I've noticed a strange phenomenon in mod calculations. I wonder whether this particular question can be answered by someone fluent with maths:

4   8   16   3   7   11   15   2   6   10   14   1   5         
5   10   15   3   8   13   1   6                        
6   12   1   7                                    
7   14   4   11   1   8                              
8   16   7   15   6   14   5   13   4   12   3   11   2   10   1   9


Above numbers are running on mod 17. If we double a number, say 4, we double it 11 times until it results in 1.
Similarly if we pick number 5 and start doubling it, we need to double it 6 times to reach 1. and two times double for 6, and 4 time doubling with 7 and so on.

I wonder, if we somehow know the starting number, which we are doubling, can we determine or calculate number of doubles it requires to reach value 1?

This is surely related to EC calculations and I believe it certainly will help minimize size of private key bay many folds, depending on doubles!!!