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Tangent lines for curves over a finite field

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This morning I was talking to one of our core crypto team when one of our engineers walked by. When he heard what we were talking about he said something like, “Wait a minute, how can you talk about things like derivatives and smooth curves when you’re dealing with curves over a finite field? Don’t you need some sort of continuous function to talk about things like that?” This led to a discussion of tangent spaces for arbitrary curves.

Here’s roughly what we talked about.

If you have a curve which has points whose coordinates come from an arbitrary field, and you want a definition of a tangent to the curve that makes sense, one way to do it is to generalize what we have for the continuous case.

Here’s a picture of the tangent to the curve y = x2 at the point (1,1). It’s the line

y = 2x – 1

Quad

At the point (1,1) if we set the two expressions for y equal to each other we get that

x2 = 2x – 1

or

x2 – 2x – 1 = 0

which we can write as

(x – 1)2 = 0

In we look at the tangent to the curve y = x3 at the point (1,1) we get that it’s the line

y = 3x – 2

Cubic

And if we set these two expressions for y equal to each other we get that

x3 = 3x – 2

or

x3 – 3x + 2 = 0

which we can write as

(x – 1)2(x + 2) = 0

In both cases we have a repeated root where the tangent line intersects the curve.

In the general case we say that a line is tangent to a curve at a point if the multiplicity of the intersection of the line and the curve at the point is greater than 1. And that even makes sense for cases where we don’t have anything close to the continuity that we have in the case of functions of a real variable. Like we have with finite fields.

And once you have the idea of a generalized tangent line, getting a generalized tangent space is easy: it's just the set of all tangent lines.



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