Okay, this one took me hours to figure out, so I figured I would put my results here for others to find.  Hopefully it will spare others the hours of frustration I went through!

If you do not know what Amazon AWS S3 is, you should read up on it.  Basically, it's super-cheap online file storage for your web site or even personal use.  You can even make it so that files are private, and can only be downloaded with a self-expiring URL (which might only be valid for, say, 1 hour).

Here is how, in PHP, to do just that.  Please note, these code snippits were pieced together from various sources online until I finally got everything working the way it is supposed to.

First things first, you will need these functions:

function gs_getStringToSign($request_type, $expires, $uri) {
   return "$request_type\n\n\n$expires\n$uri";
}

function gs_encodeSignature($s, $key) {
    $s = utf8_encode($s);
    $s = hash_hmac('sha1', $s, $key, true);
    $s = base64_encode($s);
    return urlencode($s);
}
 
function gs_prepareS3URL($file, $bucket) {
 
  $awsKeyId = "PUT ACCESS ID HERE"; // this is the non-secret key ID.
  $awsSecretKey = "PUT SECRET KEY HERE"; // this is the SECRET access key!
 
 
  $file = rawurlencode($file); 
  $file = str_replace('%2F', '/', $file);
  $path = $bucket .'/'. $file;
 
  $expires = strtotime('+1 hour');
 
  $stringToSign = gs_getStringToSign('GET', $expires, "/$path"); 
  $signature = gs_encodeSignature($stringToSign, $awsSecretKey); 

  $url = "http://$bucket.s3.amazonaws.com/$file";
  $url .= '?AWSAccessKeyId='.$awsKeyId
         .'&Expires='.$expires
         .'&Signature='.$signature;
        
  return $url;
}

Next, all you have to do is call the gs_prepareS3URL function like so:

  $file = "WordHunt.jar";
  $bucket = "bytebyte.net";
  $secure_link = gs_prepareS3URL($file, $bucket);

Isn't that easy?  Just make sure you have set the ACL's on the file in question to not be readable by anyone but you (otherwise you wouldn't need these functions at all, and you could just download the file directly).

Fine print: This code is considered "open source," and is released under the GNU Public License version 3 or any later version (http://www.gnu.org/copyleft/gpl.html).  In a nut shell, you can change this code however you want, and redistribute it, but ONLY if it is ALSO under the GNU Public License version 3 or later.  If you do use this software, I would appreciate credit and a link back to this site, though this is not required.