The lovely thing about that of course is your public key could be 32 bytes long (the hash of the 2 end states), and the signature == 2*the initial start positions to reach the public end state, say 2*64 bytes == 128 bytes.A fatal issue, aka major reason for falling down, is that while many PRNG can be "fast-fowarded" in this way, when the cycle length is known you can in effect reverse by moving ahead (cycle_length-1) steps. Thus for system to be practical, in order to sign a 256 bit value in this way you need to know that the cycle length is >2^256, and that the cycle length is unknown (and not easily knowable)As a side note, SHA256 may or may not have multiple cycles. It's also unknown.I've been looking into a few possibilities involving a vector (X,Y) multiplied into a new X,Y using a singular matrix followed by modulo. (think LCG with singular mixing). Sure enough the vector seems to have multiple/many cycles depending on the start position.As I always do, I'm putting far too much time into this without asking the appropriate questions - can quantum computing break these methods anyway using either shors, grovers or some combinatorial magic?
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