Edwards normal form

An elliptic curve is in Edwards normal form if it can be described by the equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + y^2 = 1 + dx^2y^2}

The more general original Edwards form has another parameter c

$x^{2}+y^{2}=c^{2}(1+dx^{2}y^{2})$

but it is very simple to eliminate the scale parameter c and reduce this equation to its normal form for ease of performing algebraic group operations on rational points or finite fields.

The twist

There is also a “twisted” Edwards form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ax^2 + y^2 = 1 + dx^2y^2}

And this brings us back to the ancient Greek word στραγγός with the idea of using something “twisted” for strong cryptography.