Security protocols

If you send a user-controlled number ``I'' and your peer replies with I + 1 encrypted by K and then gives you a number ``J'', is there any way to find J + 1 encrypted by K if you do not know the K?

You can do that exactly if the scheme is malleable. You can also use encryption for authentication but please don't try to write your own authentication scheme if you don't know about malleability, it will probably fail horribly in some way.