https://dev.twitter.com/docs/api/1.1/post/statuses/retweet/%3Aid
Despite what the docs say (as of this moment) not sending post data results in Could not authenticate you",“code”:32
Supplying the id of the status to retweet validates the request.
Example Using PHP class TwitterAPIExchange, where $tweet[‘id’] is the id of the tweet.
$postfields = array(
'id' => $tweet['id']
);
$url = 'https://api.twitter.com/1.1/statuses/retweet/'.$tweet['id'].'.json';
$requestMethod = 'POST';
$twitter = new TwitterAPIExchange($settings);
$json=$twitter->buildOauth($url, $requestMethod)->setPostfields($postfields)->performRequest();
$retweetdata=json_decode($json, true);