How to recover generator G of edwards curve

IlIIIlllIIllIIl

4/21/24, 6:15 PM

If we know the value of G*n, the value of n, and the value of p in Zp, is there a way to find out G?

I tried the code below on sage but the value was different from the original G.

# we know nG = (x, y)
Gx = inverse_mod(n, p)*nG[0] % p
Gy = inverse_mod(n, p)*nG[1] % p
G = (Gx, Gy)

1 + 0

elliptic-curves

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Crypto

poncho

4/21/24, 7:05 PM

If we know the value of G*n, the value of n, and the value of p in Zp, is there a way to find out G?

Sure is, assuming that you also know what the neutral element is (for example, it's $(0,1)$)

First step is recovering the curve equation. Fortunately, with Edwards curves, it's easy to do with from a single point (assuming $x, y \ne 0$), by just solving this equation for $d$:

$$x^2 + y^2 = 1 + dx^2y^2 \pmod p$$

Second step is to determine $q$ the order of the curve; this can be done by 'point counting'; I believe Sage includes such a facility.

Third step is to compute $G = [n^{-1} \bmod q](nG)$ (note: this is inverse modulo $q$, not modulo $p$); Sage also has the facilities to do that; you'll need the neutral element to compute this step, as the addition law depends on it.

There; you're done

+ 0

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