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What is the characteristic polynomial of the Frobenius?

the characteristic polynomial det(I TF | Hi(V )) of the Frobenius endomorphism F of the variety V over the field k. We use the same notation, but with a subscript c, when we deal with cohomology with compact support.

What is the Frobenius endomorphism with finite fields?

Now consider the finite field Fqf as an extension of Fq, where q = pn as above. If n 1, then the Frobenius automorphism F of Fqf does not fix the ground field Fq, but its nth iterate Fn does. The Galois group Gal(Fqf /Fq) is cyclic of order f and is generated by Fn. It is the subgroup of Gal(Fqf /Fp) generated by Fn.

What is the endomorphism of an elliptic curve?

An endomorphism of an elliptic curve E is a homomorphism from E to itself. The set of all endomorphisms of E is denoted End(E). The group structure on E makes End(E) into a ring. Addition in End(E) is defined by ( + )(P) := (P) + (P) Multiplication in End(E) is defined by := ◦ .

What is the frobenius endomorphism of the elliptic curve?

The Frobenius endomorphism of the elliptic curve has quadratic characteristic polynomial. In most cases this is irreducible and defines an imaginary quadratic order; for some supersingular curves, Frobenius is an integer a and the polynomial is ( x a ) 2 .

What is the characteristic equation of the Frobenius?

The characteristic polynomial of the Frobenius endomorphism Here tl t mod l and ql q mod l correspond to restrictions of the scalar multiplication endomorphisms [t],[q] End(E). ql = ql [1]l = [1]l + + [1]l using double-and-add, provided that we know how to explicitly compute in End(E[l]).

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What is a docHub advantage of the use of elliptic curve cryptography ECC over older forms?

The main advantage of elliptic curves cryptography is that to achieve a certain level of security shorter keys are sufficient than in case of usual cryptography. Using shorter keys can result in a considerable savings in hardware implementations.

What are the weakness of elliptic curve cryptography?

There are a docHub number of potential vulnerabilities to elliptic curves, such as side-channel attacks and twist-security attacks. These attacks threaten to invalidate the security ECC aims to provide to private keys.

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