Twin primes and RSA

I just noticed an interesting quirk in how the RSA algorithm works if you use twin primes for the p and q that are used to calculate the modulus N = pq. Twin primes are those that differ by 2, so instead of writing p and q, let’s write p and p + 2 so that N = p (p + 2) and we have that f(N) = (p -1) (p +1) = p² – 1.

Now let’s pick our public key to be e = p. Then we need that the private key d has the property that ed = 1 (mod f(N)). Note that d = p satisfies this because p² = 1 (mod p² – 1). So if we have that the modulus is the product of twin primes, then we can find a case where the public key and the private key are the same.

I doubt that this is the first time that someone’s though of this, but it was new to me.