D. J. Bernstein
Fighting patents

US patent 8045705, Antipa Poeluev, double-scalar multiplication

Text of the patent
No reported enforcement attempts. Priority date 2005.11.03. Basic claims:

Prior art:

Readers interested in understanding these techniques should consult my 2006 paper, along with Daniel Brown's ladder generalization to 3 or more scalars.

The patent specifically recommends computing mP+nQ together with (m+1)P+(n+1)Q, and says that this requires only 1 doubling and on average 1.5 additions per bit. However, many of these additions flunk the differential-addition requirement; this specific approach is clearly slower than, e.g., 2001 Akishita.