The signer creates the transaction. As part of that it creates a nonce, which is supposed to be random. What if instead of a random number it stored part of your seed encrypted under a secret only known by the signer? It would appear random, how could you tell? You can't. Then each time you put out a transaction, the nonce would leak part of your seed. After 6 of these your whole seed could be exposed on the blockchain to anybody who knows the key the signer encrypted it under.