Not How The Grinch Stole Christmas, But How Galois Saved Cybersecurity
So, for an electrical engineer, the most fundamental knowledge that every practicing engineer has is Ohm’s Law. But, what’s the most basic knowledge that a cryptographer should know? Well, I think it is the magic of Galois fields, and which has saved our cybersecurity world, and will scale into a future of lattice-based cryptography. These lattice methods are based on polynomial operations and Galois fields and will replace our existing public key methods of RSA, ECC and discrete logs.
Okay. So here’s my Christmas essay, and it’s not about how the Gringe Stole Christmas, but about how Galois saved cybersecurity. The Galois in this case is Évariste Galois, and who invented Galois fields. He lived from 1811 to 1832, and where his short life was ended from the wounds he received in a duel.
Z_p (mod p)
First, let’s start with the problem. In finite fields, we have a (mod n) operation for a list of integers (Z), and where n is a prime number. This works for our add (+) and multiply (*) operations, and where:
a (mod n) + b (mod n) = (a+b) (mod n)
a (mod n) *b (mod n) = (a*b) (mod n)
If we take Z_7, we get [here]:
Additive group for Z_7
+ | 0 1 2 3 4 5 6
…
