I'm trying to learn about elliptic curve cryptography.

Let's say you have point $P$ and 256 bit number $n$ and you want to compute $nP$. It sounds like computing additions one at a time is not computationally feasible. Is there an algorithmic "shortcut" to compute this? If so, how does it work?


It is called "double-and-add". For instance, take a point, and add it to itself. Then take the result, and add it to itself. Do that again. And again. After ten additions, what you get is $2^{10} = 1024$ times the original point: in just 10 additions, not 1023. That's the "shortcut".

More generally, with $k$ successive doublings (a "doubling" is the addition of a point with itself), starting from a point $P$, you get $2^k P$. Now, all you have to do is to consider your multiplier $n$ in binary: this is about writing it as a sum of powers of two. For instance, if $n = 224965$, then, in binary, it becomes $110110111011000101$, which means that: $$ n = 2^0 + 2^2 + 2^6 + 2^7 + 2^9 + 2^{10} + 2^{11} + 2^{13} + 2^{14} + 2^{16} + 2^{17} $$ And therefore: $$ nP = 2^0P + 2^2P + 2^6P + 2^7P + 2^9P + 2^{10}P + 2^{11}P + 2^{13}P + 2^{14}P + 2^{16}P + 2^{17}P $$ So all you have to do to compute $nP$ is to compute these $2^kP$ (with successive doublings) and then add them together.

$2^0P$ is just $P$, so you already have it. Double it twice, and you get $2^2P$. Double that one four times, and you get $2^6P$. And so on. With $17$ doublings, you'll get all $2^kP$ for $k$ up to $17$, so in particular you'll get all the values you are interested in. As the formulas above show, ten extra additions, between the eleven $2^kP$ that are needed, will yield the result. In total, you got $224965 P$ with $17$ doublings and $10$ additions, i.e. much fewer than $224964$.

In all generality, if $n$ is a number of $t$ bits (i.e. less than $2^t$), then $t-1$ doublings and at most $t-1$ extra additions are enough.

There are many variations upon this mechanism, depending on how you interleave the doublings and the additions, and possibly reuse some addition results. You'll still need to compute the $t-1$ doublings, but you may save on some of the extra additions. How many intermediate results you need to keep in RAM is also a consideration. Moreover, if $n$ is a secret value, then you'll want to avoid side-channel attacks that may result in leaking some of the bits of $n$, and this impacts the kinds of double-and-add algorithms you can tolerate.


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