How to protect against the theft of a paper? A professor requested that I exchange my paper with another student, to get their suggestions. Every student in the course must exchange their papers in this way, before the teacher will look at it. I have never encountered another professor with such a policy and am concerned of the possibility that a student might claim my paper to be their own. What steps can I take to guard against such a possibility?
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.