Embedding Degree ¶
Let \(E\) be an elliptic curve defined over a finite field \(\mathbb{F}_p\), where \(n\) is the curve order. The embedding degree of \(E\) with respect to \(n\) is the smallest integer \(k\) such that \(n\) divides \(p^k - 1\), expressed as \(p^k - 1 = n\times X\).
Why \(p^k - 1 = N \times X\)? ¶
The embedding process ensures that the group of points of interest can be perfectly replicated within the finite field \(\mathbb{F}_p\).
From this: \(P \times (n+1) = P\) , so \(P \times (p^k - 1 + 1) = P \implies P \times (p^k) = P\).