Start simple - you can't encrypt the URL, but you can encrypt the query string. I'd start with something simple like ?q=value1|value2|value3
and use a simple Base64 encoded value so that the query string becomes ?q=dmFsdWUxfHZhbHVlMnx2YWx1ZTM=
. Then you can decode and parse out the values by splitting on a separator like the pipe "|".
If that's not enough, then search on public/private key encryption. You can use a js library to encrypt client-side with the public key and decrypt server-side with the private key.