https://github.com/JeanLucPons/Kangaroo/blob/master/Kangaroo.cpp#L381-L420

jump size is a uint64_t and it's straight added to the distance.

I had a mind blackout for a moment and thought this was actually adding 64-bit numbers to the kangaroo X,Y; It's actually taking part of the distance and using that as input to a jump table full of G,2G,3G... points.

If I undersrand this snippet correctly then for the kangaroo itself, we subtract p1y (Y from the corresponding jump distance in the table) from p2y (the old y) to get another kangaroo Y, then we multiply this with p2x-p1x (same logic as p1y and p2y) which is old kangaroo old X - X point in jump table, then these are multiplied together to get a 128-bit value which is then squared (so I was wrong about jump search being limited to 64 bits)

Then it looks like it subtracts the kangaroo and the jump distance from this squared value to get new X (also 128 bits), then for Y it subtracts that from the jump distance and multiplied that result by the aforementioned 128-bit value which potentially gives a 256-bit value which is then subtracted from old kangaroo Y to get new Y.

And each operation is taken to the modulus of secp256k1 group order.

Hard to remember our starting points and terms are (X,Y) pairs and not single numbers since we are solving elliptic curves and not descrete logs.  

I wonder what happens if more points are generated in the jump table...