<?php require_once('Connections/TOM.php'); ?>
<?php
if (isset($_POST['pwd'])) { $_POST['pwd'] = sha1($_POST['pwd']); }
if (!function_exists("GetSQLValueString")) {
function GetSQLValueString($theValue, $theType, $theDefinedValue = "", $theNotDefinedValue = "") 
{
  $theValue = get_magic_quotes_gpc() ? stripslashes($theValue) : $theValue;

  $theValue = function_exists("mysql_real_escape_string") ? mysql_real_escape_string($theValue) : mysql_escape_string($theValue);

  switch ($theType) {
    case "text":
      $theValue = ($theValue != "") ? "'" . $theValue . "'" : "NULL";
      break;    
    case "long":
    case "int":
      $theValue = ($theValue != "") ? intval($theValue) : "NULL";
      break;
    case "double":
      $theValue = ($theValue != "") ? "'" . doubleval($theValue) . "'" : "NULL";
      break;
    case "date":
      $theValue = ($theValue != "") ? "'" . $theValue . "'" : "NULL";
      break;
    case "defined":
      $theValue = ($theValue != "") ? $theDefinedValue : $theNotDefinedValue;
      break;
  }
  return $theValue;
}
}
?>
<?php
// *** Validate request to login to this site.
if (!isset($_SESSION)) {
  session_start();
}

$loginFormAction = $_SERVER['PHP_SELF'];
if (isset($_GET['accesscheck'])) {
  $_SESSION['PrevUrl'] = $_GET['accesscheck'];
}

if (isset($_POST['username'])) {
  $loginUsername=$_POST['username'];
  $password=$_POST['username'];
  $MM_fldUserAuthorization = "";
  $MM_redirectLoginSuccess = "success.php";
  $MM_redirectLoginFailed = "loginfailed.php";
  $MM_redirecttoReferrer = false;
  mysql_select_db($database_TOM, $TOM);
  
  $LoginRS__query=sprintf("SELECT username, username FROM users WHERE username=%s AND username=%s",
    GetSQLValueString($loginUsername, "text"), GetSQLValueString($password, "text")); 
   
  $LoginRS = mysql_query($LoginRS__query, $TOM) or die(mysql_error());
  $loginFoundUser = mysql_num_rows($LoginRS);
  if ($loginFoundUser) {
     $loginStrGroup = "";
    
    //declare two session variables and assign them
    $_SESSION['MM_Username'] = $loginUsername;
    $_SESSION['MM_UserGroup'] = $loginStrGroup;	      

    if (isset($_SESSION['PrevUrl']) && false) {
      $MM_redirectLoginSuccess = $_SESSION['PrevUrl'];	
    }
    header("Location: " . $MM_redirectLoginSuccess );
  }
  else {
    header("Location: ". $MM_redirectLoginFailed );
  }
}
?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
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<title>GC1D4JV Start</title>
<link href="admin.css" rel="stylesheet" type="text/css" />
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<body>
<div align="center">
  <h1>GC1D4JV</h1>
</div>
<div align="left">
  <h5>Welcome to The Prime Factor geocaching site. If you are an existing user, please login to the site. <img mmdw="3"  src="publickey.jpg" alt="public key" width="178" height="241" align="bottom" class="floatRight" /><br />
If you have not registered, please follow this link: <a mmdw="4"  href="newuser.php">NEW USER</a></h5>
</div>
<form id="login" name="login" method="POST" action="<?php echo $loginFormAction; ?>">
  <p>
    <label for="textfield">Geocaching Log Name:</label>
    <br />
    <input type="text" name="username" id="username" />
  </p>
  <p>
    <input name="doLogin" type="submit" id="doLogin" value="Login" />
</p>
</form>
<p mmdw="11"  align="center">RSA Encryption</p>
<p>Encryption has been used to  exchange information between two parties and keep that information from a third  party. The problem with classical symmetric encryption is that the key used to  encrypt the information must be distributed to the people who are supposed to  be able to read the message before they can decrypt it. When a lot of messages  are to be exchanged, lots of keys must be distributed which can be a logistical  problem. Computers can efficiently encrypt and decrypt messages with symmetric  ciphers. </p>
<p>In 1976, Whitfield Diffie  along with Martin Hellman discovered public key cryptography and purposed the  Diffie-Hellman key exchange which is still a commonly used key exchange.</p>
<p>In 1977, Ron Rivest, Adi  Shamir, and Leonard Aldeman at MIT publicized the algorithm named from the  initials of the three inventors. The algorithm is based on the fact that it is  easy to multiply two large prime numbers together to get a product but it is  very difficult to reverse the process. Given the product, it is difficult to  calculate what two prime numbers were multiplied together initially.</p>
<p>In 1991, Senate Bill 266 was  introduced by the US  government. This anti-crime bill had a measure in it requiring all encryption  software must have a back door in it to allow the government to obtain the  plaintext of voice, data and other communications when authorized by law. This  bill prompted Phil Zimmerman to develop Pretty Good Privacy (PGP). In the  pgpguide.1st in PGP 1.0 was the following:</p>
<p><em>The 17 Apr 1991 New York Times reports on an unsettling US Senate  proposal that is part of a counterterrorism bill. If this nonbinding resolution  became real law, it would force manufacturers of secure communications  equipment to insert special &quot;trap doors&quot; in their products, so that  the Government can read anyone's encrypted messages. It reads: &quot;It is the  sense of Congress that providers of electronic communications services and  manufacturers of electronic communications service equipment shall insure that  communications systems permit the Government to obtain the plain text contents  of voice, data, and other communications when appropriately authorized by  law.&quot;</em></p>
<p>The problem with public key  encryption systems with includes RSA is that they are computationally intensive  and runs slowly as compared to symmetric key ciphers when dealing with large pieces  of data. PGP utilized the RSA public key/private key system to exchange the  symmetric key securely. The gist of a public key/private key system is that the  key you want to exchange is encrypted with the recipient&rsquo;s public key and the  only way to decrypt the key is to have the corresponding private key. Knowing  the public key does not help someone in trying to decrypt the message. PGP uses  the RSA public key system to encrypt the session key that the rest of the  message is encrypted with. RSA can run fairly quickly with the small amount of  data that makes up the session key. So the message is encrypted with the IDEA  cipher using a randomly generated key. The key is encrypted with the public key  of the recipient using RSA. The recipient receives the message and uses his  private RSA key to decrypt what the IDEA key was and that is used to decrypt  the message itself.</p>
<p><em>I was involved in the porting of PGP to A/UX  (Apple&rsquo;s Unix) and so I thought a puzzle based on a simplified RSA would be  interesting for my contribution to the TOM Creative Group puzzle.</em></p>
<p mmdw="12"  align="center"><em>The RSA  algorithm in detail</em></p>
<p><em>RSA involves a public key and a private key. The  public key can be known by everyone and be used to encrypt messages being sent  to the recipient who uses the private key to decrypt the message. The keys for  the RSA algorithm are generated the following way:</em></p>
<ol>
  <li><em>Choose  two large prime numbers </em><em>p</em><em> and </em><em>q</em></li>
  <li><em>Compute </em><em>n</em><em>=</em><em>pq</em></li>
  <li><em>Compute the totient</em><em> &#966;(n) = (p-1)(q-1) </em></li>
  <li><em>Choose an integer </em><em>e </em><em>such that 1 &lt; e &lt; &#966;(n) &nbsp;and </em><em>e </em><em>and phi(n)&nbsp;are coprime (share no  common factors other than 1) and frequently is chosen as </em><em>3</em><em> or </em><em>17</em><em> or 65537</em><em></em><em>.</em></li>
  <li><em>Compute </em><em>d </em><em>such that 1 &lt; d &lt;&#966;(n) &nbsp;and ed &#8801; 1 mod &#966;(n) &nbsp;(</em>To compute the value for d, use the Extended  Euclidean Algorithm this is known as modular inversion.) </li>
  <li>The public key is  (<em>n,e</em>) and the private key is (<em>n,d</em>). The values of <em>p</em>,<em>q</em>, and &#966;(n) &nbsp;should be kept secret.</li>
</ol>
<p> &#8729; <em>n</em> is known as the modulus<br />
  &#8729; <em>e</em> is known as the public exponent or  encryption exponent<br />
  &#8729; <em>d</em> is known as the private exponent or  decryption exponent</p>
<p>&nbsp;</p>
<p mmdw="13"  align="center">A very simple example of RSA encryption</p>
<p mmdw="14"  align="center">This example can be done using a calculator or by  hand.</p>
<ol>
  <li>Choose p=3 and  q=11</li>
  <li>n=pq  =3*11=33&nbsp; and &#966;(n) = (p-1)(q-1)=(2)(10)=20</li>
  <li>choose e=3 which  is &gt;1 and coprime to &#966;(n)=20 ie they have no common factors </li>
  <li>compute d such  that ed &#8801; 1 mod(&#966;(n))&nbsp;<br />
    or d = e <sup>-1</sup>mod &#966;(n)<br />
    or find d such that  &#966;(n) divides (ed &ndash; 1)&nbsp;<br />
    or find d such that 20 divides 3d &ndash; 1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</li>
  <li> by brute force d=7 or (3*7-1)/20 = (21-1)/20 = 20/20=1<br />
    the public key is  (33,3) and the private key is (33,7)</li>
</ol>
<p>&nbsp;</p>
<p>If we want to encrypt the  message m = 7<br />
  c = m<sup>e </sup>mod n = 7<sup>3</sup> mod 33 = 343 mod 33 = 13</p>
<p>If we want to decrypt the  ciphertext c = 13<br />
  m = c<sup>d</sup> mod n = 13<sup>7</sup> mod 33 = 7  which is the original message text.</p>
<p>If  you decide you want to try  this cache, you will need to register with my site by following the link at the top of the page.  This registration is needed to link your geocaching log name with a unique puzzle. You will be given a choice of 10 puzzles that have not been used to pick from.. You will then be asked to &quot;log in&quot; again with the geocaching name and you will be presented with your unique puzzle and prompted to enter the pricate key which you will have to calculate. If you enter the correct private key for that puzzle, you will be shown the coordinates for the cache and be allowed to log the cache.  If you enter the wrong private key, you will be allowed to continue trying. If anyone logs the cache without first calculating the correct private key (and everyone will have theor own unique modulus  and public key), your log will be deleted. </p>
<h2 align="center">A</h2>
<p align="center"><img src="images/Tom Creative Group.jpg" alt="TOM" width="288" height="184" /></p>
<h2 align="center">Production</h2>
<p>&nbsp;</p>
</body>
</html>
