Elliptic curve addition formulas

The geometric description of addition of points P and Q on an elliptic curve involves four logical branches:

If one of P or Q is the point at infinity …

Else if P = Q

Else if P and Q lie on a vertical line …

Else …

It would seem that an algorithm for adding points would have to have the same structure, which is unfortunate for at least a couple reasons: it adds complexity, and it raises the possibility of a timing attack since some branches will execute faster than others.

However, an algebraic formula for addition need not mirror its geometric motivation.

Jacobian coordinates

Jacobian coordinates are a different way to describe elliptic curves. They have the advantage that addition formulas have fewer logic branches than the geometric description. They may have one test of whether a point is the point at infinity but they don’t have multiple branches.

Jacobian coordinates also eliminate the need to do division. The geometric description of addition on an elliptic curve involves calculating where the line through two points intersects the curve, and this naturally involves calculating slopes. But in Jacobian coordinates, addition is done using only addition and multiplication, no division. When you’re working over the field of integers modulo some gigantic prime, multiplication and addition are much faster than division.

You can find much more on Jacobian coordinates, including explicit formulas and operation counts, here.

Complete formulas

Sometimes it’s possible to completely eliminate logic branches as well as division operations. An elliptic curve addition formula is called complete if it is valid for all inputs. The first surprise is that complete addition formulas sometimes exist. The second surprise is that complete addition formulas often exist.

You can find much more on complete addition formulas in the paper “Complete addition formulas for prime order elliptic curves” available here.

McCabe versus Kolmogorov

Whether the Jacobian coordinate formulas or complete formulas are more or less complex than direct implementation of the geometric definition depends on how you define complexity. The formulas are definitely less complex in terms of McCabe complexity, also known as cyclomatic complexity. But they are more complex in terms of Kolmogorov complexity.

The McCabe complexity of a function is essentially the number of independent paths through the function. A complete addition formula, one that could be implemented in software with no branching logic, has the smallest possible McCabe complexity.

I’m using the term Kolmogorov complexity to mean simply the amount of code it takes to implement a function. It’s intuitively clear what this means. But if you make it precise, you end up with a measure of complexity that is only useful in theory and cannot be calculated in practice.

Counting operations

Instead of literally computing Kolmogorov complexity you’d count the number of multiplications and additions necessary to execute a formula. The links above do just that. If you know how many cycles it takes to execute an addition or multiplication (in the field you’re working over), and how many cycles it would take to carry out addition (on the elliptic curve) implemented directly from the definition, include the division required, then you could estimate whether these alternative formulas would save time.

It used to be common to count how many floating point operations it would take to execute an algorithm. I did that back when I took numerical analysis in college. But this became obsolete not long after I learned it. Counting floating point operations no longer tells you as much about an algorithm’s runtime as it used to due to changes in the speed of arithmetic relative to memory access and changes in computer architecture. However, in the context of this post, counting operations still matters because each operation—such as adding two 512-bit numbers modulo a 512-bit prime—involves a lot of more basic operations.

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