# Elliptic Curve Over Extension Field ## Metadata **Status**:: #x **Zettel**:: #zettel/literature **Created**:: [[2024-03-25]] **Topic**:: [[♯ Cryptography]] ## Synopsis Elliptic Curve over extension field $F_{p^k}$ uses the non-prime field $\mathrm {GF} (p^k)$ (**see**:: [[Finite Field]]) to define the curve polynomial. Taking the [[Elliptic Curve|Weierstrass Curve]] as an example $ y^2 = ax^3 + bx + c $ If the curve is defined over $\mathrm{GF}(p^k)$, all the variables $x, y$, and coefficients $a, b, c$ are polynomials in $\mathrm{GF}(p^k)$. Since $\mathrm{GF}(p^k)$ is a field, all the operations defined in [[Elliptic Curve]] and [[Elliptic Curve Over Finite Field]] also works for the extension field. Elliptic Curve Over Extension Field must use [[Extension Field Arithmetic]] for all computations.