SPDZ- How to do secure equality comparison
One way of doing this is to subtract the non-secret value from the secret value (in the secure computation) and to test that the result is 0. There are a number of ways of testing 0; generically, you can generate a shared unknown random value, multiply it with the value you want to test and then open. If the opened value is 0 then you know that you started with 0. Otherwise, it's a random value and reveals nothing (since the shared random value is unknown to all parties and so masks the original value).