Abstract
We use two parametrizations of points on elliptic curves in generalized Edwards form x 2 + y 2 = c 2 (1 + d x 2 y 2) that omit the x-coordinate. The first parametrization leads to a differential addition formula that can be computed using 6M + 4S, a doubling formula using 1M + 4S and a tripling formula using 4M + 7S. The second one yields a differential addition formula that can be computed using 5M + 2S and a doubling formula using 5S. All formulas apply also for the case c ≠ 1 and arbitrary curve parameter d. This generalizes formulas from the literature for the special case c = 1 or d being a square in the ground field.
For both parametrizations the formula for recovering the missing X-coordinate is also provided.
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Justus, B., Loebenberger, D. (2010). Differential Addition in Generalized Edwards Coordinates. In: Echizen, I., Kunihiro, N., Sasaki, R. (eds) Advances in Information and Computer Security. IWSEC 2010. Lecture Notes in Computer Science, vol 6434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16825-3_21
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DOI: https://doi.org/10.1007/978-3-642-16825-3_21
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