# What does signed fixed window method mean in ECC?

I am studying (sliding) window method in Elliptic Curve Cryptography (ECC) but I am confused by the term, signed fixed window method. By the way term is used in a research paper and not in the book that I am reading.

• It is always helpful to provide additional context, e.g. a link to the paper. May 4, 2015 at 20:10
• Sure, here is the link of the paper cr.yp.to/papers.html#curve41417 May 4, 2015 at 20:18

In the basic fixed window method of performing point multiplication, we compute the value $nP$ (where $n$ is the integer we're multiplying by, and $P$ is the basis point) by finding the base $b$ representation $n = d_k b^k + d_{k-1} b^{k-1} + ... + d_1 b^1 + d_0 b^0$ (where $0 \le d_i < b$), and then computing first $1P, 2P, ..., (b-1)P$ and then $nP = d_0 P + b ( d_1 P + b ( d_2 P + ... + b(d_kP))) ...)$ (where the computation of $d_iP$ is done by a lookup of the precomputed values).
One is, not, though. It turns out that Elliptic Curves has one cheap operation that the group $Z_p^*$ does not; it is cheap to compute the inverse. So, one improvement to this basic method we can do is represent $n$ in balanced notation, where we allow negative digits; $-b/2 \le d_i \le b/2$. We then compute the values $1P, 2P, ..., (b/2)P$, and then when we need to do a lookup of a value $d_iP$, and $d_i < 0$, we just use the value $-(-d_i)P$, which is a lookup and an inverse.
What this allows us to do is double the size of $b$ at essentially no cost; the table we need to precompute is the same, and the number of additions we need to do is reduced.