The correct response to this problem was to download a CA Bundle and tell PHP to use it in the php.ini file, e.g.
[curl] curl.cainfo="C:\xampp\php\extras\ssl\cacert.pem"
However, even after downloading a pem file from http://curl.haxx.se/docs/caextract.html, setting the above, and restarting apache, I was still getting the same error. Since I'm using a phar file for the AWS SDK, debugging the problem was difficult.
A post by GeoffGordon finally solved this for me... downloading a zip file of the pem, and then using the extracted version, fixed this for me. Thanks, Geoff!
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When using WampServer, I had the same problem, with a different cause. The php.ini file indicated by a php file served from my local server was different than the php.ini file being used by PHPStorm for debugging. I extracted the php.ini used by PHPStorm by debugging a file with the following code.
<?php ob_start(); phpinfo(); $phpinfo = ob_get_contents();