TL;DR: you possibly can to a restricted extent outline what level addition between two distinct curves is, by simply reusing the formulation, however there is not going to be any relation between this and the corresponding personal keys.
Allow us to think about elliptic curves of the shape
Y2 = X3 + aX + b, modulo p
(these are known as brief Weierstrass curves)
For secp256k1, p = 2256 – 232 – 977, a = 0, b = 7.
Probably the most impactful parameter is p. Should you change p, you alter what kind of numbers X and Y are. Whereas they’re written identically, “the integer 37“, “the integer 37 within the set of numbers modulo p“, and “the integer 37 within the set of numbers modulo q ≠p” are all three very completely different beasts, with very completely different mathematical properties. Due to this, I’ll keep on with the identical modulus p going ahead. Should you change the modulus, I do not see the way you’d outline what operations on elliptic curve factors appear like. Which modulus do you utilize?
As an alternative, let’s prohibit ourselves to modulus p. Now take a look at the elliptic curve level addition equation (which doesn’t apply when including a degree to itself or its negation, however ignore these) for including the purpose (X1,Y1) to (X2,Y2), leading to (X3,Y3):
X3 = λ2 – X1 – X2, and Y3 = λ(X1 – X3) – Y1, the place λ = (X2 – X1)-1(Y2 – Y1), all modulo p.
Be aware that ()-1 refers back to the modular inverse right here, not the common inverse.
Now observe that neither the curve coefficients a or b seem on this equation. Which means in principle, there isn’t a problem with appropriating the identical system to attempt to do a cross-chain level addition. We’ll get some numbers out; the query is simply if these are significant numbers.
Because you’re asking a couple of relation between the personal keys, we have now an extra requirement. The personal keys reside in yet one more type of quantity house, the integers modulo n, the place n is the order of the curve, and this order depends upon the opposite parameters (p, a, and b). If we hold a=0 like in secp256k1, we discover the next orders:
- b=1: n = p – 671331852483699643819086596696745227419
- b=2: n = p + 432420386565659656852420866390673177328
- b=3: n = p – 238911465918039986966665730306072050092
- b=4: n = p + 238911465918039986966665730306072050094
- b=5: n = p – 238911465918039986966665730306072050092 (identical as b=3)
- b=6: n = p + 671331852483699643819086596696745227421
- b=7: n = p – 432420386565659656852420866390673177326 (
secp256k1) - b=8: n = p – 671331852483699643819086596696745227419 (identical as b=1)
- …
- b=12: n = p – 671331852483699643819086596696745227419 (identical as
secp256k1, b=7)
Non-public keys should not comparable when working with curves of distinct orders, so let’s choose two equal-order curves:
- E1 : Y2 = X3 + 7 (
secp256k1) - E2 : Y2 = X3 + 12 (identical order as
secp256k1).
And let’s choose a degree on every:
- P1 = (1, √8) ∊ E1 (word that √ right here refers back to the modular sq. root).
- P2 = (3, √39) ∊ E2
If we apply the addition system to those factors, we get (95199522409000127469965119215597403251155081608447464174576216444950477495870, 17817862784113219767619204370558314933049610986520000131582270812967486670101) which lies on yet one more curve, Y2 = X3 + 113806959772543662429059682568787218964422045791894232716808486305727155222289, which has order n = p + 671331852483699643819086596696745227421. Because the order is completely different from secp256k1, this implies its personal key shall be incomparable to secp256k1 personal keys.
Let’s attempt once more:
- Q1 = (4, √71) ∊ E1
- Q2 = (8, √524) ∊ E2
Including these, we get (83860777440659491102688138897119789724824336349874362905090427965396639347581,94100849044437599780075883675011580302479606607256518328981974291822837065288), which lies on curve Y2 = X3 + 49979308264443915896639903731913831145103210634021312327468991256158292998092, which has order n = p + 432420386565659656852420866390673177328.
So, even once we choose maximally comparable curves, and attempt to add factors throughout them, we find yourself with factors that land on numerous distinct unrelated curves, on which the personal keys are meaningless to match with secp256k1.
