With all of the mathematicians on this list, these functions have likely been 
implemented before in J.

elyptic curve point add, multiplication and double
a python reference implementation: 
https://github.com/warner/python-ecdsa/blob/master/ecdsa/ellipticcurve.py

the functions are: __add__  __mul__ and double

if I may suggest J explicit signatures for the first 2 functions as:

F =: 4 : 0
'yx yy yo' =. y
'xx xy xo' =. x
)

Some other methods than the python reference could be considered here:

http://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication


also appreciated if you have in implementation of inverse_mod 
for reference function of same nate at:  
https://github.com/warner/python-ecdsa/blob/master/ecdsa/numbertheory.py
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