TL; PhD: Just reusing the formula is limited in adding points between two different curves, but there is no relationship between this and the corresponding private key.
Consider the elliptic curve of a form
y2 = x3 + ax + bmodulo p
(These are called short weierStrass curves)
for secp256k1, p = 2256 -232 -977, a = 0, b = 7.
The most impactful parameter is p. If you change it pwhat will you change Something like numbers x and y teeth. They are written in the same way, but “integers” 37“,” integer 37 for set of numbers p“,” integer 37 for set of numbers Q≠p“All three very Different beasts, with very different mathematical properties. For this reason, I’m going to stick to the same modulus p from now on. When I change the modulus, I don’t know how to define what the elliptic curve manipulation looks like. Which modulus are you using?
Instead, limit yourself to modulus p. Next, to add points, look at the elliptic curve point addition equation (which does not apply when adding points to itself, but ignores these). (x1y1)) In (x2y2))the result (x3y3)):
x3 =λ2 -X1 -X2and y3 =λ(x1 -X3) – y1where λ=(x2 -X1))-1(y2 – y1))all modulo p.
note that ()-1 Here we refer to the inverse of modular rather than the reverse of normal.
Here we observe that there is no curve coefficient a or b It appears in this equation. This means that in theory there is no problem repurposing the same equation that attempts to add a cross-chain point. We’ll get Some I’ll give the numbers. The only question is whether these are meaningful numbers.
As you are asking about the relationship between private keys, there are additional requirements. Private keys live in spaces of yet another type, Integers modulo nwhere n It is the order of the curves, and this order depends on other parameters (p, aand b). If we hold a = 0 Like secp256k1find the next order.
- b = 1: N = P -671331852483699643819086596696745227419
- b = 2: N = P + 432420386565659656852420866390673177328
- b = 3: N = P -23891146591803998696665730306072050092
- b = 4: N = P + 23891146591803998696665730306072050094
- b = 5: N = P -23891146591803998696665730306072050092 (same as b = 3))
- b = 6: N = P + 671331852483699643819086596696745227421
- b = 7: N = P -432420386565659656852420866390673177326 (
secp256k1)) - b = 8: N = P -671331852483699643819086596696745227419 (same as b = 1))
- …
- b = 12: N = P -671331852483699643819086596696745227419 (same as
secp256k1, b = 7))
Private keys are not comparable when using curves from different orders, so choose two equal curves.
- e1 :y2 = x3 +7 (
secp256k1)) - e2 :y2 = x3 + 12 (same order
secp256k1).
Then, choose a point for each one:
- p1 =(1,√8)∊ e1 (note that √ Here we refer to the modular square root).
- p2 =(3,√39)∊ e2
Applying an additional formula to these points: (95199522409000127469965119215597403251155081608447444174576216444444444495047495870, 1781786278411321976761920437055831493304961098652000013158227081296748670101) This is in yet another curve, y2 = x3 + 113806959725436624290596825687872189644220457918942327168084848630572715522289there is an order N = P + 671331852483699643819086596696745227421. Because the order is different secp256k1which means that private key cannot be compared secp256k1 Private key.
Let’s try again:
- Q1 =(4,√71)∊ e1
- Q2 =(8,√524)∊ e2
Add these and you’ll see (8386077744065949110268813889711978972482433634987436290509042796539639347581, 9410084904444375978007588367501158030247960607256518328981974291822837065288)on the curve y2 = x3 + 49979308264444391589663903731913831145103210634021312327468912561582998092there is an order N = P + 432420386565659656852420866390673177328.
So even if you choose the largest similar curve and try to add points to them, you’ll end up with different points. clear Unrelated curves, private keys are pointless to compare secp256k1.
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