Pair encodings and predicate encodings, recently introduced by Attrapadung (Eurocrypt 2014) and Wee (TCC 2014) respectively, greatly simplify the process of designing and analyzing predicate and attribute-based encryption schemes. However, they are still somewhat limited in that they are restricted to composite order groups, and the information theoretic properties are not sufficient to argue about many of the schemes. Here we focus on pair encodings, as the more general of the two. We first study the structure of these objects, then propose a new relaxed but still information theoretic security property. Next we show a generic construction for predicate encryption in prime order groups from our new property; it results in either semi-adaptive or full security depending on the encoding, and gives security under SXDH or DLIN. Finally, we demonstrate the range of our new property by using it to design the first semi-adaptively secure CP-ABE scheme with constant size ciphertexts.