ_pX Q;
   ZZ_pXMatrix M2;

   DivRem(Q, U, U, V);
   swap(U, V);

   XHalfGCD(M2, U, V, d2);

   ZZ_pX t(INIT_SIZE, deg(M1(1,1))+deg(Q)+1);

   mul(t, Q, M1(1,0));
   sub(t, M1(0,0), t);
   swap(M1(0,0), M1(1,0));
   swap(M1(1,0), t);

   t.kill();

   t.SetMaxLength(deg(M1(1,1))+deg(Q)+1);

   mul(t, Q, M1(1,1));
   sub(t, M1(0,1), t);
   swap(M1(0,1), M1(1,1));
   swap(M1(1,1), t);

   t.kill();

   mul(M_out, M2, M1); 
}

void HalfGCD(ZZ_pX& U, ZZ_pX& V)
{
   long d_red = (deg(U)+1)/2;

   if (IsZero(V) || deg(V) <= deg(U) - d_red) {
      return;
   }

   long du = deg(U);


   long d1 = (d_red + 1)/2;
   if (d1 < 1) d1 = 1;
   if (d1 >= d_red) d1 = d_red - 1;

   ZZ_pXMatrix M1;

   HalfGCD(M1, U, V, d1);
   mul(U, V, M1);

   long d2 = deg(V) - du + d_red;

   if (IsZero(V) || d2 <= 0) {
      return;
   }

   M1(0,0).kill();
   M1(0,1).kill();
   M1(1,0).kill();
   M1(1,1).kill();


   ZZ_pX Q;

   DivRem(Q, U, U, V);
   swap(U, V);

   HalfGCD(M1, U, V, d2);

   mul(U, V, M1); 
}


void GCD(ZZ_pX& d, const ZZ_pX& u, const ZZ_pX& v)
{
   ZZ_pX u1, v1;

   u1 = u;
   v1 = v;

   if (deg(u1) == deg(v1)) {
      if (IsZero(u1)) {
         clear(d);
         return;
      }

      rem(v1, v1, u1);
   }
   else if (deg(u1) < deg(v1)) {
      swap(u1, v1);
   }

   // deg(u1) > deg(v1)

   while (deg(u1) > NTL_ZZ_pX_GCD_CROSSOVER && !IsZero(v1)) {
      HalfGCD(u1, v1);

      if (!IsZero(v1)) {
         rem(u1, u1, v1);
         swap(u1, v1);
      }
   }

   PlainGCD(d, u1, v1);
}



void XGCD(ZZ_pX& d, ZZ_pX& s, ZZ_pX& t, const ZZ_pX& a, const ZZ_pX& b)
{
   ZZ_p w;

   if (IsZero(a) && IsZero(b)) {
      clear(d);
      set(s);
      clear(t);
      return;
   }

   ZZ_pX U, V, Q;

   U = a;
   V = b;

   long flag = 0;

   if (deg(U) == deg(V)) {
      DivRem(Q, U, U, V);
      swap(U, V);
      flag = 1;
   }
   else if (deg(U) < deg(V)) {
      swap(U, V);
      flag = 2;
   }

   ZZ_pXMatrix M;

   XHalfGCD(M, U, V, deg(U)+1);

   d = U;

   if (flag == 0) {
      s = M(0,0); 
      t = M(0,1);
   }
   else if (flag == 1) {
      s = M(0,1);
      mul(t, Q, M(0,1));
      sub(t, M(0,0), t);
   }
   else {  /* flag == 2 */
      s = M(0,1);
      t = M(0,0);
   }

   // normalize

   inv(w, LeadCoeff(d));
   mul(d, d, w);
   mul(s, s, w);
   mul(t, t, w);
}

      





void IterBuild(ZZ_p* a, long n)
{
   long i, k;
   ZZ_p b, t;

   if (n <= 0) return;

   negate(a[0], a[0]);

   for (k = 1; k <= n-1; k++) {
      negate(b, a[k]);
      add(a[k], b, a[k-1]);
      for (i = k-1; i >= 1; i--) {
         mul(t, a[i], b);