Frobenius endomorphism

English

Etymology

Named after German mathematician Template:W.

Noun

1. Template:Lb Given a commutative ring R with prime characteristic p, the endomorphism that maps x → x ^p for all x ∈ R.
   • Template:Quote-book
   • 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Template:W 1859, page 11,

     Let $k={\overline{𝔽}}_p$, and let $q$ be a power of $p$ such that the group $G$ is defined over ${𝔽}_q$. We then denote by $F:G\to G$ 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 $F:𝒢\to 𝒢$ the Frobenius endomorphism on $𝒢$.

     Template:...Assume that $H,X$ and the action of $H$ over $X$ are all defined over ${𝔽}_q$. Let $F:X\to X$ and $F:H\to H$ 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

     $E:y^2+y=x^3$.

     In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ${\varphi }_2$ (cf. Example 4.87 and Section 13.1.8), which sends $P_{\mathrm{\infty }}$ to itself and $(x_1,y_1)$ to $(x_1^2,y_1^2)$, 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 $P\in E({𝔽}_{2^d})$, we have ${\varphi }_2^2=-[2]P$.

Synonyms

• Template:Sense Template:L

Related terms

• Template:L
• Template:L
• Template:L
• Template:L

Translations

• French: Template:T
• German: Template:T
• Italian: Template:T

Further reading

• Template:Pedia
• Template:Pedia
• Template:Pedia
• Frobenius endomorphism on Template:W
• Frobenius morphism on Template:W