From 84d0ed4fd4a98943287d88cca08a9492fcbffefa Mon Sep 17 00:00:00 2001
From: Jeremy Teitelbaum <jeremy.teitelbaum@uconn.edu>
Date: Wed, 30 May 2018 10:27:50 -0400
Subject: [PATCH] Fixed minor typos in the talk

---
 ctnt2018/ModularExponentiation.ipynb | 22 +++++++++++-----------
 ctnt2018/PminusOneMethod.ipynb       | 20 ++++++++++++++++----
 ctnt2018/ctnt2018.tex                | 11 ++++++++---
 3 files changed, 35 insertions(+), 18 deletions(-)

diff --git a/ctnt2018/ModularExponentiation.ipynb b/ctnt2018/ModularExponentiation.ipynb
index c582406..3ab5570 100644
--- a/ctnt2018/ModularExponentiation.ipynb
+++ b/ctnt2018/ModularExponentiation.ipynb
@@ -2,14 +2,14 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 4,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [],
    "source": [
     "def mexp(a,x,N):\n",
     "    m,s=1,a\n",
     "    while x>0:\n",
-    "        print('{0:5d} {1:5d} {2:10d}'m,s,x)\n",
+    "        print('{0:10d} {1:10d} {2:10b}'.format(m,s,x))\n",
     "        if x % 2 ==1:\n",
     "            m=((m*s) % N)\n",
     "        s=((s*s) % N)\n",
@@ -19,20 +19,20 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "1 5 123\n",
-      "5 25 61\n",
-      "125 625 30\n",
-      "125 21028 15\n",
-      "41321 15573 7\n",
-      "23556 62697 3\n",
-      "104119 3316 1\n"
+      "         1          5    1111011\n",
+      "         5         25     111101\n",
+      "       125        625      11110\n",
+      "       125      21028       1111\n",
+      "     41321      15573        111\n",
+      "     23556      62697         11\n",
+      "    104119       3316          1\n"
      ]
     },
     {
@@ -41,7 +41,7 @@
        "55006"
       ]
      },
-     "execution_count": 6,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
diff --git a/ctnt2018/PminusOneMethod.ipynb b/ctnt2018/PminusOneMethod.ipynb
index 9b322d9..41f484b 100644
--- a/ctnt2018/PminusOneMethod.ipynb
+++ b/ctnt2018/PminusOneMethod.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -61,7 +61,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 40,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
@@ -77,7 +77,7 @@
        "(641, -1415069, 615794)"
       ]
      },
-     "execution_count": 40,
+     "execution_count": 3,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -91,7 +91,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 36,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
@@ -127,6 +127,18 @@
     "show_factors(640)"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "N=2**128+1\n",
+    "T=mexp(3,factorial(10000),N)-1\n",
+    "print(T)\n",
+    "euclid(T,N)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
diff --git a/ctnt2018/ctnt2018.tex b/ctnt2018/ctnt2018.tex
index 080fcae..283bec5 100644
--- a/ctnt2018/ctnt2018.tex
+++ b/ctnt2018/ctnt2018.tex
@@ -182,7 +182,7 @@
 \end{frame}
 \begin{frame}{A simple example}
 	Suppose $N=F_{5}=2^{2^5}+1$ is the fifth Fermat number.  We can't use $a=2$ because clearly high powers of $a$ are going to be 
-	$-1$ mod $N$; so let's try $a=3$. That doesn't work -- but $a=5$ does. Take $M=10!$.  
+	$-1$ mod $N$. Let's try $a=5$ instead. Take $M=10!$.  
 	$$
 	5^M-1 \equiv 1869036133\pmod{F_{5}}
 	$$
@@ -190,7 +190,7 @@
 \end{frame}
 
 \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 $B$.
+	For the $p-1$ method to work, we have to be lucky enough to have a prime factor $p$ of $N$ so that $p-1$ 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.
@@ -281,7 +281,7 @@
 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
+\item Draw three elements $a,x_0,y_0$ at random modulo $N$.  Let $b=y_0^2-x_0^3-ax_0$.  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}
@@ -375,3 +375,8 @@
 \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}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% End: