Cryptosystems which support computation on encrypted data. They might be partially homomorphic (support for one operation such as + or *) or they might be fully homomorphic (any sequence of + and *).

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How to construct encrypted functions (with either public or private data)?

Homomorphic encryption is often touted for its ability to Compute on encrypted data with public functions Compute an encrypted function on public (or private) data I feel I have a good grasp of #1 ...
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How close is homomorphic encryption to handling regular expressions?

Is there any reasonable homomorphic encryption protocol that supports some meaningful fragment of regular languages/expressions and/or edit distance bounds? I'm suspicious that homomorphic encryption ...