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In section 4 (protocol 4.1) of the paper by Naor and Pinkas [1], why did the authors decide to operate in a subgroup? When they say "the messages are in the subgroup" does that mean $x, y, z_0, z_1$, $w_0, w_1$ and the encryption keys are in the subgroup but the private values such as $a, b, r_0, s_0$, etc. are not? Does the construction work for any subgroup or is it more typical to use group of quadratic residues modulo prime?

Sorry if this is a bit elementary, any references regarding how subgroups are used in DH-based protocols are also appreciated.

[1] "Efficient oblivious transfer protocols" http://www.pinkas.net/PAPERS/effot.ps

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The protocol's security comes from the DDH assumption, which is only secure when operating in groups of prime order [1]. The more typical groups, e.g. $\mathbb{Z}_p^*$ is not of prime order, but of order $p - 1$. Fortunately, a subgroup of $\mathbb{Z}_p^*$, namely the set of quadratic residues modulo $p$ doesn't have this problem. This answers the first question.

To answer the second question, the public values such as $x, y$ and so on are in the subgroup. The private values are selected uniformly at random from $\mathbb{Z}_q$, where $q$ is the order of the subgroup (and the generator $g$).

Finally, the set of quadratic residues is one of the simpler subgroups to understand, but others are also possible as long as DDH is hard. Some alternatives are outlined in [2].

[1] Katz and Lindell, Introduction to modern cryptography, Chapter 7.3.3

[2] Dan Boneh, The decision diffie-hellman problem, Section 1.1, http://crypto.stanford.edu/~dabo/pubs/papers/DDH.pdf

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