(2026-06) Decomposition of compressions on elliptic curves and point recovery
2026-06-22
Let E be an elliptic curve over a perfect field K. A function f∈K(E) is a compression of degree 2 on E if f(−P)=f(P) for all P∈E, and the field extension K(f)⊂K(E) is of degree 2. For a finite subgroup G⊂E over K a function w∈K(E) we will call a G-compression if w(±P+G)=w(P) for all P∈E, and the field extension K(w)⊂K(E) is of degree 2|G|. We will show that w∈K(E) is a G-compression if and only if w=f∘Φ for a separable isogeny Φ:E→E′ over K with kerΦ=G, an elliptic curve E′/K, and a compression f∈K(E′) of degree 2 on E′. This allows to obtain a doubling, a differential addition, and a method for point recovery for G-compressions using known properties of compressions of degree 2. For G-compressions w studied in the literature on an extended Jacobi quartic, a twisted Edwards curve, a twisted Jacobi intersection, and a twisted Hessian curve (for the first and third model additional conditions on coefficients are assumed) we will give the decomposition w=f∘Φ as above, and the function induced by the dual isogeny Φ^ and compressions of degree 2, which can be used for point recovery. For the first three models this isogeny Φ is to a Montgomery curve over K, and has the first coordinate x(Φ)=1/w. We also give isomorphisms from some models of elliptic curves to a
Montgomery curve.