# Can encrypted information be sent back in time while avoiding the Unproven Theorem Paradox?

Background: The Unproven Paradox involves sending a mathematical proof to individuals in the past that is obtained from the fact that there is a recipient in the past that publicly reveals the proof. David Deutsch asks the question: Where did the proof originally come from? Another interpretation of this scenario simply leads to the conclusion that "information" cannot be sent back in time (probability of self-inconsistent events is 0).

Scenario: Suppose, in the future, we generate a one-time pad from a source of perfectly random bits and encrypt (bitwise XOR) a desired mathematical proof. Assuming its existence, we use a communications channel (ex. CTCs) to send this ciphertext back to ourselves in the past. From the perspective of participants in the past, this information should be structurally equivalent to and indistinguishable from random data.

Note that this communications channel can be noisy. Assume that this noise factor makes it so that the probability of a successful transmission of "unproven" information is zero. For example, sending a proof in the clear that has not been proven should not be possible due to noise. This is intended to be a "direct" way of combating the Unproven Theorem paradox.

Question 1. Should ciphertexts be classified as information that could violate the unproven theorem paradox? What various models of randomness are most appropriate for this situation? Should the generation of truly random bits be conditioned on events that occur in the future? If so, does this not technically provide information about the future itself?

Now consider the following modification of the protocol:

Extension: Let us instead generate the pad in the past, then wait on the other end of the communications channel for a transmission. When we have the desired proof in the future (from whatever source), we can encrypt it and send it back through our communications channel to the past.

Our past selves decrypt the information using the pad to recover the original plaintext. Assume that the participants in the protocol operate in good faith and that we have sufficient error correction abilities. Depending on the theorem itself, we may be able to formally verify its correctness.

This situation may be alternatively viewed as us generating two random strings and XORing them together to wishfully produce any desired proof that exists.

Question 2. Are there problems with this protocol that are not dependent on our model of randomness and the mere existence of our communications channel? Does this situation technically violate acceptable modern interpretations of the self consistency of time? I prefer that our analysis focus on a single timeline rather than some multiple worlds perspective.

If desired, let me know what I need to clarify in the question via a comment.

• Perhaps people could suggest how I could improve the question?
– mdxn
Commented Nov 19, 2013 at 17:53
• I have a question about the premises. If I look in a math history book, and it says that Fred Bloggs proved Blogg's theorem in 1823; and Fred Blogg's private notebook says: "I'm glad I found this proof from the future under the old oak tree. But I'd have rather had next week's stock prices," then we are to take this as evidence of time travel? What is the mechanism of verifying the facts of the situation? Commented Jan 29, 2014 at 1:53
• Just to make my usual terminological amendment to these issues, the existence of closed temporal loops of intensional information concerning an object doesn't form a paradox about the object's existence. The proof existed even before the information was received in the "past" because extensional identification of proofs is a matter of mathematical fact, not a contingent aspect of human mathematical history; what is new and threatens temporal paradox is this particular information about the proof (that it can be written thus). Commented Mar 2, 2014 at 22:37