Is it permissible for a couple who have officially divorced to remarry without witnesses, knowing that their close associates are unaware of their previous divorce?

If a man divorces his wife with the first or second divorce, he is permitted to take her back as long as she is within her waiting period (iddah (definition)). If the waiting period ends, she becomes irrevocably separated from him, and she is not lawful for him again except through a new marriage contract (aqd) that fulfills the conditions, such as the guardian's permission and the presence of witnesses. It is sufficient for the contract to be concluded with the guardian's permission and the presence of two witnesses; public announcement of it is not necessary.