➿Elliptic Curve Cryptography

Elliptic curve cryptography is a type of asymmetric or public key cryptography based on the discrete logarithm problem as expressed by addition and multiplication on the points of an elliptic curve.

A visualization of an elliptic curve is an example of an elliptic curve, similar to that used by Unloqen.

Note	Unloqen uses the elliptic curve, called bn254. That makes it possible to reuse many of the elliptic curve libraries and tools.

Figure 1. A visualization of an elliptic curve

Unloqen uses a specific elliptic curve and set of mathematical constants, as defined in a standard called bn254. The bn256 curve is defined by the following function, which produces an elliptic curve:

$$y^2 = ( x^3 +b )$$

over a prime field $F_p$ where the prime $p$ is given by

$$p = 36x^4 + 36x^3 + 24x^2 + 6x + 1$$

for a parameter $x$

The parameter $x$ also determines other constants attached to the curve $E$ of interest:

$$r = 36x^4 + 36x^3 + 18x^2 + 6x + 1$$

$$t = 6x^2 + 1$$

The embedding degree of a BN curve is always 12, meaning that 12 is the smallest integer $k$ such that $E(F_(p^k))[r] = E(F^{-1}_p)[r]$

BN curves are constructed (using the CM method) to always have CM discriminant − 3 and $i$ -invariant equal to 0 .