The fastest known differential-addition formulas for elliptic curves (with small parameters) were introduced by Gaudry in 2006, beating the well-known 1987 Montgomery formulas. Computer-verified versions of Gaudry's formulas, reinterpreted as formulas for (Y:Z) and (Y^2:Z^2) coordinates on Edwards curves, have been online since June 2009 as part of the Explicit-Formulas Database: [hyperelliptic.org
One can also find slightly different Edwards (Y:Z) formulas in eprint 2008/218. I don't know any situations where 2008/218 is useful: for small parameters it's faster than 1987 Montgomery but slower than 2006 Gaudry, and for large parameters 1987 Montgomery is best.
Paper 2009/523 by Justus and Loebenberger claims to "introduce" (Y:Z) and (Y^2:Z^2) coordinates for Edwards curves. The paper cites EFD but nevertheless claims, falsely, that these coordinates are new. The central formulas in the paper are identical to previously published formulas. This is inexcusable; the paper has to be withdrawn.
---D. J. Bernstein
Research Professor, Computer Science, University of Illinois at Chicago