Symetric and assymmetric encryption systems
"Functional encryption is designed to solve the hassle tied to traditional public-key encryption resulting from distributing and managing thousands or millions of private keys authorized people need to decrypt protected data. If 1,000 people in an organization need to securely share their public key with their co-workers, that requires close to one million separate exchanges."
You seem to be getting confused between symmetric and assymmetric encryptions. In a symmetric system you have a seperate secret key that you exchange with everybody. Total number of keys needed is n(n+1)/2 which does get very near to a million for 1000 employees.
In an assymmetric system you have one public key that is available to everybody. There is no need to secure your public key, as it's a public key: there is a clue in the name after all. In an assymmetric system you only need n keys.