}

   long r = DivRem(qq, a, b);
   if (r) return 0;
   q = qq;
   return 1;
}

long divide(const ZZ& a, long b)
{
   if (!b) return IsZero(a);
   if (b == 1) {
      return 1;
   }

   long r = rem(a,  b);
   return (r == 0);
}
   


long RandomPrime_long(long l, long NumTrials)
{
   if (l <= 1 || l >= NTL_BITS_PER_LONG)
      Error("RandomPrime: length out of range");

   long n;
   do {
      n = RandomLen_long(l);
   } while (!ProbPrime(n, NumTrials));

   return n;
}


PrimeSeq::PrimeSeq()
{
   movesieve = 0;
   movesieve_mem = 0;
   pshift = -1;
   pindex = -1;
   exhausted = 0;
}

PrimeSeq::~PrimeSeq()
{
   if (movesieve_mem)
      free(movesieve_mem);
}

long PrimeSeq::next()
{
   if (exhausted) {
      return 0;
   }

   if (pshift < 0) {
      shift(0);
      return 2;
   }

   for (;;) {
      char *p = movesieve;
      long i = pindex;

      while ((++i) < NTL_PRIME_BND) {
         if (p[i]) {
            pindex = i;
            return pshift + 2 * i + 3;
         }
      }

      long newshift = pshift + 2*NTL_PRIME_BND;

      if (newshift > 2 * NTL_PRIME_BND * (2 * NTL_PRIME_BND + 1)) {
         /* end of the road */
         exhausted = 1;
         return 0;
      }

      shift(newshift);
   }
}

static char *lowsieve = 0;

void PrimeSeq::shift(long newshift)
{
   long i;
   long j;
   long jstep;
   long jstart;
   long ibound;
   char *p;

   if (!lowsieve)
      start();

   pindex = -1;
   exhausted = 0;

   if (newshift < 0) {
      pshift = -1;
      return;
   }

   if (newshift == pshift) return;

   pshift = newshift;

   if (pshift == 0) {
      movesieve = lowsieve;
   } 
   else {
      if (!movesieve_mem) {
         movesieve_mem = (char *) NTL_MALLOC(NTL_PRIME_BND, 1, 0);
         if (!movesieve_mem) 
            Error("out of memory in PrimeSeq");
      }

      p = movesieve = movesieve_mem;
      for (i = 0; i < NTL_PRIME_BND; i++)
         p[i] = 1;

      jstep = 3;
      ibound = pshift + 2 * NTL_PRIME_BND + 1;
      for (i = 0; jstep * jstep <= ibound; i++) {
         if (lowsieve[i]) {
            if (!((jstart = (pshift + 2) / jstep + 1) & 1))
               jstart++;
            if (jstart <= jstep)
               jstart = jstep;
            jstart = (jstart * jstep - pshift - 3) / 2;
            for (j = jstart; j < NTL_PRIME_BND; j += jstep)
               p[j] = 0;
         }
         jstep += 2;
      }
   }
}


void PrimeSeq::start()
{
   long i;
   long j;
   long jstep;
   long jstart;
   long ibnd;
   char *p;

   p = lowsieve = (char *) NTL_MALLOC(NTL_PRIME_BND, 1, 0);
   if (!p)
      Error("out of memory in PrimeSeq");

   for (i = 0; i < NTL_PRIME_BND; i++)
      p[i] = 1;
      
   jstep = 1;
   jstart = -1;
   ibnd = (SqrRoot(2 * NTL_PRIME_BND + 1) - 3) / 2;
   for (i = 0; i <= ibnd; i++) {
      jstart += 2 * ((jstep += 2) - 1);
      if (p[i])
         for (j = jstart; j < NTL_PRIME_BND; j += jstep)
            p[j] = 0;
   }
}

void PrimeSeq::reset(long b)
{
   if (b > (2*NTL_PRIME_BND+1)*(2*NTL_PRIME_BND+1)) {
      exhausted = 1;
      return;
   }

   if (b <= 2) {
      shift(-1);
      return;
   }

   if ((b & 1) == 0) b++;

   shift(((b-3) / (2*NTL_PRIME_BND))* (2*NTL_PRIME_BND));
   pindex = (b - pshift - 3)/2 - 1;
}
 
long Jacobi(const ZZ& aa, const ZZ& nn)
{
   ZZ a, n;
   long t, k;
   long d;

   a = aa;
   n = nn;
   t = 1;

   while (a != 0) {
      k = MakeOdd(a);
      d = trunc_long(n, 3);
      if ((k & 1) && (d == 3 || d == 5)) t = -t;

      if (trunc_long(a, 2) == 3 && (d & 3) == 3) t = -t;
      swap(a, n);
      rem(a, a, n);
   }

   if (n == 1)
      return t;
   else
      return 0;
}


void SqrRootMod(ZZ& x, const ZZ& aa, const ZZ& nn)
{
   if (aa == 0 || aa == 1) {
      x = aa;
      return;
   }

   // at this point, we msut have nn >= 5

   long i, k;
   ZZ ma, n, t, u, v, e;
   ZZ t1, t2, t3;

   n = nn;
   NegateMod(ma, aa, n);

   // find t such that t^2 - 4*a is not a square

   MulMod(t1, ma, 4, n);
   do {
      RandomBnd(t, n);
      SqrMod(t2, t, n);
      AddMod(t2, t2, t1, n);
   } while (Jacobi(t2, n) != -1);

   // compute u*X + v = X^{(n+1)/2} mod f, where f = X^2 - t*X + a

   add(e, n, 1);
   RightShift(e, e, 1);

   u = 0;
   v = 1;

   k = NumBits(e);

   for (i = k - 1; i >= 0; i--) {
      SqrMod(t1, u, n);
      SqrMod(t2, v, n);
      MulMod(t3, u, v, n);
      MulMod(t3, t3, 2, n);
      MulMod(u, t1, t, n);
      AddMod(u, u, t3, n);
      MulMod(v, t1, ma, n);
      AddMod(v, v, t2, n);

      if (bit(e, i)) {
         MulMod(t1, u, t, n);
         AddMod(t1, t1, v, n);
         MulMod(v, u, ma, n);
         u = t1;
      }

   }

   x = v;
}



// Chinese Remaindering.
//
// This version in new to v3.7, and is significantly
// simpler and faster than the previous version.
//
// This function takes as input g, a, G, p,
// such that a > 0, 0 <= G < p, and gcd(a, p) = 1.
// It computes a' = a*p and g' such that 
//   * g' = g (mod a);
//   * g' = G (mod p);
//   * -a'/2 < g' <= a'/2.
// It then sets g := g' and a := a', and returns 1 iff g has changed.
//
// Under normal use, the input value g satisfies -a/2 < g <= a/2;
// however, this was not documented or enforced in earlier versions,
// so to maintain backward compatability, no restrictions are placed
// on g.  This routine runs faster, though, if -a/2 < g <= a/2,
// and the first thing the routine does is to make this condition
// hold.
//
// Also, under normal use, both a and p are odd;  however, the routine
// will still work even if this is not so.
//
// The routine is based on the following simple fact.
//
// Let -a/2 < g <= a/2, and let h satisfy
//   * g + a h = G (mod p);
//   * -p/2 < h <= p/2.
// Further, if p = 2*h and g > 0, set
//   g' := g - a h;
// otherwise, set
//   g' := g + a h.
// Then g' so defined satisfies the above requirements.
//
// It is trivial to see that g's satisfies the congruence conditions.
// The only thing is to check that the "balancing" condition
// -a'/2 < g' <= a'/2 also holds.


long CRT(ZZ& gg, ZZ& a, long G, long p)
{
   if (p >= NTL_SP_BOUND) {
      ZZ GG, pp;
      conv(GG, G);
      conv(pp, p);
      return CRT(gg, a, GG, pp);
   }

   long modified = 0;

   ZZ g;

   if (!CRTInRange(gg, a)) {
      modified = 1;
      ZZ a1;
      rem(g, gg, a);
      RightShift(a1, a, 1);
      if (g > a1) sub(g, g, a);
   }
   else
      g = gg;


   long p1;
   p1 = p >> 1;

   long a_inv;
   a_inv = rem(a, p);
   a_inv = InvMod(a_inv, p);

   long h;
   h = rem(g, p);
   h = SubMod(G, h, p);
   h = MulMod(h, a_inv, p);
   if (h > p1)
      h = h - p;

   if (h != 0) {
      modified = 1;
      ZZ ah;
      mul(ah, a, h);

      if (!(p & 1) && g > 0 && (h == p1))
         sub(g, g, ah);
      else
         add(g, g, ah);
   }

   mul(a, a, p);
   gg = g;

   return modified;
}

long CRT(ZZ& gg, ZZ& a, const ZZ& G, const ZZ& p)
{
   long modified = 0;

   ZZ g;

   if (!CRTInRange(gg, a)) {
      modified = 1;
      ZZ a1;
      rem(g, gg, a);
      RightShift(a1, a, 1);
      if (g > a1) sub(g, g, a);
   }
   else
      g = gg;


   ZZ p1;
   RightShift(p1, p, 1);

   ZZ a_inv;
   rem(a_inv, a, p);
   InvMod(a_inv, a_inv, p);

   ZZ h;
   rem(h, g, p);
   SubMod(h, G, h, p);
   MulMod(h, h, a_inv, p);
   if (h > p1)
      sub(h, h, p);

   if (h != 0) {
      modified = 1;
      ZZ ah;
      mul(ah, a, h);

      if (!IsOdd(p) && g > 0 &&  (h == p1))
         sub(g, g, ah);
      else
         add(g, g, ah);
   }

   mul(a, a, p);
   gg = g;

   return modified;
}



void sub(ZZ& x, const ZZ& a, long b)
{
   static ZZ B;
   conv(B, b);
   sub(x, a, B);
}

void sub(ZZ& x, long a, const ZZ& b)
{
   static ZZ A;
   conv(A, a);
   sub(x, A, b);
}


void power2(ZZ& x, long e)
{
   if (e < 0) Error("power2: negative exponent");
   set(x);
   LeftShift(x, x, e);
}

   
void conv(ZZ& x, const char *s)
{
   long c;
   long cval;
   long sign;
   long ndigits;
   long acc;
   long i = 0;

   static ZZ a;

   if (!s) Error("bad ZZ input");

   if (!iodigits) InitZZIO();

   a = 0;

   c = s[i];
   while (IsWhiteSpace(c)) {
      i++;
      c = s[i];
   }

   if (c == '-') {
      sign = -1;
      i++;
      c = s[i];
   }
   else
      sign = 1;

   cval = CharToIntVal(c);
   if (cval < 0 || cval > 9) Error("bad ZZ input");

   ndigits = 0;
   acc = 0;
   while (cval >= 0 && cval <= 9) {
      acc = acc*10 + cval;
      ndigits++;

      if (ndigits == iodigits) {
         mul(a, a, ioradix);
         add(a, a, acc);
         ndigits = 0;
         acc = 0;
      }

      i++;
      c = s[i];
      cval = CharToIntVal(c);
   }

   if (ndigits != 0) {
      long mpy = 1;
      while (ndigits > 0) {
         mpy = mpy * 10;
         ndigits--;
      }

      mul(a, a, mpy);
      add(a, a, acc);
   }

   if (sign == -1)
      negate(a, a);

   x = a;
}



void bit_and(ZZ& x, const ZZ& a, long b)
{
   static ZZ B;
   conv(B, b);
   bit_and(x, a, B);
}

void bit_or(ZZ& x, const ZZ& a, long b)
{
   static ZZ B;
   conv(B, b);
   bit_or(x, a, B);
}

void bit_xor(ZZ& x, const ZZ& a, long b)
{
   static ZZ B;
   conv(B, b);
   bit_xor(x, a, B);
}


long power_long(long a, long e)
{
   if (e < 0) Error("power_long: negative exponent");

   if (e == 0) return 1;

   if (a == 1) return 1;
   if (a == -1) {
      if (e & 1)
         return -1;
      else
         return 1;
   }

   // no overflow check --- result is computed correctly
   // modulo word size

   unsigned long res = 1;
   unsigned long aa = a;
   long i;

   for (i = 0; i < e; i++)
      res *= aa;

   return to_long(res);
}

//  RANDOM NUMBER GENERATION

// Idea for this PRNG.  Iteratively hash seed using md5 
// to get 256 bytes to initialize arc4.
// Then use arc4 to get a pseudo-random byte stream.

// I've taken care that the pseudo-random numbers generated by
// the routines RandomBnd, RandomBits, and RandomLen 
// are completely platform independent.

// I make use of the md5 compression function,
// which I've modified to work on 64-bit machines


/*
 *  BEGIN RSA's md5 stuff
 *
 */

/*
 **********************************************************************
 ** md5.c                                                            **
 ** RSA Data Security, Inc. MD5 Message Digest Algorithm             **
 ** Created: 2/17/90 RLR                                             **
 ** Revised: 1/91 SRD,AJ,BSK,JT Reference C Version                  **
 **********************************************************************
 */

/*
 **********************************************************************
 ** Copyright (C) 1990, RSA Data Security, Inc. All rights reserved. **
 **                                                                  **
 ** License to copy and use this software is granted provided that   **
 ** it is identified as the "RSA Data Security, Inc. MD5 Message     **
 ** Digest Algorithm" in all material mentioning or referencing this **
 ** software or this function.                                       **
 **                                                                  **
 ** License is also granted to make and use derivative works         **
 ** provided that such works are identified as "derived from the RSA **
 ** Data Security, Inc. MD5 Message Digest Algorithm" in all         **
 ** material mentioning or referencing the derived work.             **
 **                                                                  **
 ** RSA Data Security, Inc. makes no representations concerning      **
 ** either the merchantability of this software or the suitability   **
 ** of this software for any particular purpose.  It is provided "as **
 ** is" without express or implied warranty of any kind.             **
 **                                                                  **
 ** These notices must be retained in any copies of any part of this **
 ** documentation and/or software.                                   **
 **********************************************************************
 */


#if (NTL_BITS_PER_LONG <= 32)
#define TRUNC32(x) (x)
#else
#define TRUNC32(x) ((x) & ((1UL << 32)-1UL))
#endif

/* F, G and H are basic MD5 functions: selection, majority, parity */
#define F(x, y, z) (((x) & (y)) | ((~x) & (z)))
#define G(x, y, z) (((x) & (z)) | ((y) & (~z)))
#define H(x, y, z) ((x) ^ (y) ^ (z))
#define I(x, y, z) (TRUNC32((y) ^ ((x) | (~z)))) 

/* ROTATE_LEFT rotates x left n bits */
#define ROTATE_LEFT(x, n) (TRUNC32(((x) << (n)) | ((x) >> (32-(n)))))

/* FF, GG, HH, and II transformations for rounds 1, 2, 3, and 4 */
/* Rotation is separate from addition to prevent recomputation */
#define FF(a, b, c, d, x, s, ac) \
  {(a) = TRUNC32((a) + F((b), (c), (d)) + (x) + (ac)); \
   (a) = ROTATE_LEFT((a), (s)); \
   (a) = TRUNC32((a) + (b)); \
  }
#define GG(a, b, c, d, x, s, ac) \
  {(a) = TRUNC32((a) + G((b), (c), (d)) + (x) + (ac)); \
   (a) = ROTATE_LEFT((a), (s)); \
   (a) = TRUNC32((a) + (b)); \
  }
#define HH(a, b, c, d, x, s, ac) \
  {(a) = TRUNC32((a) + H((b), (c), (d)) + (x) + (ac)); \
   (a) = ROTATE_LEFT((a), (s)); \
   (a) = TRUNC32((a) + (b)); \
  }
#define II(a, b, c, d, x, s, ac) \
  {(a) = TRUNC32((a) + I((b), (c), (d)) + (x) + (ac)); \
   (a) = ROTATE_LEFT((a), (s)); \
   (a) = TRUNC32((a) + (b)); \
  }



static
void MD5_default_IV(unsigned long *buf)
{
   buf[0] = 0x67452301UL;
   buf[1] = 0xefcdab89UL;
   buf[2] = 0x98badcfeUL;
   buf[3] = 0x10325476UL;
}



/* Basic MD5 step. Transform buf based on in.
 */

static
void MD5_compress(unsigned long *buf, unsigned long *in)
{
  unsigned long a = buf[0], b = buf[1], c = buf[2], d = buf[3];

  /* Round 1 */
#define S11 7
#define S12 12
#define S13 17
#define S14 22
  FF ( a, b, c, d, in[ 0], S11, 3614090360UL); /* 1 */
  FF ( d, a, b, c, in[ 1], S12, 3905402710UL); /* 2 */
  FF ( c, d, a, b, in[ 2], S13,  606105819UL); /* 3 */
  FF ( b, c, d, a, in[ 3], S14, 3250441966UL); /* 4 */
  FF ( a, b, c, d, in[ 4], S11, 4118548399UL); /* 5 */
  FF ( d, a, b, c, in[ 5], S12, 1200080426UL); /* 6 */
  FF ( c, d, a, b, in[ 6], S13, 2821735955UL); /* 7 */
  FF ( b, c, d, a, in[ 7], S14, 4249261313UL); /* 8 */
  FF ( a, b, c, d, in[ 8], S11, 1770035416UL); /* 9 */
  FF ( d, a, b, c, in[ 9], S12, 2336552879UL); /* 10 */
  FF ( c, d, a, b, in[10], S13, 4294925233UL); /* 11 */
  FF ( b, c, d, a, in[11], S14, 2304563134UL); /* 12 */
  FF ( a, b, c, d, in[12], S11, 1804603682UL); /* 13 */
  FF ( d, a, b, c, in[13], S12, 4254626195UL); /* 14 */
  FF ( c, d, a, b, in[14], S13, 2792965006UL); /* 15 */
  FF ( b, c, d, a, in[15], S14, 1236535329UL); /* 16 */

  /* Round 2 */
#define S21 5
#define S22 9
#define S23 14
#define S24 20
  GG ( a, b, c, d, in[ 1], S21, 4129170786UL); /* 17 */
  GG ( d, a, b, c, in[ 6], S22, 3225465664UL); /* 18 */
  GG ( c, d, a, b, in[11], S23,  643717713UL); /* 19 */
  GG ( b, c, d, a, in[ 0], S24, 3921069994UL); /* 20 */
  GG ( a, b, c, d, in[ 5], S21, 3593408605UL); /* 21 */
  GG ( d, a, b, c, in[10], S22,   38016083UL); /* 22 */
  GG ( c, d, a, b, in[15], S23, 3634488961UL); /* 23 */
  GG ( b, c, d, a, in[ 4], S24, 3889429448UL); /* 24 */
  GG ( a, b, c, d, in[ 9], S21,  568446438UL); /* 25 */
  GG ( d, a, b, c, in[14], S22, 3275163606UL); /* 26 */
  GG ( c, d, a, b, in[ 3], S23, 4107603335UL); /* 27 */
  GG ( b, c, d, a, in[ 8], S24, 1163531501UL); /* 28 */
  GG ( a, b, c, d, in[13], S21, 2850285829UL); /* 29 */