Programming and the ECDSA equation

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I've been implementing point addition into a c++ program I've written but I don't see how this can be done right. When I do slope = (y1 - y2)/(x1 - x2) I get a freaking decimal, which doesn't produce the proper points when applied to the other parts of the equation due to it not retaining its fractional qualities. Anyone have any ideas how how to get past that?

Point Addition being defined by the following equation:

slope = (y1 - y2) / (x1 - x2)

xsum = slope ^ 2 - (x1 + x2)

ysum = slope * (x1 - xsum) - y1

Whereby Private Address x02 with x,y coordinates respectively:

89565891926547004231252920425935692360644145829622209833684329913297188986597
12158399299693830322967808612713398636155367887041628176798871954788371653930

with the Point Addition of Private Address x01 with x,y coordinates respectively:

55066263022277343669578718895168534326250603453777594175500187360389116729240
32670510020758816978083085130507043184471273380659243275938904335757337482424

applied to the above equation produce the result of Private Address x03 with x,y coordinates respectively:

112711660439710606056748659173929673102114977341539408544630613555209775888121
25583027980570883691656905877401976406448868254816295069919888960541586679410

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

EDIT:

I put this C++ program together, and I've modified in every way I can think of (moving %p around, doing it too many times, breaking up the equations and the like). I can't get it to result in the proper results. Anyone mind checking it out and see what you can find please?

http://coliru.stacked-crooked.com/a/26f9ed24ed5a86ed

Mine

Posted 2014-06-15T22:27:28.023

Reputation: 1 142

1You should probably give a pointer to the description you're reading, since those I have seen don't have anything called slope. At a guess, though, perhaps the division is meant to be done using modular arithmetic? – Nate Eldredge – 2014-06-15T22:36:01.780

Wow, first response and its already challenged my limited understanding of c++. how would I make it a ..."pointer"? – Mine – 2014-06-15T22:38:53.693

1Sorry, I just mean a URL to the description of the algorithm. – Nate Eldredge – 2014-06-15T22:41:36.137

1you use ph1^2 which is XOR in C++, you should write ph1*ph1, – Willem Hengeveld – 2014-06-18T09:30:29.420

and you use way to many brackets. and the %p applies only to (gx1 + gx2), not to the whole expression – Willem Hengeveld – 2014-06-18T09:33:44.080

and you use unsigned ints, i think you need signed integers. – Willem Hengeveld – 2014-06-18T09:46:25.370

and the while (a>1) in your mul_inv should be: while (b!=0) – Willem Hengeveld – 2014-06-18T10:43:02.640

On the bottom right corner of the website there is an "edit" button anyone can use to try it (then you compile and share it, the share will produce a new url u can share). http://coliru.stacked-crooked.com/a/7368ec065cd78b8c is the newest one. I tried all your suggestions (any many variations thereof) to no avail.

– Mine – 2014-06-18T14:01:34.507

http://coliru.stacked-crooked.com/a/3f9112304c3c9c44 – Willem Hengeveld – 2014-06-19T02:25:36.217

Wilem, tyvm, I can't even make sense of why one of my variations didn't work (as I "know" I tried that). Maybe I'm making a mistake where my programs aren't compiling in an updated form or something.... would you by chance mind showing me how you'd get the proper y coordinate too in that program? i really don't understand it; but mine isn't working for the y value. – Mine – 2014-06-19T05:22:31.050

http://coliru.stacked-crooked.com/a/fdaee6a813614d51 – Mine – 2014-06-19T05:43:15.913

Okay, I got it....I don't get why I had to do y2 = (ph1 gx1) - (ph1 x2) - gy1 with %p of each parenthesis rather than y2 = ph1 * (gx1 - x2) - y1 and then y2 %= p. – Mine – 2014-06-19T06:02:48.577

your ph1 result somehow has the wrong sign, if you replace it with 'p-ph1' you do arrive at the correct result. maybe something is wrong with your mul_inv still? – Willem Hengeveld – 2014-06-19T10:50:20.613

That seems possible suddenly, I just checked some of my results, the program spits out the proper x,y coordinates for only a couple addresses. It suddenly gets the wrong y coordinate, then that skews all subsequent values. I'm assessing it further – Mine – 2014-06-21T08:06:20.307

http://coliru.stacked-crooked.com/a/74648b16c2692525 Resulting values of x are valid up to 0x05, at 0x06 they are wrong. Values of y are only valid for 0x03, and for no others. – Mine – 2014-06-21T09:22:27.870

Answers

2

The magic phrase on that page is in a finite field. Here the finite field is the integers mod p, where p is the number 2256 - 232 - 29 - 28 - 27 - 26 - 24 - 1 (see here). So all the arithmetic in your equations isn't ordinary arithmetic of integers or real numbers; it needs to be done mod p. See http://en.wikipedia.org/wiki/Modular_arithmetic. For addition, subtraction and multiplication, you can use ordinary integer arithmetic and compute the remainder mod p at the end. For division, you will need something like the extended Euclidean algorithm. Of course, you will also need to be using arbitrary precision arithmetic if you are not already, since numbers of this size are much too large for standard C++ types like long int and double.

Nate Eldredge

Posted 2014-06-15T22:27:28.023

Reputation: 21 420

So basically there is no single short simple method by which to achieve this? – Mine – 2014-06-15T23:16:29.917

2No, I don't believe there are any shortcuts that simplify it significantly more than what you've already read. In particular, you can't avoid the use of modular arithmetic. – Nate Eldredge – 2014-06-16T00:34:33.200

Asked: 2014-06-15T22:27:28.023

Viewed: 269 times

Active: 2014-06-18T07:35:41.873