Plaintext block chaining, bad idea why?

Question

Ahead of the question, little warning. Question is not about what are the better options. Question is why it is bad idea?

Ok, suppose we have plaintext $p$, which divided into blocks: $p_1,p_2,p_3,...,p_n$

CBC (cipher block chaining) can be then defined as $c_1=E_k(p_1 \oplus IV)$,$c_2=E_k(p_2 \oplus c_1)$,$c_{i+1}=E_k(p_{i+1}\oplus c_i)$

Ok. What would be bad about this version $c_{i+1}=E_k(p_{i+1}\oplus p_i)$?

($E_k$ is some encryption function)

Answer 1

1. Patterns aren't hidden. Assume you've got a pair of plaintext blocks $p_1, p_2$ then the encryption of $p_2$ will always be the same: $c_2=E_k(p_1 \oplus p_2)$ as it only depends on the plaintext. So you haven't basically solved the basic problem of ECB (patterns remain) but rather moved it (to the more rare case) that two blocks repeat.
2. Plaintext isn't random. In most cases an attacker can predict plaintext. As you basically use plaintext as IV for each block you make the scheme vulnerable to attacks like BEAST, where attackers use predictable IVs to gain information about target connections. See this related question.

Corrected my speed concerns. Encryption and decryption can be parallelized, speed should be comparable with CTR.