Hello Hicham Yousfi,
Thank you for contacting Apple Support Communities.
You can start with this article to troubleshoot your Game Center sign in issues:
Firstly, I wanna say that I'm Chinese. I'm not very good at English.
I have found the solution finally.
I saw someone solve the problem by changing the language so I realized that the cause may be related with the language.
And then I found the folders in /var/mobile/Library/GameKit/Data are catagrated by language.
They are only the languages I have tried to set so I guess they may be downloaded automatically to cache data from the Game Center server at the first time when we logged in the Game Center. I thought the cause leading to the problem was the invalid cache data. The cocoa threw the exception information because it cannot parse the cache data required.
Just RENAME /var/mobile/Library/GameKit/Data to /var/mobile/Library/GameKit/Data1 or DELETE it directly.
And then you will log in the Game Center successfully!
If someone can get what I meant, please help me to rewrite the solution in native English in order to help more people. Thank you!