diff --git a/.gitignore b/.gitignore
index 3d12d3f..637f77b 100644
--- a/.gitignore
+++ b/.gitignore
@@ -19,7 +19,7 @@
 # these rules might exclude image files for figures etc.
 # *.ps
 # *.eps
-# *.pdf
+*.pdf
 
 ## Generated if empty string is given at "Please type another file name for output:"
 .pdf
@@ -244,3 +244,6 @@ TSWLatexianTemp*
 
 # generated if using elsarticle.cls
 *.spl
+
+#
+.ipynb_checkpoints/
diff --git a/ctnt2018/Untitled.ipynb b/ctnt2018/Untitled.ipynb
new file mode 100644
index 0000000..564eeb2
--- /dev/null
+++ b/ctnt2018/Untitled.ipynb
@@ -0,0 +1,1402 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "-20\n",
+      "-19\n",
+      "-18\n",
+      "-17\n",
+      "-16\n",
+      "-15\n",
+      "try the following: a= -15  and k= 8377\n",
+      "-14\n",
+      "-13\n",
+      "-12\n",
+      "try the following: a= -12  and k= 3407\n",
+      "-11\n",
+      "-10\n",
+      "-9\n"
+     ]
+    },
+    {
+     "ename": "KeyboardInterrupt",
+     "evalue": "",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m    115\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    116\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m149185656432189838133\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 117\u001b[0;31m \u001b[0mecm_trial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m20\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m10000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    118\u001b[0m \u001b[0;31m#N=2**128+1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    119\u001b[0m \u001b[0;31m#ecm_trial(N,arange=100,krange=10000)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mecm_trial\u001b[0;34m(N, arange, krange)\u001b[0m\n\u001b[1;32m    100\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    101\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 102\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    103\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    104\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'try the following: a='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m' and k='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     81\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     82\u001b[0m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 83\u001b[0;31m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     84\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     85\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mtwo_p\u001b[0;34m(x, y, a, b, N)\u001b[0m\n\u001b[1;32m     56\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     57\u001b[0m    \u001b[0;31m# print(Lu)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 58\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     59\u001b[0m   \u001b[0;31m#  print(Lb)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     60\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     47\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     48\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 49\u001b[0;31m     \u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meuclid\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     50\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m==\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     51\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36meuclid\u001b[0;34m(u, v)\u001b[0m\n\u001b[1;32m     37\u001b[0m         \u001b[0mq\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     38\u001b[0m         \u001b[0mx2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 39\u001b[0;31m         \u001b[0ma2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0ma0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     40\u001b[0m         \u001b[0mb2\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mb0\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mq\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     41\u001b[0m         \u001b[0mx0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb0\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb1\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
+     ]
+    }
+   ],
+   "source": [
+    "# %load ECM.py\n",
+    "from math import factorial, gcd, log\n",
+    "import numpy as np\n",
+    "P=np.zeros(10000)\n",
+    "P[0]=1\n",
+    "P[1]=1\n",
+    "for i in range(2,10000):\n",
+    "    if P[i]==0:\n",
+    "        j=2\n",
+    "        while i*j<100:\n",
+    "            P[i*j]=1\n",
+    "            j=j+1\n",
+    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  \n",
+    "B1000=1\n",
+    "B10000=1\n",
+    "for x in Primes10000:\n",
+    "    if x<1000:\n",
+    "        B1000=B1000*(x**(int(log(1000)/log(x))))\n",
+    "    B10000=B10000*(x**(int(log(1000)/log(x))))\n",
+    "\n",
+    "def mexp(a,x,N):\n",
+    "    m,s=1,a\n",
+    "    while x>0:\n",
+    "        if x % 2 ==1:\n",
+    "            m=((m*s) % N)\n",
+    "        s=((s*s) % N)\n",
+    "        x=x//2\n",
+    "    return m\n",
+    "\n",
+    "def euclid(u,v):\n",
+    "    if v==0:\n",
+    "        raise ArithmeticError('Division by Zero')\n",
+    "    x0,x1=u,v\n",
+    "    a0,a1=1,0\n",
+    "    b0,b1=0,1\n",
+    "    while x1!=0:\n",
+    "        q=x0//x1\n",
+    "        x2=x0-q*x1\n",
+    "        a2=a0-q*a1\n",
+    "        b2=b0-q*b1\n",
+    "        x0,a0,b0=x1,a1,b1\n",
+    "        x1,a1,b1=x2,a2,b2\n",
+    "    if x0<0:\n",
+    "        return -x0,-a0,-b0\n",
+    "    else:\n",
+    "        return x0,a0,b0\n",
+    "\n",
+    "def mod_inv(u,N):\n",
+    "    d,a,b=euclid(u,N)\n",
+    "    if d==1:\n",
+    "        return a\n",
+    "    else:\n",
+    "        raise ArithmeticError('Common factor is '+str(d))\n",
+    "\n",
+    "def two_p(x,y,a,b,N):\n",
+    "    Lu=(3*x**2+a) % N\n",
+    "   # print(Lu)\n",
+    "    Lb=mod_inv(2*y,N)\n",
+    "  #  print(Lb)\n",
+    "    L=Lu*Lb % N\n",
+    "    x_two=(L*L-2*x) % N\n",
+    "    y_two=(L*(x-x_two)-y) %N\n",
+    "    return x_two,y_two\n",
+    "\n",
+    "def sum_p(x1,y1,x2,y2,a,b,N):\n",
+    "    Lu=(y2-y1) % N\n",
+    "    Lb=mod_inv(x2-x1,N)\n",
+    "    L=(Lu*Lb) % N\n",
+    "    x_sum=(L*L-x1-x2) %N\n",
+    "    y_sum=(L*(x1-x_sum)-y1) %N\n",
+    "    return x_sum,y_sum\n",
+    "    \n",
+    "def exp_p(x,y,a,b,m,N):\n",
+    "    sx,sy=x,y\n",
+    "    first=True\n",
+    "    while m>0:\n",
+    "        if m%2==1:\n",
+    "            if first:\n",
+    "                xm,ym=sx,sy\n",
+    "                first=False\n",
+    "            else:\n",
+    "                xm,ym=sum_p(xm,ym,sx,sy,a,b,N)\n",
+    "        sx,sy=two_p(sx,sy,a,b,N)\n",
+    "        m=m//2\n",
+    "    return xm,ym\n",
+    "\n",
+    "#def mexp(a,x,N):\n",
+    " #   m,s=1,a\n",
+    " #   while x>0:\n",
+    " #       if x % 2 ==1:\n",
+    " #           m=((m*s) % N)\n",
+    " #       s=((s*s) % N)\n",
+    " #       x=x//2\n",
+    " #   return m\n",
+    "\n",
+    "def ecm_trial(N,arange=50,krange=30):\n",
+    "    for a in range(-arange,arange):\n",
+    "        xm,ym=0,1\n",
+    "        print(a) \n",
+    "        for k in range(2,krange):\n",
+    "            try:\n",
+    "                xm,ym=exp_p(xm,ym,a,1,k,N)\n",
+    "            except ArithmeticError:\n",
+    "                print('try the following: a=',a,' and k=',k)\n",
+    "                break\n",
+    "\n",
+    "def ecm_trial_2(N,arange=50,B=B1000):\n",
+    "    for a in range(-arange,arange):\n",
+    "        xm,ym=0,1\n",
+    "        try:\n",
+    "            xm,ym=exp_p(xm,ym,a,1,B,N)\n",
+    "        except ArithmeticError:\n",
+    "            print('try the following: a=',a)\n",
+    "            break\n",
+    "                    \n",
+    "N=149185656432189838133\n",
+    "ecm_trial(N,arange=20,krange=10000)\n",
+    "#N=2**128+1\n",
+    "#ecm_trial(N,arange=100,krange=10000)\n",
+    "#xm,ym=exp_p(0,1,-91,1,factorial(7883),N)\n",
+    "                                                                                                                                    \n",
+    "                                            \n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "ArithmeticError",
+     "evalue": "Common factor is 16029560051",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mArithmeticError\u001b[0m                           Traceback (most recent call last)",
+      "\u001b[0;32m<ipython-input-61-1561928d1db5>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m15\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mfactorial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m8377\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     80\u001b[0m                 \u001b[0mfirst\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     81\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 82\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     83\u001b[0m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     84\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36msum_p\u001b[0;34m(x1, y1, x2, y2, a, b, N)\u001b[0m\n\u001b[1;32m     65\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     66\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 67\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     68\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     69\u001b[0m     \u001b[0mx_sum\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-60-32002a3ff6a1>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     51\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     52\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 53\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Common factor is '\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     54\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     55\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mArithmeticError\u001b[0m: Common factor is 16029560051"
+     ]
+    }
+   ],
+   "source": [
+    "exp_p(0,1,-15,1,factorial(8377),N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "try the following: a= -91\n"
+     ]
+    }
+   ],
+   "source": [
+    "P=np.zeros(10000)\n",
+    "P[0]=1\n",
+    "P[1]=1\n",
+    "for i in range(2,10000):\n",
+    "    if P[i]==0:\n",
+    "        j=2\n",
+    "        while i*j<100:\n",
+    "            P[i*j]=1\n",
+    "            j=j+1\n",
+    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  \n",
+    "B1000=1\n",
+    "B10000=1\n",
+    "for x in Primes10000:\n",
+    "    if x<1000:\n",
+    "        B1000=B1000*(x**(int(log(1000)/log(x))))\n",
+    "    B10000=B10000*(x**(int(log(10000)/log(x))))\n",
+    "ecm_trial_2(N,arange=100,B=B10000)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "B=1\n",
+    "for x in Primes10000:\n",
+    "    B=B*(x**(int(log(1000000)/log(x))))\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(272092429413788558381560766739333595442,\n",
+       " 304592951138784758364937297795729494958)"
+      ]
+     },
+     "execution_count": 51,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "exp_p(0,1,-91,1,B10000,N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "ArithmeticError",
+     "evalue": "Common factor is 59649589127497217",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mArithmeticError\u001b[0m                           Traceback (most recent call last)",
+      "\u001b[0;32m<ipython-input-32-ea587d3a9f9d>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m91\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mB\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     76\u001b[0m                 \u001b[0mfirst\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mFalse\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     77\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 78\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     79\u001b[0m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     80\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36msum_p\u001b[0;34m(x1, y1, x2, y2, a, b, N)\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     62\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0my1\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 63\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     64\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     65\u001b[0m     \u001b[0mx_sum\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mL\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx1\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mx2\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-26-e41ba6d1881a>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     48\u001b[0m     \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 49\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Common factor is '\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0md\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     50\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     51\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mArithmeticError\u001b[0m: Common factor is 59649589127497217"
+     ]
+    }
+   ],
+   "source": [
+    "exp_p(0,1,-91,1,B,N)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Primes1000"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from math import factorial, gcd, log\n",
+    "import numpy as np\n",
+    "P=np.zeros(10000)\n",
+    "P[0]=1\n",
+    "P[1]=1\n",
+    "for i in range(2,10000):\n",
+    "    if P[i]==0:\n",
+    "        j=2\n",
+    "        while i*j<10000:\n",
+    "            P[i*j]=1\n",
+    "            j=j+1\n",
+    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[2,\n",
+       " 3,\n",
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+       " 6269,\n",
+       " 6271,\n",
+       " 6277,\n",
+       " 6287,\n",
+       " 6299,\n",
+       " 6301,\n",
+       " 6311,\n",
+       " 6317,\n",
+       " 6323,\n",
+       " 6329,\n",
+       " 6337,\n",
+       " 6343,\n",
+       " 6353,\n",
+       " 6359,\n",
+       " 6361,\n",
+       " 6367,\n",
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+       " 6427,\n",
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+       " 6553,\n",
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+       " 6619,\n",
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+       " 6691,\n",
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+       " 6703,\n",
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+       " 6719,\n",
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+       " 6737,\n",
+       " 6761,\n",
+       " 6763,\n",
+       " 6779,\n",
+       " 6781,\n",
+       " 6791,\n",
+       " 6793,\n",
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+       " 6823,\n",
+       " 6827,\n",
+       " 6829,\n",
+       " 6833,\n",
+       " 6841,\n",
+       " 6857,\n",
+       " 6863,\n",
+       " 6869,\n",
+       " 6871,\n",
+       " 6883,\n",
+       " 6899,\n",
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+       " 6911,\n",
+       " 6917,\n",
+       " 6947,\n",
+       " 6949,\n",
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+       " 6967,\n",
+       " 6971,\n",
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+       " 6983,\n",
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+       " 7001,\n",
+       " 7013,\n",
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+       " 7027,\n",
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+       " 7079,\n",
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+       " 7129,\n",
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+       " 7177,\n",
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+       " 7193,\n",
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+       " 7211,\n",
+       " 7213,\n",
+       " 7219,\n",
+       " 7229,\n",
+       " 7237,\n",
+       " 7243,\n",
+       " 7247,\n",
+       " 7253,\n",
+       " 7283,\n",
+       " 7297,\n",
+       " 7307,\n",
+       " 7309,\n",
+       " 7321,\n",
+       " 7331,\n",
+       " 7333,\n",
+       " 7349,\n",
+       " 7351,\n",
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+       " 7411,\n",
+       " 7417,\n",
+       " 7433,\n",
+       " 7451,\n",
+       " 7457,\n",
+       " 7459,\n",
+       " 7477,\n",
+       " 7481,\n",
+       " 7487,\n",
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+       " 7499,\n",
+       " 7507,\n",
+       " 7517,\n",
+       " 7523,\n",
+       " 7529,\n",
+       " 7537,\n",
+       " 7541,\n",
+       " 7547,\n",
+       " 7549,\n",
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+       " 7561,\n",
+       " 7573,\n",
+       " 7577,\n",
+       " 7583,\n",
+       " 7589,\n",
+       " 7591,\n",
+       " 7603,\n",
+       " 7607,\n",
+       " 7621,\n",
+       " 7639,\n",
+       " 7643,\n",
+       " 7649,\n",
+       " 7669,\n",
+       " 7673,\n",
+       " 7681,\n",
+       " 7687,\n",
+       " 7691,\n",
+       " 7699,\n",
+       " 7703,\n",
+       " 7717,\n",
+       " 7723,\n",
+       " 7727,\n",
+       " 7741,\n",
+       " 7753,\n",
+       " 7757,\n",
+       " 7759,\n",
+       " 7789,\n",
+       " 7793,\n",
+       " 7817,\n",
+       " 7823,\n",
+       " 7829,\n",
+       " 7841,\n",
+       " 7853,\n",
+       " 7867,\n",
+       " 7873,\n",
+       " 7877,\n",
+       " 7879,\n",
+       " 7883,\n",
+       " 7901,\n",
+       " 7907,\n",
+       " 7919,\n",
+       " ...]"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Primes10000"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 54,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "U=59649589127497217"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "V=N//U"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 56,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "5704689200685129054721"
+      ]
+     },
+     "execution_count": 56,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "V\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/ctnt2018/Untitled1.ipynb b/ctnt2018/Untitled1.ipynb
new file mode 100644
index 0000000..26bc131
--- /dev/null
+++ b/ctnt2018/Untitled1.ipynb
@@ -0,0 +1,374 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "U=59649589127497217"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "N=2**(128)+1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "V=N//U"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "P=np.zeros(10000)\n",
+    "P[0]=1\n",
+    "P[1]=1\n",
+    "for i in range(2,10000):\n",
+    "    if P[i]==0:\n",
+    "        j=2\n",
+    "        while i*j<10000:\n",
+    "            P[i*j]=1\n",
+    "            j=j+1\n",
+    "Primes10000=[i for i,x in enumerate(P) if x==0 ]  "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "K=U-1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "2\n"
+     ]
+    }
+   ],
+   "source": [
+    "for x in Primes10000:\n",
+    "    if K % x ==0:\n",
+    "        print(x)\n",
+    "        K=K//x"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def mexp(a,x,N):\n",
+    "    m,s=1,a\n",
+    "    while x>0:\n",
+    "        if x % 2 ==1:\n",
+    "            m=((m*s) % N)\n",
+    "        s=((s*s) % N)\n",
+    "        x=x//2\n",
+    "    return m"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "59649589127497216"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "mexp(3,K,U)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# %load ECM.py\n",
+    "# %load ec3.py\n",
+    "from math import factorial, gcd, log\n",
+    "import numpy as np\n",
+    "P=np.zeros(1000)\n",
+    "P[0]=1\n",
+    "P[1]=1\n",
+    "for i in range(2,100):\n",
+    "    if P[i]==0:\n",
+    "        j=2\n",
+    "        while i*j<100:\n",
+    "            P[i*j]=1\n",
+    "            j=j+1\n",
+    "Primes1000=[i for i,x in enumerate(P) if x==0 ]  \n",
+    "\n",
+    "\n",
+    "def mexp(a,x,N):\n",
+    "    m,s=1,a\n",
+    "    while x>0:\n",
+    "        if x % 2 ==1:\n",
+    "            m=((m*s) % N)\n",
+    "        s=((s*s) % N)\n",
+    "        x=x//2\n",
+    "    return m\n",
+    "\n",
+    "def euclid(u,v):\n",
+    "    if v==0:\n",
+    "        raise ArithmeticError('Division by Zero')\n",
+    "    x0,x1=u,v\n",
+    "    a0,a1=1,0\n",
+    "    b0,b1=0,1\n",
+    "    while x1!=0:\n",
+    "        q=x0//x1\n",
+    "        x2=x0-q*x1\n",
+    "        a2=a0-q*a1\n",
+    "        b2=b0-q*b1\n",
+    "        x0,a0,b0=x1,a1,b1\n",
+    "        x1,a1,b1=x2,a2,b2\n",
+    "    if x0<0:\n",
+    "        return -x0,-a0,-b0\n",
+    "    else:\n",
+    "        return x0,a0,b0\n",
+    "\n",
+    "def mod_inv(u,N):\n",
+    "    d,a,b=euclid(u,N)\n",
+    "    if d==1:\n",
+    "        return a\n",
+    "    else:\n",
+    "        raise ArithmeticError('Common factor is '+str(d))\n",
+    "\n",
+    "def two_p(x,y,a,b,N):\n",
+    "    Lu=(3*x**2+a) % N\n",
+    "   # print(Lu)\n",
+    "    Lb=mod_inv(2*y,N)\n",
+    "  #  print(Lb)\n",
+    "    L=Lu*Lb % N\n",
+    "    x_two=(L*L-2*x) % N\n",
+    "    y_two=(L*(x-x_two)-y) %N\n",
+    "    return x_two,y_two\n",
+    "\n",
+    "def sum_p(x1,y1,x2,y2,a,b,N):\n",
+    "    Lu=(y2-y1) % N\n",
+    "    Lb=mod_inv(x2-x1,N)\n",
+    "    L=(Lu*Lb) % N\n",
+    "    x_sum=(L*L-x1-x2) %N\n",
+    "    y_sum=(L*(x1-x_sum)-y1) %N\n",
+    "    return x_sum,y_sum\n",
+    "    \n",
+    "def exp_p(x,y,a,b,m,N):\n",
+    "    sx,sy=x,y\n",
+    "    first=True\n",
+    "    while m>0:\n",
+    "        if m%2==1:\n",
+    "            if first:\n",
+    "                xm,ym=sx,sy\n",
+    "                first=False\n",
+    "            else:\n",
+    "                xm,ym=sum_p(xm,ym,sx,sy,a,b,N)\n",
+    "        sx,sy=two_p(sx,sy,a,b,N)\n",
+    "        m=m//2\n",
+    "    return xm,ym\n",
+    "\n",
+    "#def mexp(a,x,N):\n",
+    " #   m,s=1,a\n",
+    " #   while x>0:\n",
+    " #       if x % 2 ==1:\n",
+    " #           m=((m*s) % N)\n",
+    " #       s=((s*s) % N)\n",
+    " #       x=x//2\n",
+    " #   return m\n",
+    "\n",
+    "def ecm_trial(N,arange=50,krange=30):\n",
+    "    for a in range(-arange,arange):\n",
+    "        xm,ym=0,1\n",
+    "        print(a) \n",
+    "        for k in range(2,krange):\n",
+    "            try:\n",
+    "                xm,ym=exp_p(xm,ym,a,1,k,N)\n",
+    "            except ArithmeticError:\n",
+    "                print('try the following: a=',a,' and k=',k)\n",
+    "                break\n",
+    "\n",
+    "                        \n",
+    "                    \n",
+    "#N=149185656432189838133\n",
+    "#ecm_trial(N,arange=20,krange=10000)\n",
+    "N=2**128+1\n",
+    "#ecm_trial(N,arange=100,krange=10000)\n",
+    "#xm,ym=exp_p(0,1,-91,1,factorial(7883),N)\n",
+    "                                                                                                                                    \n",
+    "                                            \n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "-100\n",
+      "-99\n",
+      "-98\n",
+      "-97\n",
+      "-96\n",
+      "-95\n",
+      "-94\n",
+      "-93\n",
+      "-92\n",
+      "-91\n",
+      "try the following: a= -91  and k= 7883\n",
+      "-90\n",
+      "-89\n",
+      "-88\n",
+      "-87\n",
+      "-86\n",
+      "-85\n",
+      "-84\n",
+      "-83\n",
+      "-82\n",
+      "-81\n",
+      "-80\n",
+      "-79\n",
+      "-78\n",
+      "-77\n",
+      "-76\n",
+      "-75\n",
+      "-74\n",
+      "-73\n",
+      "-72\n",
+      "-71\n",
+      "-70\n",
+      "-69\n",
+      "-68\n",
+      "-67\n",
+      "-66\n",
+      "-65\n",
+      "-64\n",
+      "-63\n",
+      "-62\n",
+      "-61\n",
+      "-60\n",
+      "-59\n",
+      "-58\n",
+      "-57\n",
+      "-56\n",
+      "-55\n",
+      "-54\n",
+      "-53\n",
+      "-52\n",
+      "-51\n",
+      "-50\n",
+      "-49\n",
+      "-48\n",
+      "-47\n",
+      "-46\n",
+      "-45\n",
+      "-44\n",
+      "-43\n",
+      "-42\n"
+     ]
+    },
+    {
+     "ename": "KeyboardInterrupt",
+     "evalue": "",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
+      "\u001b[0;32m<ipython-input-19-221451d1d92a>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mecm_trial\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mU\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0marange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m100\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;36m10000\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
+      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mecm_trial\u001b[0;34m(N, arange, krange)\u001b[0m\n\u001b[1;32m     96\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mk\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mkrange\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     97\u001b[0m             \u001b[0;32mtry\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 98\u001b[0;31m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mexp_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     99\u001b[0m             \u001b[0;32mexcept\u001b[0m \u001b[0mArithmeticError\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    100\u001b[0m                 \u001b[0mprint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'try the following: a='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m' and k='\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mexp_p\u001b[0;34m(x, y, a, b, m, N)\u001b[0m\n\u001b[1;32m     77\u001b[0m             \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     78\u001b[0m                 \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msum_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 79\u001b[0;31m         \u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mtwo_p\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0msy\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     80\u001b[0m         \u001b[0mm\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m//\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     81\u001b[0m     \u001b[0;32mreturn\u001b[0m \u001b[0mxm\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mym\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mtwo_p\u001b[0;34m(x, y, a, b, N)\u001b[0m\n\u001b[1;32m     52\u001b[0m     \u001b[0mLu\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     53\u001b[0m    \u001b[0;31m# print(Lu)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 54\u001b[0;31m     \u001b[0mLb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     55\u001b[0m   \u001b[0;31m#  print(Lb)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     56\u001b[0m     \u001b[0mL\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mLu\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mLb\u001b[0m \u001b[0;34m%\u001b[0m \u001b[0mN\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;32m<ipython-input-13-94d85eebd18b>\u001b[0m in \u001b[0;36mmod_inv\u001b[0;34m(u, N)\u001b[0m\n\u001b[1;32m     43\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mmod_inv\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m     \u001b[0md\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meuclid\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mN\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0md\u001b[0m\u001b[0;34m==\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
+      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
+     ]
+    }
+   ],
+   "source": [
+    "ecm_trial(U,arange=100,krange=10000)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "mexp(11,V-1,V)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.6.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/ctnt2018/ctnt2018.tex b/ctnt2018/ctnt2018.tex
index fce1b44..bf14711 100644
--- a/ctnt2018/ctnt2018.tex
+++ b/ctnt2018/ctnt2018.tex
@@ -1,9 +1,11 @@
 \documentclass{beamer}
-\usepackage{minted}
+%\usepackage{minted}
 \newtheorem{proposition}{Proposition}
 \newtheorem{algorithm}{Algorithm}
-\begin{document}
+\hypersetup{colorlinks,linkcolor=,urlcolor=blue}
 
+%\usepackage[colorlinks=true]{hyperref}
+\begin{document}
 \begin{frame}
 	\begin{center}
 		Lenstra's Elliptic Curve Factoring Method \\
@@ -17,35 +19,35 @@
 	\begin{problem} Given a positive composite integer $N$, find a proper prime divisor of $N$.
 	\end{problem}
 \end{frame}
-\begin{frame}{Factoring is important}
-
-	{\small
-	Problema, numeros primos a compositis dignoscendi, 
-	hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, 
-	...Praetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem 
-	problematis tam elegantis ac celebris sedulo excolantur.
-	
-	\medskip\noindent
-	The problem of distinguishing prime numbers from composite numbers and of
-	resolving the latter into their prime factors is known to be one of the most
-	important and useful in arithmetic. ..Further, the dignity of the
-	science itself seems to require that every possible means be explored for
-	the solution of a problem so elegant and so celebrated. 
-	
-	\medskip\noindent
-	{\it Gauss,  Disquisitiones Arithmeticae (1801): Article 329}
-	}		
-			
-\end{frame}
-\begin{frame}{Cryptography}
-	\begin{block}{}
-	Gauss thought factoring was important and he was unaware of the role it
-	plays in the security of widely used public-key cryptographic systems.
-	\end{block}
-	\begin{block}{}
-		Although a major reason for current work on the problem, we won't get into the cryptographic applications in this talk.
-	\end{block}
-\end{frame}
+%\begin{frame}{Factoring is important}
+%
+%	{\small
+%	Problema, numeros primos a compositis dignoscendi, 
+%	hosque in factores suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, 
+%	...Praetereaque scientiae dignitas requirere videtur, ut omnia subsidia ad solutionem 
+%	problematis tam elegantis ac celebris sedulo excolantur.
+%	
+%	\medskip\noindent
+%	The problem of distinguishing prime numbers from composite numbers and of
+%	resolving the latter into their prime factors is known to be one of the most
+%	important and useful in arithmetic. ..Further, the dignity of the
+%	science itself seems to require that every possible means be explored for
+%	the solution of a problem so elegant and so celebrated. 
+%	
+%\medskip\noindent
+%	{\it Gauss,  Disquisitiones Arithmeticae (1801): Article 329}
+%	}		
+%			
+%\end{frame}
+%\begin{frame}{Cryptography}
+%	\begin{block}{}
+%	Gauss thought factoring was important and he was unaware of the role it
+%	plays in the security of widely used public-key cryptographic systems.
+%	\end{block}
+%	\begin{block}{}
+%		Although a major reason for current work on the problem, we won't get into the cryptographic applications in this talk.
+%	\end{block}
+%\end{frame}
 	
 \begin{frame}{Trial division is impractical}
 	
@@ -65,8 +67,8 @@
 \begin{frame}{Overview of factoring methods}
 	Modern methods of factoring fall into two categories:
 	\begin{itemize}
-		\item Methods based on algebraic groups (such as the $p-1$ method, the elliptic curve method, and generalizations)
-		\item Sieve methods (such as the quadratic and number field sieves)
+		\item Methods based on algebraic groups, such as the $p-1$ method, the elliptic curve method, and generalizations.
+		\item Sieve methods, such as the quadratic and number field sieves.
 	\end{itemize}
 \end{frame}
 \begin{frame}{Overview of factoring}
@@ -87,14 +89,11 @@
 	There is a polynomial time algorithm for a ``quantum computer.''
 \end{frame}
 \begin{frame}{First make sure your number is composite}
-	The complexity of factoring means factoring algorithms should only be applied to composite numbers.
-
 	\begin{theorem}[Fermat] Suppose that $N$ and $a$ are integers with $(a,N)=1$.  If 
 		$$a^{N-1}\not\equiv 1\pmod{N}.$$
 	then $N$ is composite.
 	\end{theorem}
-\end{frame}
-\begin{frame}{The Fermat Test}
+
 	Fermat's theorem allows for a quick test of compositeness.  
 	
 	\begin{block}{The Fermat Test}
@@ -135,7 +134,7 @@
 \end{algorithm}
 \end{frame}
 \begin{frame}{The $p-1$ algorithm}
-	Suppose $N$ is composite.   Then the multiplicative group of units $(\mathbf{Z}/N\mathbf{Z})^*$ is not cyclic, so it is a product of cyclic 
+	Suppose $N$ is (odd and) composite.   Then the multiplicative group of units $(\mathbf{Z}/N\mathbf{Z})^*$ is not cyclic, so it is a product of cyclic 
 	groups by the fundamental theorem of abelian groups. 
 
 	\bigskip\noindent
@@ -192,8 +191,12 @@
 
 \begin{frame}{The Elliptic Curve Method}
 	For the $p-1$ method to work, we have to be lucky enough to have a prime factor that is $B$-powersmooth for a relatively small $B$.
+
+        \bigskip\noindent
 	If the number $N$ we are trying to factor doesn't have this property, then the $p-1$ method won't work.
 
+
+        \bigskip\noindent
 	The elliptic curve method opens the door to more situations in which we can apply the idea of the $p-1$ method.
 \end{frame}
 \begin{frame}{ECM, cont'd}
@@ -204,7 +207,7 @@
 	Suppose that we can find a point $P$ on this curve mod $N$ so that a multiple $K$ of $P$ is zero in the first factor but not the second.
 	
 	\bigskip\noindent
-	If we were to write $E$ in Weierstrass form, and the point $P$ in homogeneous coordinates $[x(P):y(P):z(P)]$, then this condition
+	If we were to write $E$ in Weierstrass form, and the point $P$ in (reduced) homogeneous coordinates $[x(P):y(P):z(P)]$, then this condition
 	would mean that $z(KP)$ is divisible by $U$ but not by $V$.  
 	
 	\bigskip\noindent
@@ -213,25 +216,162 @@
 \begin{frame}{ECM,3}
 
 	If we were fortunate enough that (say) the order of the  first of the two factor groups
-	$n=|E(\mathbf{Z}/U\mathbf{Z})|$ were $B$-powersmooth for a (relatively) small $B$.
+	$n=|E(\mathbf{Z}/U\mathbf{Z})|$ were $B$-powersmooth for a (relatively) small $B$,
 	Then we could use the trick of the $p-1$ method and choose our $K$ to hopefully be divisible by $n$.
 
+        \bigskip\noindent
 	The Riemann hypothesis for elliptic curves over finite fields tells us that if $U$ is prime then $n$ is roughly $p$.
+
+        \bigskip\noindent
+        It seems reasonable to assume that these group orders are essentially random numbers of size roughly $p$.
 	So the chance that $n$ is $B$-powersmooth is the same order as $p-1$ having that property.
 
+        \bigskip\noindent
 	But there are many elliptic curves! 
 \end{frame}
+\begin{frame}{ECM: An example}
+Consider the $7^{th}$ Fermat number 
+$$
+F_{7}=2^{128}+1=340282366920938463463374607431768211457.
+$$
+A Fermat test to base $3$ confirms that $N$ is composite:
+$$
+3^{N-1}\equiv 47511664169441434718291075092691853899\not\equiv 1\pmod{N}
+$$
+\end{frame}
+\begin{frame}{ECM: An example, cont'd}
+Try many elliptic curves $E$ and a large but not hopelessly large $B$ so that, if $U$ is a factor of $N$, then
+$|E(\mathbf{Z}/U\mathbf{Z})|$ is $B$-powersmooth.
 
+\bigskip\noindent
+Consider the family of elliptic curves $E_{a}:y^2=x^3+ax+1$ which has the obvious point $P=(0,1)$ on it.  
 
+\bigskip\noindent
+Consider $a$ as integers (for example in the range $[-100,100]$) and try $B=10000$.  Let
+$$
+B_{4}=\prod_{p<10000} p^{[\log (10000)/log(p)]}
+$$
+For each $a$, compute
+$[B_4](P)$ in $E_{a}(\mathbf{Z}/N\mathbf{Z})$.  Use the same idea as in modular exponentiation to compute $[B_4](P)$ (repeated doubling of the point).  
+
+\bigskip
+In computing this, you have to do modular inversion mod $N$.  If that fails: you've found a factor of $N$.
+\end{frame}
+\begin{frame}{ECM: an example, cont'd}
+A very naive search finds that 
+$$
+E_{-91}: y^2=x^3-91x+1
+$$  
+gives us our factorization and finds the factor 
+$$N=UV=(59649589127497217)(5704689200685129054721)$$
 
+\bigskip\noindent
+Some quick pseudoprime tests suggests that this is prime (it is) and $U-1$ has a prime divisor with 15 digits and $V-1$
+has a prime factor with $12$ digits,  so this factorization is unlikely to have been found by the $p-1$ method.
+\end{frame}
+\begin{frame}{Lenstra's ECM for theoretical purposes}
+Let $N$ be an integer that is not a prime power and is relatively prime to
+$6$. 
+\begin{enumerate} 
+
+\item Choose integers $v$ and $w$ and set
+$$
+k=\prod_{2}^{w} r^{e(r)}
+$$
+where $e(r)$ is maximal so that 
+$r^{e(r)}\le v+2\sqrt{v}+1.
+$
+\item Draw three elements $a,x_0,y_0$ at random modulo $N$.  Let $b=y_0^2-x_0^3-ax$.  Then $P=(x_0,y_0)$ is a point on the curve
+$E$ with equation $y^2=x^3+ax+b$.  
+\item Try to compute $kP$.  If you find a divisor, cheer! Otherwise, try again until you've tried $h$ times, then give up.
+\end{enumerate}
+Here $v$ serves as bound on the smallest prime factor of $N$ and $w$ a bound on the largest prime divisor of the group of points mod that prime. 
+\end{frame}
+\begin{frame}{Analysis of ECM}
+The method described above will find a divisor provided that the following conditions hold:
+\begin{enumerate}
+\item $N$ has a prime divisor $p\le v$.
+\item The elliptic curve chosen is non-singular mod $p$.
+\item The order of the group of points on that curve is $w$-smooth.
+\item For some \textit{other} divisor $q$ of $N$, $E$ is non-singular mod $q$ and the order of $P$ mod $q$ is NOT divisible by the
+largest prime divisor of the order of $P$ mod $p$. 
+\end{enumerate}
+
+How likely is this to happen?  There is a trade off between time spent on one curve $v,w$ and number of times you try $h$.
 
+\end{frame}
+\begin{frame}{Analysis of ECM, cont'd}
+Lenstra uses  properties of elliptic curves to show the following. Let
+$$
+u=|\{ s: |s-p-1|<\sqrt{p} \hbox{\rm\ and $s$ is $w$-smooth}\}| 
+$$
+So $u$ is the number of $w$-smooth numbers that could possibly be the order of an elliptic curve mod $p$.
+Then the chance that a single attempt at Lenstra's ECM method works (i.e. that a particular $(a,x,y)$ succeeds) is closely related to
+the chance of a smooth group order, or $u/(2\sqrt{p}+1)$. 
 
+\bigskip\noindent
+Conjecture: The chance that a randomly chosen integer in the range $(x-\sqrt{x}+1,x+\sqrt{x}+1)$ is $w$-smooth is the same
+as the chance that a randomly chosen integer of that size is $w$-smooth.
+\end{frame}
+\begin{frame}{Analysis of ECM, cont'd}
+\begin{theorem} Suppose $N$ has two prime divisors, that $p$ is the smallest prime divisor of $N$, and that $N$ is prime to $6$.
+Let $f(w)$ be the probability that a randomly chosen integer in the range $$(p+1-\sqrt{p},p+1+\sqrt{p})$$
+is $w$-smooth.  Assuming $v\ge p$,
+then there is an explicitly computable constant $c$ so that the chance of success of the ECM with parameters $v,w,h$
+is 
+$$1-c^{-hf(w)/\log(v)}.$$
+\end{theorem}
+\end{frame}
+\begin{frame}{Analysis of ECM, cont'd}
+\bigskip\noindent
+Meanwhile, the running time for the algorithm is 
+$$O(hw\log(v)M(N))$$
+ because you have to compute $kP$,
+and 
+$$
+\log k\le w\log(v).
+$$
+\end{frame}
+\begin{frame}{Analysis of ECM, cont'd}
+Let $$
+L(x)=e^{\sqrt{\log(x)\log\log(x)}}.
+$$
 
+\bigskip\noindent
+Lenstra invokes a famous result of Canfield, Erd\"{o}s, and Pomerance that states that the probability that a random positive integer
+$s\le x$  is $L(x)^{\alpha}$-smooth is (asymptotically) 
+$$
+L(x)^{\frac{1}{2\alpha}+o(1)}
+$$
+\end{frame}
+\begin{frame}{Analysis of ECM, cont'd}
+One way to think of the problem is: for a fixed chance of success, minimize running time.  In other words,
+for a fixed value of $hf(w)$, minimize $hw$.  This boils down to minimizing $w/f(w)$.
 
-	
-
-
-
-
+\bigskip\noindent
+The CEP result says that
+$$
+w/f(w)= O(L(p)^{\frac{1}{2\alpha}+\alpha+o(1)}
+$$
+so the minimum value happens when $\alpha=\frac{1}{\sqrt{2}}$, and, in that case.
+$$
+w/f(w)=O(L(p)^{\sqrt{2}}).
+$$
 
+\bigskip\noindent
+In other words, for a fixed chance of success, the running time behaves like
+$$
+L(p)^{\sqrt{2}}=e^{\sqrt{((2+o(1))\log p\log\log p}}
+$$
+where $p$ is the smallest prime factor of $N$.
+\end{frame}
+\begin{frame}{Further Discussion}
+To make ECM work in practice there are many refinements, including:
+\begin{enumerate}
+\item Using highly optimized elliptic curve operations and modular arithmetic;
+\item Careful choice of parameters (smoothness bound, for example).
+\end{enumerate}
+
+\href{http://swc.math.arizona.edu/aws/2006/notes.html}{Bernstein's notes} from the 2006 Winter School address some of these optimizations.
+\end{frame}
 \end{document}
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