The Beauty of Elliptic Curves: A Real-life Example of an Isogeny
The world of cryptography could be changed using a relatively technique: isogenies (“equal origin”). One of the best usages of isogenies is within quantum robust key exchange (using SIDH —Supersingular isogeny key exchange) An isogeny provides a mapping from one elliptic curve to another. The standard form of an elliptic curve equation is:
y²=x³+ax+b
In this case we will follow the example defined by on Page 277 of [1]:
This defines a mapping of:
E2: y²=x³+1132x+278
to:
E4: y²=x³+500x+1005
and using 𝔽_2003. The mapping from one curve to the next is then defined with:
In [1], we defined two points on E2 of P2=(1120,1391) and Q2=(894,1452) — see the sample run below. These then map to: P4=(565,302) and Q4=(1818,1002) on E4. E2 and E4 are isogenous, but not isomorphic (and where we cannot reverse the mapping from E4 to E2.
Coding
The following is the coding required for this isogeny [here]:
