In the end, any valid point has a unique 7-dimensional scalar. I think a trial and error search of common coefficients on lower dimensions (same orbit, or finding the distance between orbits) may simplify the ECDLP (with some sort of pollard rho). It might also be nothin'.
Can you elaborate more on this? Send me a DM. I am working on this idea.
Let's drill through the ECC torus hyper-dimensions and crack open the orbitals faster than a MIT graduate can re-engineer a cat's laser toy to emit 256-bit math in a 16x16 dots grid light in real-time!
The trick is easy: let's say we have some valid k as a private key. It's gonna have this form because of the n - 1 upper bound:
k = c1 * a + c2 * b + c3 * c + c4 *d + c5 * e + c6 * f + c7
c1 to c7 are the unknown ranks of each "digit". Every digit has an upper bound (product of the lower-level rank factors).
Mission? Find k somehow, given all the properties we can take advantage of (endomorphisms for the 3-orbital, symmetry for the 2-orbital).
Build it up as a lattice or some sort of HNP. Use heuristics to search for k + delta to optimize your equations system (e.g. move around the public key, maybe in a really smart way!).
I ain't got time for this though. Some screws didn't fit back anymore, and my cat is angry that I broke its laser again. BTW never try to solder 256 diodes by hand.
You can mention me in the thank you notes inside the first page's footer of your breakthrough paper, though.