Even tho Unity has implemented a game-center api (Social api) it doesn't have a way to generate a verification signature, which you need to use game-center as a authentication method.
Luckily the objective-c code required is pretty straight forward. (Apple documentation: generateIdentityVerificationSignature)
#import <Foundation/Foundation.h>
#import <GameKit/GameKit.h>
typedef void (*IdentityVerificationSignatureCallback)(const char * publicKeyUrl, const char * signature, int signatureLength, const char * salt, int saltLength, const uint64_t timestamp, const char * error);