void CBitPatternTreeMethod::convertToIntegers(CMatrix< C_FLOAT64 > & values)
{
  bool Problems = false;

  static const C_FLOAT64 limit = 10.0 / std::numeric_limits< C_INT32 >::max();

  size_t Size = values.size();
  size_t Columns = values.numCols();

  C_FLOAT64 * pColumn = values.array();
  C_FLOAT64 * pColumnEnd = pColumn + Columns;
  C_FLOAT64 * pValue = values.array();
  C_FLOAT64 * pValueEnd = pColumn + Size;

  for (; pColumn < pColumnEnd; ++pColumn)
    {
      unsigned C_INT32 Multiplier = 1;
      unsigned C_INT32 m00, m01, m10, m11;
      unsigned C_INT32 maxden = 10000000;
      C_INT32 GCD1, GCD2;
      unsigned C_INT32 ai;

      C_FLOAT64 x;

      for (pValue = pColumn; pValue < pValueEnd; pValue += Columns)
        {
          x = fabs(*pValue);

          if (x < 100 * std::numeric_limits< C_FLOAT64 >::epsilon())
            {
              continue;
            }

          /*
           * Find rational approximation to given real number
           * David Eppstein / UC Irvine / 8 Aug 1993
           *
           * With corrections from:
           *   Arno Formella, May 2008
           *   Stefan Hoops, Sept 2009
           *
           * Based on the theory of continued fractions
           * if x = a1 + 1/(a2 + 1/(a3 + 1/(a4 + ...)))
           * then best approximation is found by truncating this series
           * (with some adjustments in the last term).
           *
           * Note the fraction can be recovered as the first column of the matrix
           *  (a1 1 ) (a2 1 ) (a3 1 ) ...
           *  (1  0 ) (1  0 ) (1  0)
           * Instead of keeping the sequence of continued fraction terms,
           * we just keep the last partial product of these matrices.
           */

          /* initialize matrix */
          m00 = m11 = 1;
          m01 = m10 = 0;

          /* loop finding terms until denom gets too big */
          while (m10 *(ai = (unsigned C_INT32) x) + m11 <= maxden)
            {
              C_INT32 t;
              t = m00 * ai + m01;
              m01 = m00;
              m00 = t;

              t = m10 * ai + m11;
              m11 = m10;
              m10 = t;

              if (fabs(x - (C_FLOAT64) ai) < limit)
                break;     // SH: We reached the numerical precision of the machine;

              x = 1 / (x - (C_FLOAT64) ai);
            }

          if ((C_FLOAT64) m10 *(C_FLOAT64) ai + (C_FLOAT64) m11 > (C_FLOAT64) maxden)
            {
              Problems = true;
            }

          if (fabs(fabs(*pValue) - ((C_FLOAT64) m00) / ((C_FLOAT64) m10)) > 100.0 * std::numeric_limits< C_FLOAT64 >::epsilon())
            {
              ai = (maxden - m11) / m10;
              m00 = m00 * ai + m01;
              m10 = m10 * ai + m11;
            }

          // Find the greatest common divisor (GCD) of the multiplier and the current denominator.
          // Euclidean algorithm
          GCD1 = m10;
          GCD2 = Multiplier;

          GCD(GCD1, GCD2);

          // Calculate the least common multiplier: LCM = v1 * v2 / GCD(v1, v2)
          Multiplier *= m10 / GCD1;
        }

      for (pValue = pColumn; pValue < pValueEnd; pValue += Columns)
        {
//.........这里部分代码省略.........

// Euclid's algorithm
int GCD(int x, int y) {
    if (y == 0)
        return x;
    return GCD(y, x%y);
}

LL LCM(LL a, LL b)
{
    return a * b / GCD(a, b);
}

static int LCMDenoCalculator( int n, int m )
{
	int gcd = GCD( n, m );
	return m/gcd;
}

int main()
{
   long n;
   GF2X a, b, c, c1, ss, ss1, tt, tt1;
   double t;
   long iter, i;

   cout << WD(12,"n") << WD(12,"OldGCD") <<  WD(12,"GCD") << WD(12,"OldXGCD")
        << WD(12, "XGCD") << "\n";

   cout.precision(3);
   cout.setf(ios::scientific);


   for (n = 32; n <= (1L << 18); n = n << 3) {
      random(a, n);
      random(b, n);
      OldGCD(c, a, b);
      GCD(c1, a, b);
      OldXGCD(c, ss, tt, a, b);
      XGCD(c1, ss1, tt1, a, b);
      if (c1 != c || ss1 != ss || tt1 != tt) {
         cerr << "**** GF2XTest FAILED!\n";
         return 1;
      }

      cout << WD(12,n); 

      iter = 0;
      do {
         iter = iter ? (2*iter) : 1;
         t = GetTime();
         for (i = 0; i < iter; i++)
            OldGCD(c, a, b);
         t = GetTime()-t;
      } while (t < 0.5);

      cout << WD(12,t/iter);

      iter = 0;
      do {
         iter = iter ? (2*iter) : 1;
         t = GetTime();
         for (i = 0; i < iter; i++)
            GCD(c, a, b);
         t = GetTime()-t;
      } while (t < 0.5);

      cout << WD(12,t/iter);

      iter = 0;
      do {
         iter = iter ? (2*iter) : 1;
         t = GetTime();
         for (i = 0; i < iter; i++)
            OldXGCD(c, ss, tt, a, b);
         t = GetTime()-t;
      } while (t < 0.5);

      cout << WD(12,t/iter);

      iter = 0;
      do {
         iter = iter ? (2*iter) : 1;
         t = GetTime();
         for (i = 0; i < iter; i++)
            XGCD(c, ss, tt, a, b);
         t = GetTime()-t;
      } while (t < 0.5);

      cout << WD(12,t/iter);

      cout << "\n";
   }

   return 0;
}

int LCM(int x, int y){    
    return x * y / GCD(x, y);
}

long IterIrredTest(const ZZ_pEX& f)
{
   if (deg(f) <= 0) return 0;
   if (deg(f) == 1) return 1;

   ZZ_pEXModulus F;

   build(F, f);
   
   ZZ_pEX h;

   FrobeniusMap(h, F);

   long CompTableSize = 2*SqrRoot(deg(f));

   ZZ_pEXArgument H;

   build(H, h, F, CompTableSize);

   long i, d, limit, limit_sqr;
   ZZ_pEX g, X, t, prod;


   SetX(X);

   i = 0;
   g = h;
   d = 1;
   limit = 2;
   limit_sqr = limit*limit;

   set(prod);


   while (2*d <= deg(f)) {
      sub(t, g, X);
      MulMod(prod, prod, t, F);
      i++;
      if (i == limit_sqr) {
         GCD(t, f, prod);
         if (!IsOne(t)) return 0;

         set(prod);
         limit++;
         limit_sqr = limit*limit;
         i = 0;
      }

      d = d + 1;
      if (2*d <= deg(f)) {
         CompMod(g, g, H, F);
      }
   }

   if (i > 0) {
      GCD(t, f, prod);
      if (!IsOne(t)) return 0;
   }

   return 1;
}

unsigned int LCM( unsigned int a, unsigned int b ) {

  unsigned int tmp=a/GCD(a,b);
  return tmp*b;

}

//.........这里部分代码省略.........
    {14112, 18415, 28, 11278},
    {15004, 15709, 22,  3867},
    {15360, 20485, 24, 12767},
 // {16384, 21845, 16, 12798},
    {17208 ,21931, 24, 18387},
    {18000, 18631, 25,  4208},
    {18816, 24295, 28, 16360},
    {19200, 21607, 40, 35633},
    {21168, 27305, 28, 15407},
    {23040, 23377, 48,  5292},
    {24576, 24929, 48,  5612},
    {27000, 32767, 15, 20021},
    {31104, 31609, 71,  5149},
    {42336, 42799, 21,  5952},
    {46080, 53261, 24, 33409},
    {49140, 57337, 39,  2608},
    {51840, 59527, 72, 21128},
    {61680, 61681, 40,  1273},
    {65536, 65537, 32,  1273},
    {75264, 82603, 56, 36484},
    {84672, 92837, 56, 38520}
  };

#if 0

  for (long i = 0; i < 25; i++) {
    long m = ms[i][1];
    PAlgebra alg(m);
    alg.printout();
    cout << "\n";
    // compute phi(m) directly
    long phim = 0;
    for (long j = 0; j < m; j++)
      if (GCD(j, m) == 1) phim++;

    if (phim != alg.phiM()) cout << "ERROR\n";
  }

  exit(0);


#endif



  // find the first m satisfying phi(m)>=N and d | ord(2) in Z_m^*
  long m = 0;
  for (unsigned i=0; i<sizeof(ms)/sizeof(long[3]); i++) 
    if (ms[i][0]>=N && (ms[i][2] % d) == 0) {
      m = ms[i][1];
      c_m = 0.001 * (double) ms[i][3];
      break;
    }
  if (m==0) Error("Cannot support this L,d combination");
#endif
  //  m = 257;
  FHEcontext context(m);
#if 0
  context.stdev = to_xdouble(0.5); // very low error
#endif
  activeContext = &context; // Mark this as the "current" context

  context.zMstar.printout();
  cout << endl;

// Generate the representation of Z_m^* for a given odd integer m
// and plaintext base p
PAlgebra::PAlgebra(unsigned long mm, unsigned long pp)
{
    m = mm;
    p = pp;

    assert( (m&1) == 1 );
    assert( ProbPrime(p) );
    // replaced by Mahdi after a conversation with Shai
    // assert( m > p && (m % p) != 0 );	// original line
    assert( (m % p) != 0 );
    // end of replace by Mahdi
    assert( m < NTL_SP_BOUND );

    // Compute the generators for (Z/mZ)^*
    vector<unsigned long> classes(m);
    vector<long> orders(m);

    unsigned long i;
    for (i=0; i<m; i++) { // initially each element in its own class
        if (GCD(i,m)!=1)
            classes[i] = 0; // i is not in (Z/mZ)^*
        else
            classes[i] = i;
    }

    // Start building a representation of (Z/mZ)^*, first use the generator p
    conjClasses(classes,p,m);  // merge classes that have a factor of 2

    // The order of p is the size of the equivalence class of 1
    ordP = (unsigned long) count (classes.begin(), classes.end(), 1);

    // Compute orders in (Z/mZ)^*/<p> while comparing to (Z/mZ)^*
    long idx, largest;
    while (true) {
        compOrder(orders,classes,true,m);
        idx = argmax(orders);      // find the element with largest order
        largest = orders[idx];

        if (largest <= 0) break;   // stop comparing to order in (Z/mZ)^*

        // store generator with same order as in (Z/mZ)^*
        gens.push_back(idx);
        ords.push_back(largest);
        conjClasses(classes,idx,m); // merge classes that have a factor of idx
    }
    // Compute orders in (Z/mZ)^*/<p> without comparing to (Z/mZ)^*
    while (true) {
        compOrder(orders,classes,false,m);
        idx = argmax(orders);      // find the element with largest order
        largest = orders[idx];

        if (largest <= 0) break;   // we have the trivial group, we are done

        // store generator with different order than (Z/mZ)^*
        gens.push_back(idx);
        ords.push_back(-largest);  // store with negative sign
        conjClasses(classes,idx,m);  // merge classes that have a factor of idx
    }

    nSlots = qGrpOrd();
    phiM = ordP * nSlots;

    // Allocate space for the various arrays
    T.resize(nSlots);
    dLogT.resize(nSlots*gens.size());
    Tidx.assign(m,-1);    // allocate m slots, initialize them to -1
    zmsIdx.assign(m,-1);  // allocate m slots, initialize them to -1
    for (i=idx=0; i<m; i++) if (GCD(i,m)==1) zmsIdx[i] = idx++;

    // Now fill the Tidx and dLogT translation tables. We identify an element
    // t\in T with its representation t = \prod_{i=0}^n gi^{ei} mod m (where
    // the gi's are the generators in gens[]) , represent t by the vector of
    // exponents *in reverse order* (en,...,e1,e0), and order these vectors
    // in lexicographic order.

    // buffer is initialized to all-zero, which represents 1=\prod_i gi^0
    vector<unsigned long> buffer(gens.size()); // temporaty holds exponents
    i = idx = 0;
    do {
        unsigned long t = exponentiate(buffer);
        for (unsigned long j=0; j<buffer.size(); j++) dLogT[idx++] = buffer[j];
        T[i] = t;       // The i'th element in T it t
        Tidx[t] = i++;  // the index of t in T is i

        // increment buffer by one (in lexigoraphic order)
    } while (nextExpVector(buffer)); // until we cover all the group

    PhimX = Cyclotomic(m); // compute and store Phi_m(X)

    // initialize prods array
    long ndims = gens.size();
    prods.resize(ndims+1);
    prods[ndims] = 1;
    for (long j = ndims-1; j >= 0; j--) {
        prods[j] = OrderOf(j) * prods[j+1];
    }
}

/**************************
//Paillier HE system
//len: length of params p,q
**************************/
void Paillier(int len=512){
  ZZ n, n2, p, q, g, lamda;
  ZZ p1, q1, p1q1;
  ZZ miu;
  
  ZZ m1, m2;
  ZZ BSm, HEm; //baseline and HE result
  ZZ c, c1, c2, cm1, cm2, r;

  //key gen
  start = std::clock();

  GenPrime(p, len);
  GenPrime(q, len);
  mul(n, p, q);
  mul(n2, n, n);

  sub(p1,p,1);
  sub(q1,q,1);
  GCD(lamda,p1,q1);
  mul(p1q1,p1,q1);
  div(lamda, p1q1, lamda);

  RandomBnd(g, n2);

  PowerMod(miu,g,lamda,n2);
  sub(miu, miu, 1);
  div(miu,miu,n); //should add 1?
  InvMod(miu, miu, n);
  
  duration = ( std::clock() - start ) / (double) CLOCKS_PER_SEC;
  cout<<"Pailler Setup:"<< duration <<'\n';

  //Enc
  start = std::clock();
  RandomBnd(m1,n);
  RandomBnd(m2,n);

  RandomBnd(r,n); //enc m1
  PowerMod(c1, g,m1,n2);
  PowerMod(c2, r,n,n2);
  MulMod(cm1, c1,c2, n2);

  RandomBnd(r,n); //enc m2
  PowerMod(c1, g,m2,n2);
  PowerMod(c2, r,n,n2);
  MulMod(cm2, c1,c2, n2);

  duration = ( std::clock() - start ) / (double) CLOCKS_PER_SEC;
  cout<<"Pailler Enc:"<< duration/2 <<'\n';

  //Evaluation
  start = std::clock();
  c=cm1;
  for(int i=0; i<TIMES; i++)
  	MulMod(c,c,cm2,n2);
  duration = ( std::clock() - start ) / (double) CLOCKS_PER_SEC;
  cout<<"Pailler Eval:"<< duration <<'\n';

  //c=cm2;
  //Dec  
  start = std::clock();
  PowerMod(c,c,lamda,n2);
  sub(c,c,1);
  div(c,c,n); //should add 1?
  MulMod(HEm, c, miu, n);  

  duration = ( std::clock() - start ) / (double) CLOCKS_PER_SEC;
  cout<<"Pailler Dec:"<< duration <<'\n';

  //baseline
  BSm=m1;
  for(int i=0; i<TIMES; i++)
  	AddMod(BSm,BSm,m2,n);

  assert(BSm==HEm);
}

int GCD(int a, int b) {
	//printf(">> %d %d\n", a, b);
	if (b==0) return a;
	else return GCD(b, a%b);
};

void main(void){
  printf("%d\n",GCD(6,12));
}

//.........这里部分代码省略.........

          C_INT64 TMP = Identity[CurrentRowIndex];
          Identity[CurrentRowIndex]  = Identity[NonZeroIndex];
          Identity[NonZeroIndex] = TMP;
        }

      if (*(pActiveRowStart + CurrentColumnIndex) < 0)
        {
          for (pRow = pActiveRowStart; pRow < pActiveRowEnd; ++pRow)
            {
              *pRow *= -1;
            }

          Identity[CurrentRowIndex] *= -1;
        }

      // For each row
      pRow = pActiveRowStart + NumCols;
      pIdentity = Identity.array() + CurrentRowIndex + 1;

      C_INT64 ActiveRowValue = *(pActiveRowStart + CurrentColumnIndex);
      *(pActiveRowStart + CurrentColumnIndex) = Identity[CurrentRowIndex];

      for (; pRow < pRowEnd; pRow += NumCols, ++pIdentity)
        {
          C_INT64 RowValue = *(pRow + CurrentColumnIndex);

          if (RowValue == 0)
            continue;

          *(pRow + CurrentColumnIndex) = 0;

          // compute GCD(*pActiveRowStart, *pRow)
          C_INT64 GCD1 = abs64(ActiveRowValue);
          C_INT64 GCD2 = abs64(RowValue);

          GCD(GCD1, GCD2);

          C_INT64 alpha = ActiveRowValue / GCD1;
          C_INT64 beta = RowValue / GCD1;

          // update rest of row
          pActiveRow = pActiveRowStart;
          pCurrent = pRow;
          *pIdentity *= alpha;

          GCD1 = abs64(*pIdentity);

          for (; pActiveRow < pActiveRowEnd; ++pActiveRow, ++pCurrent)
            {
              // Assert that we do not have a numerical overflow.
              assert(fabs(((C_FLOAT64) alpha) *((C_FLOAT64) * pCurrent) - ((C_FLOAT64) beta) *((C_FLOAT64) * pActiveRow)) < std::numeric_limits< C_INT64 >::max());

              *pCurrent = alpha * *pCurrent - beta * *pActiveRow;

              // We check that the row values do not have any common divisor.
              if (GCD1 > 1 &&
                  (GCD2 = abs64(*pCurrent)) > 0)
                {
                  GCD(GCD1, GCD2);
                }
            }

          if (GCD1 > 1)
            {
              *pIdentity /= GCD1;

// TODO: Add the ability to set a maximum number of iterations
inline BigInt FindFactor
( const BigInt& n,
  Int a,
  const PollardRhoCtrl& ctrl )
{
    if( a == 0 || a == -2 )
        Output("WARNING: Problematic choice of Pollard rho shift");
    BigInt tmp, gcd;
    BigInt one(1);

    auto xAdvance =
      [&]( BigInt& x )
      {
        if( ctrl.numSteps == 1 )
        {
            // TODO: Determine if there is a penalty to x *= x
            /*
            tmp = x;
            tmp *= x;
            tmp += a;
            x = tmp;
            x %= n;
            */
            x *= x;
            x += a;
            x %= n;
        }
        else
        {
            PowMod( x, 2*ctrl.numSteps, n, x );
            x += a;
            x %= n;
        }
      };

    auto QAdvance =
      [&]( const BigInt& x, const BigInt& x2, BigInt& Q )
      {
        tmp = x2;
        tmp -= x;
        Q *= tmp;
        Q %= n;
      };

    Int gcdDelay = ctrl.gcdDelay;
    BigInt xi=ctrl.x0;
    BigInt x2i(xi);
    BigInt xiSave=xi, x2iSave=x2i;
    BigInt Qi(1);
    Int k=1, i=1; // it is okay for i to overflow since it is just for printing
    while( true )
    {
        // Advance xi once
        xAdvance( xi );

        // Advance x2i twice
        xAdvance( x2i );
        xAdvance( x2i );

        // Advance Qi
        QAdvance( xi, x2i, Qi );

        if( k >= gcdDelay )
        {
            GCD( Qi, n, gcd );
            if( gcd > one )
            {
                // NOTE: This was not suggested by Pollard's original paper
                if( gcd == n )
                {
                    if( gcdDelay == 1 )
                    {
                        RuntimeError("(x) converged before (x mod p) at i=",i);
                    }
                    else
                    {
                        if( ctrl.progress )
                            Output("Backtracking at i=",i);
                        i = Max( i-(gcdDelay+1), 0 );
                        gcdDelay = 1;
                        xi = xiSave;
                        x2i = x2iSave;
                    }
                }
                else
                {
                    if( ctrl.progress )
                        Output("Found factor ",gcd," at i=",i); 
                    return gcd;
                }
            }

            // NOTE: This was not suggested by Pollard's original paper
            k = 0;
            xiSave = xi;
            x2iSave = x2i;
            Qi = 1;
        }
        ++k;
//.........这里部分代码省略.........

// Reduce plaintext space to a divisor of the original plaintext space
void Ctxt::reducePtxtSpace(long newPtxtSpace)
{
  long g = GCD(ptxtSpace, newPtxtSpace);
  assert (g>1);
  ptxtSpace = g;
}

bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod)
{
	assert(!equiv.IsNegative() && equiv < mod);

	Integer gcd = GCD(equiv, mod);
	if (gcd != Integer::One())
	{
		// the only possible prime p such that p%mod==equiv where GCD(mod,equiv)!=1 is GCD(mod,equiv)
		if (p <= gcd && gcd <= max && IsPrime(gcd))
		{
			p = gcd;
			return true;
		}
		else
			return false;
	}

	BuildPrimeTable();

	if (p <= primeTable[primeTableSize-1])
	{
		word *pItr;

		--p;
		if (p.IsPositive())
			pItr = std::upper_bound(primeTable, primeTable+primeTableSize, p.ConvertToLong());
		else
			pItr = primeTable;

		while (pItr < primeTable+primeTableSize && *pItr%mod != equiv)
			++pItr;

		if (pItr < primeTable+primeTableSize)
		{
			p = *pItr;
			return p <= max;
		}

		p = primeTable[primeTableSize-1]+1;
	}

	assert(p > primeTable[primeTableSize-1]);

	if (mod.IsOdd())
		return FirstPrime(p, max, CRT(equiv, mod, 1, 2, 1), mod<<1);

	p += (equiv-p)%mod;

	if (p>max)
		return false;

	PrimeSieve sieve(p, max, mod);

	while (sieve.NextCandidate(p))
	{
		if (FastProbablePrimeTest(p) && IsPrime(p))
			return true;
	}

	return false;
}

// Encrypts plaintext, result returned in the ciphertext argument. The
// returned value is the plaintext-space for that ciphertext. When called
// with highNoise=true, returns a ciphertext with noise level~q/8.
long FHEPubKey::Encrypt(Ctxt &ctxt, const ZZX& ptxt, long ptxtSpace,
			bool highNoise) const
{
  FHE_TIMER_START;
  assert(this == &ctxt.pubKey);

  if (ptxtSpace != pubEncrKey.ptxtSpace) { // plaintext-space mistamtch
    ptxtSpace = GCD(ptxtSpace, pubEncrKey.ptxtSpace);
    if (ptxtSpace <= 1) Error("Plaintext-space mismatch on encryption");
  }

  // generate a random encryption of zero from the public encryption key
  ctxt = pubEncrKey;  // already an encryption of zero, just not a random one

  // choose a random small scalar r and a small random error vector e,
  // then set ctxt = r*pubEncrKey + ptstSpace*e + (ptxt,0)
  DoubleCRT e(context, context.ctxtPrimes);
  DoubleCRT r(context, context.ctxtPrimes);
  r.sampleSmall();

  for (size_t i=0; i<ctxt.parts.size(); i++) {  // add noise to all the parts
    ctxt.parts[i] *= r;

    if (highNoise && i == 0) {
      // we sample e so that coefficients are uniform over 
      // [-Q/(8*ptxtSpace)..Q/(8*ptxtSpace)]

      ZZ B;
      B = context.productOfPrimes(context.ctxtPrimes);
      B /= ptxtSpace;
      B /= 8;
      e.sampleUniform(B);
    }
    else { 
      e.sampleGaussian();
    }

    e *= ptxtSpace;
    ctxt.parts[i] += e;
  }

  // add in the plaintext
  // FIXME: This relies on the first part, ctxt[0], to have handle to 1
  if (ptxtSpace==2) ctxt.parts[0] += ptxt;

  else { // The general case of ptxtSpace>2: for a ciphertext
         // relative to modulus Q, we add ptxt * Q mod ptxtSpace.
    long QmodP = rem(context.productOfPrimes(ctxt.primeSet), ptxtSpace);
    ctxt.parts[0] += MulMod(ptxt,QmodP,ptxtSpace); // MulMod from module NumbTh
  }

  // fill in the other ciphertext data members
  ctxt.ptxtSpace = ptxtSpace;

  if (highNoise) {
    // hack: we set noiseVar to Q^2/8, which is just below threshold 
    // that will signal an error

    ctxt.noiseVar = xexp(2*context.logOfProduct(context.ctxtPrimes) - log(8.0));

  }
  else {
    // We have <skey,ctxt>= r*<skey,pkey> +p*(e0+e1*s) +m, where VAR(<skey,pkey>)
    // is recorded in pubEncrKey.noiseVar, VAR(ei)=sigma^2*phi(m), and VAR(s) is
    // determined by the secret-key Hamming weight (skHwt). 
    // VAR(r)=phi(m)/2, hence the expected size squared is bounded by:
    // E(X^2) <= pubEncrKey.noiseVar *phi(m) *stdev^2
    //                               + p^2*sigma^2 *phi(m) *(skHwt+1) + p^2
  
    long hwt = skHwts[0];
    xdouble phim = to_xdouble(context.zMStar.getPhiM());
    xdouble sigma2 = context.stdev * context.stdev;
    xdouble p2 = to_xdouble(ptxtSpace) * to_xdouble(ptxtSpace);
    ctxt.noiseVar = pubEncrKey.noiseVar*phim*0.5 
                    + p2*sigma2*phim*(hwt+1)*context.zMStar.get_cM() + p2;
  }
  return ptxtSpace;
}

void TranslateBetweenGrids
( const DistMatrix<T,MC,MR>& A, DistMatrix<T,MC,MR>& B ) 
{
    DEBUG_ONLY(CSE cse("copy::TranslateBetweenGrids [MC,MR]"))

    B.Resize( A.Height(), A.Width() );
    // Just need to ensure that each viewing comm contains the other team's
    // owning comm. Congruence is too strong.

    // Compute the number of process rows and columns that each process
    // needs to send to.
    const Int colStride = B.ColStride();
    const Int rowStride = B.RowStride();
    const Int colRank = B.ColRank();
    const Int rowRank = B.RowRank();
    const Int colStrideA = A.ColStride();
    const Int rowStrideA = A.RowStride();
    const Int colGCD = GCD( colStride, colStrideA );
    const Int rowGCD = GCD( rowStride, rowStrideA );
    const Int colLCM = colStride*colStrideA / colGCD;
    const Int rowLCM = rowStride*rowStrideA / rowGCD;
    const Int numColSends = colStride / colGCD;
    const Int numRowSends = rowStride / rowGCD;

    const Int colAlign = B.ColAlign();
    const Int rowAlign = B.RowAlign();
    const Int colAlignA = A.ColAlign();
    const Int rowAlignA = A.RowAlign();

    const bool inBGrid = B.Participating();
    const bool inAGrid = A.Participating();
    if( !inBGrid && !inAGrid )
        return;

    const Int maxSendSize =
        (A.Height()/(colStrideA*numColSends)+1) *
        (A.Width()/(rowStrideA*numRowSends)+1);

    // Translate the ranks from A's VC communicator to B's viewing so that
    // we can match send/recv communicators. Since A's VC communicator is not
    // necessarily defined on every process, we instead work with A's owning
    // group and account for row-major ordering if necessary.
    const int sizeA = A.Grid().Size();
    vector<int> rankMap(sizeA), ranks(sizeA);
    if( A.Grid().Order() == COLUMN_MAJOR )
    {
        for( int j=0; j<sizeA; ++j )
            ranks[j] = j;
    }
    else
    {
        // The (i,j) = i + j*colStrideA rank in the column-major ordering is
        // equal to the j + i*rowStrideA rank in a row-major ordering.
        // Since we desire rankMap[i+j*colStrideA] to correspond to process
        // (i,j) in A's grid's rank in this viewing group, ranks[i+j*colStrideA]
        // should correspond to process (i,j) in A's owning group. Since the
        // owning group is ordered row-major in this case, its rank is
        // j+i*rowStrideA. Note that setting
        // ranks[j+i*rowStrideA] = i+j*colStrideA is *NOT* valid.
        for( int i=0; i<colStrideA; ++i )
            for( int j=0; j<rowStrideA; ++j )
                ranks[i+j*colStrideA] = j+i*rowStrideA;
    }
    mpi::Translate
    ( A.Grid().OwningGroup(), sizeA, &ranks[0],
      B.Grid().ViewingComm(), &rankMap[0] );

    // Have each member of A's grid individually send to all numRow x numCol
    // processes in order, while the members of this grid receive from all
    // necessary processes at each step.
    Int requiredMemory = 0;
    if( inAGrid )
        requiredMemory += maxSendSize;
    if( inBGrid )
        requiredMemory += maxSendSize;
    vector<T> auxBuf( requiredMemory );
    Int offset = 0;
    T* sendBuf = &auxBuf[offset];
    if( inAGrid )
        offset += maxSendSize;
    T* recvBuf = &auxBuf[offset];

    Int recvRow = 0; // avoid compiler warnings...
    if( inAGrid )
        recvRow = Mod(Mod(A.ColRank()-colAlignA,colStrideA)+colAlign,colStride);
    for( Int colSend=0; colSend<numColSends; ++colSend )
    {
        Int recvCol = 0; // avoid compiler warnings...
        if( inAGrid )
            recvCol=Mod(Mod(A.RowRank()-rowAlignA,rowStrideA)+rowAlign,
                        rowStride);
        for( Int rowSend=0; rowSend<numRowSends; ++rowSend )
        {
            mpi::Request sendRequest;
            // Fire off this round of non-blocking sends
            if( inAGrid )
            {
                // Pack the data
                Int sendHeight = Length(A.LocalHeight(),colSend,numColSends);
                Int sendWidth = Length(A.LocalWidth(),rowSend,numRowSends);
//.........这里部分代码省略.........

int GCD(int num1, int num2) {
    if ( num1 % num2 == 0)
        return num2;
    else
        return GCD( num2, num1 % num2);
}