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How Mathematical Curves Power Cryptography - Quanta Magazine

How Mathematical Curves Power Cryptography - Quanta Magazine

How Mathematical Curves Power Cryptography - Quanta Magazine
Sep 19, 2022 1 min, 41 secs

These applications all rely on a strikingly beautiful and conceptually straightforward use of interpolation: so-called Reed-Solomon codes, and the codes that build on them.

To allow the recipient to identify whether an error occurred, send the same message twice: 2, 7, 2, 7.

To let them figure that out and correct the error, send the same message three times: 2, 7, 2, 7, 2, 7.

Here’s a smarter approach: Encode the message as a curve, and send just enough information to allow the recipient to reconstruct that curve.

And to figure out where the error is, you just send one more value — meaning you’ve sent four numbers total, rather than the six required by the previous method.

Let’s say you want to send a longer message — 1,000 numbers.

The less efficient code would require sending 2,000 numbers to identify an error, and 3,000 to correct it.

But if you use the code that involves interpolating a polynomial through given points, you only need 1,001 numbers to find the error, and 1,002 to correct it.

(You can add more points to identify and correct more potential errors.) As the length of your message increases, the difference in efficiency between the two codes grows starker.

The more efficient code is called a Reed-Solomon code.

(Imagine an encryption key, for instance, or a missile launch code.) You encode the numbers in a polynomial, evaluate that polynomial at a predetermined set of points, and distribute each of the results to a different person.

To do this, mathematicians devised codes that involve interpolating a function — which lives in a special space associated to a more complicated curve — through given points on that curve.

These so-called algebraic geometry codes “came out of nowhere, and they’re better than any other code we know how to make [with a smaller alphabet],” Kopparty said.

In practice, implementing a Reed-Solomon code is much, much easier than implementing an algebraic geometry code.

For now, that’s not worrisome: In real-world applications, Reed-Solomon codes and related forms of error correction are sufficient

Certain versions of algebraic geometry codes might just work


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