# Why is this signature independent of the message?

Assume that we have the following signature scheme CL Signature:

• Choose a cyclic group $$G = \langle g \rangle$$ of order $$q$$.
• Uniformly and randomly choose two elements $$x,y \in \mathbb{Z}_q$$, and compute $$X = g^x$$ and $$Y = g^y$$.
• The secret key is $$sk = (x,y)$$, while the public key is $$pk = (q, G, g, X, Y)$$.
• On input a message $$m \in \mathbb{Z}_q$$, secret key $$sk$$ and public key $$pk$$, choose a random $$a \in G$$ and output the signature: $$\sigma = (a, a^y, a^{x + xym}).$$

In the same paper, they ensure that $$\sigma$$ is NOT information-theoretically independent of the message $$m$$ being signed and propose an alternative that achieves this independent notion $$\sigma = (a, a^z, a^y, a^{zy}, a^{x + xy(m + zr)}),$$ where $$z,r \in \mathbb{Z}_q$$ are another uniformly random elements such that $$Z = g^z$$ is also part of the public key $$pk$$.

Several questions arises on that:

1. What exactly means to be information-theoretically independent?
2. Why it is not achieved by the first scheme?
3. What happens if we change $$a^{x + xy(m + zr)}$$ with $$a^{x + xy(m + r)}$$?

Intuitively, I think that being information-theoretically independent means that the signature reveals no information about the message $$m$$. Then, why the first one reveals something about the message $$m$$?