untested - source Faruk Akgul -> http://faruk.akgul.org/blog/javas-missing-algorithm-biginteger-sqrt/

BigIntegerSqrt.java

    /** Java's BigInteger doesn't have a squared root function, sigh */
    public static BigInteger sqrt(BigInteger n) {
      BigInteger a = BigInteger.ONE;
      BigInteger b = new BigInteger(n.shiftRight(5).add(new BigInteger("8")).toString());
      while(b.compareTo(a) >= 0) {
        BigInteger mid = new BigInteger(a.add(b).shiftRight(1).toString());
        if(mid.multiply(mid).compareTo(n) > 0) b = mid.subtract(BigInteger.ONE);
        else a = mid.add(BigInteger.ONE);
      }
      return a.subtract(BigInteger.ONE);
    }