I'm reading Improved Efficiency for CCA-Secure Cryptosystems Built Using Identity-Based Encryption. In Section 4 there is a description of encapsulation schemes that says
We suggest an encapsulation scheme based on a fixed cryptographic hash function $H : \{0, 1\}^{448} → \{0, 1\}^{128}$ (constructed, e.g., by suitably modifying the output length of SHA-1)
Setup chooses a hash function $h$ from a family of pairwise independent hash functions mapping 448-bit strings to 128-bit strings, and outputs $\text{pub} = h$
The encapsulation algorithm S takes $\text{pub}$ as input, chooses a random $x ∈ \{0, 1\}^{448}$, and then outputs $(r = h(x), \text{com} = H(x), \text{dec} = x)$
What does it mean to do $h(x)$ vs $H(x)$? It's clear that $H$ is SHA, but what is an example of $h$? I'd appreciate it if someone could walk me through the details of this entire encapsulation algorithm, preferably with a concrete example.