Assume $x$ is unknown and constant, but the following quantities are/can be known:
- $g^x \pmod{p}$
- $f(x, m) = (m, a, b) = (m, (x+h) \pmod{p-1}, g^h \pmod{p})$
where $p$ is a prime, $m$ positive integer, $g$ a generator of $\mathbb{Z}^*_p$ and $h \in \{0, \dots, p-2 \}$ with the property:
$$ \mathcal{H}(m) + x + h \equiv 0 \pmod{p-1} $$
where $\mathcal{H}$ collision-resistant hash function. Is there a way to find $x$?
My attempt was to calculate $f$ for two values $m_1, m_2$ and then take:
$$ a_1 - a_2 \equiv h_1 - h_2 \pmod{p-1} $$
but $x \equiv a_i - h_i \pmod{p-1}$ and I cannot figure out a way to determine these $h_i$. Any help?