I don't think the "homomorphic encryption" idea works as advertised in that post--being able to execute arithmetic operations on encrypted data doesn't enable you to execute the operations that are encoded within that encrypted data.

A fully homomorphic encryption scheme for single-bit plaintexts (as in Gentry's scheme) gives us:

• For each public key K a field F with efficient arithmetic operations +F and *F.
• Encryption function E(K, p) = c: p∈{0,1}, c∈F
• Decryption function D(S, c) = p: p∈{0,1}, c∈F where S is the secret key for K.
• Homomorphisms E(K, a) +F E(K, b) = E(K, a ⊕ b) and E(K, a) *F E(K, b) = E(K, a * b)
• a ⊕ b equivalent to XOR over {0,1} and a * b equivalent to AND over {0,1}

Boolean logic circuits of arbitrary depth can be built from the XOR and AND equivalents allowing computation of arbitrary binary functions. Let M∈{0,1}^N be a sequence of bits representing the state of a bounded UTM with an arbitrary program on its tape. Let binary function U(M): {0,1}^N -> {0,1}^N compute the next state of M. Let E(K, B) and D(S, E) also operate element-wise over sequences of bits and elements of F, respectively. Let UF be the set of logic circuits equivalent to U (UFi calculates the ith bit of U's result) but with XOR and AND replaced by +F and *F. Now D(S, UF^t(E(K, M)) = U^t(M) shows that an arbitrary number of UTM steps can be calculated homomorphically by evaluating equivalent logic circuits over the homomorphically encrypted bits of the state.