I create a new keypair, <c_pub> with <c_priv> which I know (it can be any arbitrary key pair).  But I don't give you <c_pub>, I give you  <b_pub> = <c_pub> minus <a_pub> (which I can do because I've seen <a_pub> before doing this). 

Sure, I don't know the private key for <b_pub>, but it doesn't matter... because what

<b_pub> + <a_pub> = <c_pub> (mine)

You have no way to detect this condition, because you don't know what c_pub/c_priv I created, so you can only detect this after it's too late (after I abuse the private key)