Connecting Legendre with Kummer and Edwards
The paper under review deals with the scalar multiplication over Legendre form elliptic curves which is a basic operation in many public key cryptographic schemes. The bulk of the computation can be performed either on the associated Kummer line or on an appropriate twisted Edwards form elliptic curve. The first approach requires a method for recovering the $y$-coordinate of the result from the Kummer line computation. In this paper, a detailed explicit formula for doing this is provided. Furthermore three conversion methods from Legendre form curves to appropriate twisted Edwards form curves are given. The two methods are constructions of birational equivalences while the third a 2-isogeny. Furthermore, these methods are applied in order to obtain new concrete twisted Edwards curves from Legendre curves which correspond to known Kummer lines at the 128-bit security level. These Kummer lines provide very fast scalar multiplication on modern architectures supporting SIMD operations.
scientific article; zbMATH DE number 7024180
scientific article