Preventing access to encrypted files at all

Is there any way of preventing the adversaries from accessing the (encrypted) data at all, even after releasing the data?

For almost every single case, there's no way. The "after releasing the data" statement means your data (or a copy of it) left your hands and went to someone else.

But there's one way (not that I would recommend to this scenario): very expensive, not practical at all, and as cumbersome as carrying a safe around: an HSM (Hardware Security Module). An HSM is hardened to protect its contents at all costs, having multiple sensors to make sure they are working on a predefined set of environment values (temperature, acceleration, voltage, vibration, and so on). If any of those variables are outside of the expected range, it erases its contents.

But there's another issue: an HSM only keeps private keys. It's kind of a write-only device. After storing the private key on it, it never ever leaves. You send the data, it uses the private key internally, signs or decrypts the data, and returns the resulting decrypted or signed data.

If you are irrationally paranoid and very rich, you can convince a vendor to design a kind of HSM for you that stores multiple gigabytes of data, and receives commands signed by your key to store or serve its contents. It will be very expensive, not practical at all, but you can rent floor space on any datacenter and put it there.

Again, you don't need to go so overkill. Protect the key with your life, and release the encrypted data. Properly encrypted data with AES256 using secure methods is considered unbreakable with current technology. Having the encrypted version of anything sans the key is the same as having a file full of white noise. Or having a long, truly random password, but you don't know the username nor the server or service where that password belongs. It does not mean anything and not helps anyone gain access to anything.

The only way to be certain that your data is safe is to manage the storage, network, and transport of the data yourself.

Some questions to ask:

• Does this data need to go offsite? Can it be stored locally?
• Does the provider have any known breaches?
• Who uses the provider? IE is BIGCORP a customer?
• Could you store this data on an encrypted drive at a friend's house?

However with good hygiene with the encryption and the transport the risk is minimal, unless you are uploading state secrets...

Here is what you asked: "I want to hand the encrypted data to this party to store and process, but I don't want them to see the encrypted data."

That's not possible or realistic.

If you don't want them to have access to the data, then don't give them access. Don't give it to them.

You could try multiple levels of different encryption, but that's just encrypting in layers. They still see the encrypted data, just higher levels of it.

You could hide the encrypted data in other data so that they don't notice the data, but they could still notice it, if they looked for it.