Suppose that Alice and Bob use an encryption algorithm based on a one-way function $F$ (Eve knows it), and $r$, a secret key shared between Alice and Bob (Eve doesn't know it). Key is computed as: $K = F(r)||F(2r)||F(3r)||...$, and the ciphertext is computed as $C = M \oplus K$, and all the blocks of $F(r)$ are $1024$-bit long.
If $F(x) = x^e \mod n$, and $(n, e)$ pair is public, will the first $1024$ bits of both plaintext and ciphertext be enough for Eve to read the entire message?
Same for $F(x) = g^x \mod p$, and public $(g, p)$ pair.
Also, are the cases described above good examples of public-key cryptography? But Eve can't encrypt only having public keys, as she doesn't have $r$. So interception of $1024$-bit long plain/cipher pair is the only way for her to obtain some information from the system.