I'm currently working with BLS Signature Schemes in the field of publicly verifiable Compact Proofs of Retrievability by Shacham and Waters.
So for creating the Sigmas the following function is defined:
$$
\sigma_i \leftarrow\left(H(\text { name } \| i) \cdot u^{m_{i}}\right)^\alpha .
$$
$\left\{m_{i}\right\}_{\substack{1 \leq i \leq n}}$ are bytes derived from a File $M$ which is devided into $n$ blocks. $u$ is a generator of $G_1$. $\alpha$ is the private key with $\alpha \stackrel{\mathrm{R}}{\leftarrow} \mathbb{Z}_p$. The hash function is defined as $H:\{0,1\}^* \rightarrow G_1$
The Proofs are generated by:
$$\mu \leftarrow \sum_{\left(i, \nu_i\right) \in Q} \nu_i m_{i} \in \mathbb{Z}_p$$
and
$$\quad \sigma \leftarrow \prod_{\left(i, \nu_i\right) \in Q} \sigma_i^{\nu_i} \in G_1 $$
where $\nu_i \stackrel{\mathrm{R}}{\leftarrow} \mathbb{Z}_p$.
Finally the verification is done by the following pairing:
$$
e(\sigma, g_2) \stackrel{?}{=} e\left(\prod_{\left(i, \nu_i\right) \in Q} H(\text { name } \| i)^{\nu_i} \cdot u^{\mu}, v\right) ;
$$
Where $v$ is the public key computed by $v \leftarrow g^\alpha$ and $g_2$ is a generator of $G_2$
The problem:
As $\mu$ in the proof generation becomes a big number, my idea was to apply a modulus on $m_i$. I would choose a random prime $c \stackrel{\mathrm{R}}{\leftarrow} \mathbb{Z}_p$ and reduce $m_i$ in the sigma generation and the proof generation.
So the new construction of $\sigma_i$ and $\mu$ would look like this:
$$
\sigma_i \leftarrow\left(H(\text { name } \| i) \cdot u^{(m_{i} \bmod c)}\right)^\alpha .
$$
$$\mu \leftarrow \sum_{\left(i, \nu_i\right) \in Q} \nu_i \cdot (m_{i} \bmod c) \in \mathbb{Z}_p$$
The question:
Does applying the modulus like this lower the security of the authenticators (sigmas) in a bad way?
A solution:
As @poncho pointed out, I can just apply $\mu \bmod p$ where $p$ is the prime order of $G_1$ in order to bring the size of $\mu$ down and still have a valid pairing. So I do not need to fiddle around with the bytes I'm reading as i wrote in my idea.
Thanks and best regards.
