John’s Napier’s Legacy Lives On!

I work in the shadow of John Napier’s Tower. Each weekday I walk past the place that he created one of the greatest scientific achievements … the discovery of logarithms.

If he were alive today, I feel he would be proud of the work of the university that took his name. His work now protects those online from the prying eyes of those who wish to do harm.

John’s legacy lives on in the core of modern-day cryptography, but this time we use discrete logarithms, and where we only use integer values. We also constrain to a finite field of between 0 and a prime number (p) minus 1. The addition of the (mod p) operation allows us to apply our normal maths principles, but constrain the range of numbers produced (a finite field). And so:

a x b (mod p) = a (mod p) x b (mod p)

and where (mod p) is the remainder to a division by p. Within discrete logarithms, we have a value of g, and raise it to the power of x, and then take (mod p):

Y = g^a (mod p)

If p is large, then we do not have the computing power to work out the value of x, even though we know Y, g and p. I can see John smiling as he reviews his contribution, and then would delight at the Diffie-Hellman method:

Alice generates a, Bob generates b. Alice calculates A=g^a (mod p). Bob calculates…