Can we determine a security parameter in the hybrid argument where the number of hybrids is polynomially bounded but not known? Let $\lambda$ be a statistical security parameter. Consider a security proof that is based on hybrid argument, where there are polynomially many (say, $n = p(\lambda)$) hybrids, $H_1, ..., H_n$. Any two hybrids $H_i$ and $H_{i + 1}$ can only be efficiently distinguished with probability $2^{-\lambda}$. So, by iterating through all the hybrids, it seems to me that the probability that $H_1$ and $H_n$ can be efficiently distinguished is bounded by $(p(\lambda) - 1) \cdot 2^{-\lambda}$ using union bound. Asymptotically, this probability is negligible in $\lambda$.
My question is that, when $n = p(\lambda)$ is polynomially bounded with respect to $\lambda$ but not known, it seems impossible to choose a reasonable $\lambda$ such that $(p(\lambda) - 1) \cdot 2^{-\lambda}$ is sufficiently small (e.g., $&lt; 2^{-40}$). I think that the same issue may exist as well when it comes to some computational security parameter $\kappa$. It would be appreciated if someone could give me any hint on this question. Many thanks.
a function &quot;polynomially bounded&quot; means there exists $d\in \mathbb{N}$ such taht
$f(\lambda)\leq \lambda^d$ asymptotically.
You don't have to know $f$, but if you want to bound the advantage of your adversary asymptotically, you have to know a such $d$ which verifies this inequality.
If you don't, then you can't deduce anything. But in general, if you look the proof in details what is $n$, and why it has been claimed polynomially bounded, it's easy to find a such $d$, such that $n\leq \lambda^d$ asymptotically.