A group law for PKC purposes

02/01/2018

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by   R. Durán Díaz, et al.

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Let F be a field, let V=F^3, and let A V→ V a linear map. The polynomial P(x)= (x_1I+x_2A+x_3A^2) does not depend on A but only on its characteristic polynomial χ(X). A law of composition ⊕ V× V → V is defined and it induces an Abelian group law on FP^2∖ P^-1(0). The cubic P^-1(0) is irreducible if and only if χ is irreducible in F[X], and in this case the group law ⊕ is cyclic.