Asymmetric sign in public enviroment
"if everybody know message after it is decrypted there is" still a point in padding,

since it would (hopefully) stop people from learning the message before it is decrypted.

It would probably be better to

sign the message with your private key, rather than encrypt the message with your private key.

RSA needs padding for signatures too, although that padding is different from RSA encryption padding.

Also, although you can probably still get away with 1024-bit RSA since it's just

for a game, but you should make it easy for you to move to longer public keys.