Good Morning RetiredCoder:

I have been studying the puzzles for a year now.  I was able to comunicate with professor Teske and professor Galbraith, they both worked with professor Pollard, and pointed me in the right direction. I have some questions and some original ideas about the pollard methods. I would like to ask if you can give me some advice.

I believe the Pollard methods can be improved using this observations I have made:

The key to accelerate the Pollard method computations is to improve the efficiency of the inversion part.

Teske in her paper says it is ok to use 32 types of jumps to give enough randomness. What about using more jump types, let´s say 2**32 types of jumps? Then you select every jump by the first 32 bits of the X coordinate. You store the X and Y in a database, which index is the first 32 bits of X. Now the jump formula considers the 32 most significant bits of the actual point, and adds the corresponding point in the database, with the same 32 bits .

This has the advantage that X2-X1 for the inversion, can be selected to give a number of only 224 Bits instead of 256. When you calculate the batch inversion, let's say you make 400 inversions, you can multiply a 256 bit number times a 224 bit number for the partial product part.

I also have made an algorithm for batch inversion using instead a pairwise multiplication, where I think you could improve the efficiency a little bit more, because you will begin making 224 bit*224 bit multiplications. 

Does this make any sense? Will the big database search at every jump make the program slower vs the speedup of the multiplication of smaller numbers?

Thanks for your time.