You'd need at least one trustworthy party because you can not trust the mutually untrustworthy parties to not give similarly faked logs. If the goal is secrecy

You'd need at least one trustworthy party because you can not trust the mutually untrustworthy parties to not give similarly faked logs. If the goal is secrecy, increasing the number of parties that have access to the inputs undermines that, especially if they are untrustworthy, even if they have "contracts". While the solution you've outlined may be mathematically sound, I don't think it is very pragmatic.