Cliff Reiter's code uses a different algorithm than the python code (uses jacobian coordinate transformation to avoid inverse mod calculation). His code can be found here (but his first link is a good intro): http://webbox.lafayette.edu/~reiterc/j/vector/factor_ecj.html
For the python implementation, you want to look at __mul__ . __rmul__ I believe just overloads the * operator perhaps for the right argument to *. The general overview of elyptic curve multiplication is that it involves repeated doublings and additions. http://repl.it/languages/Python is a useful resource for figuring out python code. But here is the only part I had trouble understanding: leftmostbit =: 2&#.@:({. , 0 $~ 2 -~ #)@:(2&#. inv) NB. for some reason divides msb by 2. ________________________________ From: Raul Miller <rauldmil...@gmail.com> To: Programming forum <programm...@jsoftware.com> Sent: Thursday, January 30, 2014 12:32:59 PM Subject: Re: [Jprogramming] math requests I have been looking at this, and I did a straightforward conversion of the python code to J (not quite complete). And I have a couple issues I would like to talk through before proceeding (I've also not yet studied Cliff Reiter's code). First, I would like to quibble with what I assume is the underlying standards document. The value INFINITY behaves like a zero in the context of addition. That strikes me as a bit odd. Second, though, I would like to understand python's rules a bit better - what happens when one multiples a Point with integer? Specifically, consider this excerpt from https://github.com/warner/python-ecdsa/blob/master/ecdsa/ellipticcurve.py : p192 = Point( c192, Gx, Gy, r ) # Checking against some sample computations presented # in X9.62: d = 651056770906015076056810763456358567190100156695615665659 Q = d * p192 The definitions of p192 and d are trivial for me to translate: p192=. (c192;Gx;Gy;r) conew 'Point' d=. 651056770906015076056810763456358567190100156695615665659x But how do I translate Q? In the definition of Point, I saw the following bit of python: def __rmul__( self, other ): """Multiply a point by an integer.""" return self * other But if that is the relevant code, all it is telling me is that multiplication is defined in terms of multiplication - as far as I can see, there is no other definition for multiplying a Point and an integer, nor have I seen any constructor for constructing a Point from an integer. Before I spend too much time chasing down rabbit holes (I really should be working on some other issues), can someone explain to me what is supposed to happen here? Thanks, -- Raul On Wed, Jan 29, 2014 at 11:35 AM, Pascal Jasmin <godspiral2...@yahoo.ca> wrote: > > > 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 > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
