Question

I don’t understand how an algorithm is able to encrypt plaintext with a public key yet not be able to decrypt it with the same key. Could someone explain this process in the simplest form possible, with mathematical terms defined?

Answer 1

Think of it like this. Some mathematical operations are invertible. Consider, for example, the operation “multiplication by a non-zero real number.” Fix a non-real number s and consider the operation f(x) defined by x -> x * s. Then this operation is invertible. In fact, if you take t = 1 / s then the operation g(x) defined by g(x) = x * t has the property that g(f(x)) = x so that f is invertible. Think of x as the message, s as the public key, f as the encryption algorithm, t as the private key and g as the decryption algorithm. Of course, this is a terrible algorithm, but this is all there is to asymmetric encryption: find a parameterized invertible mathematical operation. The parameter provides the public key, and the “inverse parameter” provides the private key.

Of course, with encryption we want it to be harder to find the inverse. In fact, the mathematics of RSA, the most well-known of the asymmetric key algorithms, are quite sophisticated. It relies on the fact that a certain mathematical problem is thought to be extremely hard.