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I am interested what is the best way to hide contract function calls and contract state details from everyone except certain parties (if that is possible at all).

Encryption sounds like a possible solution so at least the state of the contract can be kept in a secure way, but from what I understand it would be very inefficient and it also doesn't solve the problem of the contract method calls being visible. Also how would you provide in a secure way the private key to a contract in order to decrypt the storage data?

Are there any other solutions?

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There's no way to hide exactly how a contract executed from the world at large. It's possible to debug, step-by-step, any transaction, as the EVM is deterministic. It is also possible to read every byte of a contract's storage. In addition, a contract may never have access to a private key, or the private key can be read.

That said, there's a number of options, but they do tend to require a specific architecture. Here is an overview by Vitalik Buterin of options, some of which are currently theoretical.

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