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When processing signals in the encrypted domain, homomorphic encryption can be used to enable linear operations on encrypted data. Comparison of encrypted data however requires an additional protocol between the parties and will be relatively expensive. A well-known and frequently used comparison protocol is by Damgard, Geisler and Kroigaard. We present two ways of improving this comparison protocol. Firstly, we reduce the computational effort of one party by roughly 50%. Secondly, we show how to achieve perfect security towards the other party without additional costs, whereas the original version with encrypted inputs only achieved statistical security. An additional advantage is that larger inputs are allowed.