I have two public/private keypairs, A and B. I want to add them together to get a new keypair AB.
I'd also like to be able to add the public keys of A and B to get the public key of AB.
How do I do this?
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Here is some python code you can reverse engineer
def add(p, q): if p % P == 0 and p % P == 0: return q if q % P == 0 and q % P == 0: return p if p == q and p == q: if p == 0: return [0, 0] l = (3 * p**2) * modInv((2 * p), P) elif p == q: return [0, 0] else: l = (p - q) * modInv((p - q), P) x = l**2 - (p + q) y = l * (p - x) - p return [x % P, y % P] def modInv(n, p): return pow(n, p - 2, p) #some constants P = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f x = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798 y = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8 #example usage of the add function g1 = [x, y] print "g1 = " + hex(g1) + " : " + hex(g1) g2 = add([x, y], [x, y]) print "g2 = " + hex(g2) + " : " + hex(g2) g3 = add([x, y], g2) print "g3 = " + hex(g3) + " : " + hex(g3) g4 = add(g2, g2) print "g4 = " + hex(g4) + " : " + hex(g4)
Most ecc libraries will have this function, but if you want to program it yourself, here's what you do:
First, compute the slope of the line containing the points A and B. Let A = (X_a, Y_a) and B = (X_b, Y_b). The equation for the slope is:
s = (Y_a - Y_b) / (X_a - X_b)
The resulting point, we'll call C = (X_c, Y_c) = A+B. Doing some math, you get:
X_c = s^2 - X_a - X_b
Y_c = Y_a + s (X_c - X_a) = Y_b + s(X_c - X_b)
If X_a == X_b, then it depends on Y_a and Y_b. If Y_a == Y_b, then A and B are the same, so really, you're just computing a point doubling (2*A). If Y_a == -Y_b (the only other possibility) then A+B = the point at infinity, or the identity. Usually, that's not a very interesting point for cryptography. Computing 2*A is a little trickier, but can be done. You're already doing that when you compute the public key from the private key with G anyway, so I'll assume you have access to something that lets you double a point.
Note all the operations are all field operations, so you have to mod by P for secp256k1 (FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F in hex).