What is the reverse formula from Coords on E11 to Coords on E1

bnsage123

6/23/24, 6:29 AM

U= 115792089237316195423570985008687907852598652813156864395638497411212089444244
a = 20412485227
E1 = EllipticCurve(GF(p), [0,1]) with order U
E11 = EllipticCurve(GF(p), [0,1]) with order a

Formula from Coords on E1 to Coords on E11 is

Q11 = (U/a) * (x,y) (x,y) on E1

But, what is the reverse formula from Coords on E11 to Coords on E1

Q1 = ?? * (x,y) (x,y) on E11

1 + 0

elliptic-curves

Score:1

Crypto

Daniel S

6/23/24, 1:31 PM

If you are asking about the map from an elliptic curve subgroup with $U$ points to a subgroup with $a$ points, the map cannot be reversed because it is many-to-one. Specifically, pick any point $(x,y)$ in $E1$ whose order is not divisible by $a$, now let $H=a*(x,y)$ then $Q11((x,y))=Q11((x,y)+H)$ for all $(x,y)$ in $E1$ and there cannot be a function that maps back to both $(x,y)$ and $(x,y)+H$.

+ 3

bnsage123

6/23/24, 1:47 PM

thanks. I'm asking about map from elliptic curve (E1's subgroup) E11 with order a to a elliptic curve group E1 with order U, when U is divisible by a. In the E1 to E11 mapping case, it can be obtained using like Lagrange's theorem. So, I am asking extension case.

Daniel S

6/23/24, 2:31 PM

If you just need a map from $E11$ to $E1$,, the identity map will work as all of the points in $E11$ already lie in $E1$.

bnsage123

6/23/24, 2:48 PM

yes, map from E11 to E1 is inclusion map, However, after using this formula Q11 = (U/a) * (x,y), I didn't know how to specifically revert it (i.e.,E11 --> E1 case). E1 --> E11 --> E1

Elon Musk

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