Frobenius endomorphism

Examples

• 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11, Let k=◌̅ F_p, and let q be a power of p such that the group G is defined over F_q. We then denote by F:G→G the corresponding Frobenius endomorphism. The Lie algebra G and the adjoint action of G on G are also defined over F_q and we still denote by F:G→G the Frobenius endomorphism on G. [...] Assume that H,X and the action of H over X are all defined over F_q. Let F:X→X and F:H→H be the corresponding Frobenius endomorphisms.
• 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 (MEVA 1900) using the curve E:y2+y=x3. In this case, the characteristic polynomial of the Frobenius endomorphism denoted by φ2 (cf. Example 4.87 and Section 13.1.8), which sends P_∞ to itself and (x_1,y_1) to (x,y), is χ_E(T)=T2+2. Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P∈E( F_2d), we have φ=-[2]P.