And again in 1997: Fermigier, Stéfane - Une courbe elliptique définie sur Q de rang ≥22. (French) [An elliptic curve defined over Q of rank ≥22] Acta Arith. 82 (1997), no. 4, 359–363.
Noam Elkies was already a major contributor to the field at the time. I dropped math to do computer stuff :)
Determining the halting behavior of each successive Turing machine generally becomes harder and harder until eventually we reach a machine with Collatz-like behavior.
The two problems are equivalent in some sense, but I wonder if there's an easy way to "port" over the work between the two projects.
For every Diophantine equation P(x, y) = 0, where x is a tuple of integer parameters and y is a tuple of unknowns, there is a Turing machine that takes the parameters x as input and halts iff there is an integer solution y satisfying the equation (this direction is easy, just have the Turing machine test all tuples of integers in some order and halt iff it found a solution)
and for every Turing machine, there is a Diophantine equation so that if x is an encoding of the Turing machine input, there is an integer solution y to the equation iff the the Turing machine halts for input x. (This direction is hard, you need to have y encode the execution history of the Turing machine somehow, and enforce the transition rules with the structure of the equation.)
Also, the talk about polynomial parametrizations reminded me of the first Diophantine equation I solved in high school: (a^2+b^2)/(a+b) = (c^2+d^2)/(c+d). I had initially thought it had finitely many solutions, but then Nikos Tzanakis corrected me and told me I am missing many. So I toiled for two entire days and found the complete 2-parameter polynomial family of solutions.
This is not real encryption, it picks only one byte of shared secret and XORs it into the plaintext. Therefore, there are only 256 possible decryption keys to check, which is trivial.
Instead, you'd want to use the shared secret as a key to something strong and symmetric like AES.