Frobenius endomorphism

Etymology
Named after German mathematician.

Noun

 * 1)  Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xp for all x ∈ R.
 * 2) * 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, 1859, page 11,
 * Let $$k = \overline\mathbb F_p$$, and let $$q$$ be a power of $$p$$ such that the group $$G$$ is defined over $$\mathbb F_q$$. We then denote by $$F: G\to G$$ the corresponding Frobenius endomorphism. The Lie algebra $$\mathcal G$$ and the adjoint action of $$G$$ on $$\mathcal G$$ are also defined over $$\mathbb F_q$$ and we still denote by $$F: \mathcal G\to\mathcal G$$ the Frobenius endomorphism on $$\mathcal G$$.
 * Assume that $$H, X$$ and the action of $$H$$ over $$X$$ are all defined over $$\mathbb F_q$$. Let $$F: X\to X$$ and $$F: H\to H$$ be the corresponding Frobenius endomorphisms.
 * 1) * 2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356,
 * The first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (M E V A 1900) using the curve
 * $$E: y^2+y=x^3$$.
 * In this case, the characteristic polynomial of the Frobenius endomorphism denoted by $$\phi_2$$ (cf. Example 4.87 and Section 13.1.8), which sends $$P_\infty$$ to itself and $$(x_1, y_1)$$ to $$(x^2_1, y^2_1)$$, is
 * $$\chi_E(T) = T^2 + 2$$.
 * Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points $$P\in E(\mathbb F_{2^d})$$, we have $$\phi^2_2 =-[2]P$$.
 * Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points $$P\in E(\mathbb F_{2^d})$$, we have $$\phi^2_2 =-[2]P$$.

Translations

 * French: endomorphisme de Frobenius
 * German: Frobeniushomomorphismus
 * Italian: endomorfismo di Frobenius