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Groups, Subgroups and Orders Of An Elliptic Curve
Can I get two groups of points from an elliptic curve? Well, with a well-designed elliptic curve, if I pick one base point on the curve, I get one group of points, and if I pick another one, I get another group of points.
I know that the discrete logarithm problem that is used in Elliptic Curve Cryptography (ECC) can be broken by Shor’s algorithm. But against our classical computing methods, it is still as strong as ever after more than two decades of trying to break it. To me, ECC is beautiful, and one of the most beautiful structures of the Twisted Edwards curve:
As the order defines the number of possible points, this will define the security of the curve we are using. If we had an order of 256, then the security of the curve would be 8 bits.
Finding the points
Within an elliptic curve using (mod p), not all the points are possible. For this, we define the order of the curve (n) and which is equal to the number of possible points. In this case, we will use the Baby Jubjub curve, and which uses a Twisted Edwards curve of:
