**Differential Addition in generalized Edwards Coordinates**

*Benjamin Justus and Daniel Loebenberger*

**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.

For both parametrizations the formula for recovering the missing X-coordinate is also provided.

**Category / Keywords: **foundations /

**Date: **received 28 Oct 2009, last revised 4 Mar 2010

**Contact author: **daniel at bit uni-bonn de

**Available format(s): **PDF | BibTeX Citation

**Version: **20100304:162037 (All versions of this report)

**Short URL: **ia.cr/2009/523

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