Meaning of Frobenius endomorphism | Babel Free

Definitions

Given a commutative ring R with prime characteristic p, the endomorphism that maps x → xᵖ for all x ∈ R.

Equivalents

Examples

“Section 3 concerns what properties of the ring⟳ other than regularity are reflected by the homological properties of the Frobenius endomorphism.”

“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.”

CEFR level