Would it be possible to get the browser to generate a random key, use the current mechanism to encrypt that key and then upload a file, that has been encrypted

Would it be possible to get the browser to generate a random key, use the current mechanism to encrypt that key and then upload a file, that has been encrypted with the original key, to a third party? The reason I ask is that something like that could be used to build a file sharing service where the content is only known to those people with the correct links (with the # anchor information). This would be super useful for distributing private data.