How to protect against the theft of a paper?
This seems like a reasonable request by the instructor. Without understanding why you are feeling vulnerable, it is not possible to provide an answer. It is worth talking to someone (possibly the instructor or maybe your advisor or a therapist) about why you are feeling vulnerable.

As for protecting yourself, emailing the document to/from your university email account is probably sufficient as documentation. The next level up would be to email the student the document with the instructor cc'd. This way you are proving you shared your paper (and gaining documentation). Anything more and you will likely be revealing your insecurities, which might lead to more insecurities.