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Computing Isogenies Between Montgomery Curves Using the Action of (0, 0)

2018; Springer Science+Business Media; Linguagem: Inglês

10.1007/978-3-319-79063-3_11

1611-3349

Joost Renes,

Advanced Numerical Analysis Techniques

A recent paper by Costello and Hisil at Asiacrypt’17 presents efficient formulas for computing isogenies with odd-degree cyclic kernels on Montgomery curves. We provide a constructive proof of a generalization of this theorem which shows the connection between the shape of the isogeny and the simple action of the point $$(0,0)$$ . This generalization removes the restriction of a cyclic kernel and allows for any separable isogeny whose kernel does not contain $$(0,0)$$ . As a particular case, we provide efficient formulas for 2-isogenies between Montgomery curves and show that these formulas can be used in isogeny-based cryptosystems without expensive square root computations and without knowledge of a special point of order 8. We also consider elliptic curves in triangular form containing an explicit point of order 3.