The following answers are based upon general U.S. law, I have not personally confirmed that they apply in Illinois in particular under its adoption law, or each of its welfare benefits programs, or its wrongful death statute.
There is no affirmative duty of a mother to inform a father or potential father of the existence of his child, at least outside (1) the context of relinquishment of parental rights in anticipation of an adoption (where there may be a statutory notice requirement in some cases), (2) applications for certain kinds of welfare benefits (where identification of a father is required for some programs), and (3) possibly in the context of a lawsuit for the wrongful death of a child in which both parents are frequently vested with authority to bring the suit.
Identification of a father on a birth certificate is not mandatory (and not infrequently omitted).
In an adoption case, due process often requires some effort "reasonably calculated to give notice" to the father of the proceeding relinquishing parental rights, although failure to do so is generally not a jurisdictional defect that renders the adoption void ab initio. There is also generally a requirement to affirmatively state in adoption proceeding if the father is, or might be, a member of a Native American tribe, as that goes to the jurisdictional issue of the applicability of the federal Indian Child Welfare Act, which requires notice, at a minimum, to the father's Indian tribe, even if notice cannot be given to the father.
There is a presumption that a child born to a woman married to a husband is the child of the husband and after a certain number of years, for certain purposes (such as child support). This presumption is usually conclusive and cannot be rebutted even by DNA evidence to the contrary, on the theory that the husband is the psychological parent of the child and has assumed that responsibility, and in order to discourage after the fact denials of paternity simply to avoid child support when paternity was never disputed.
There is also a presumption that a birth certificate's statement of paternity is correct, although this is in often not a conclusive presumption. And, birth certificates are a matter of public record, so a man can search the vital statistics records of a state to see if anyone has identified him as a father even if he has not been told of that fact, arguably giving him constructive knowledge of his paternity in that case.
But, the identity of the person who is the child's father, or of persons who could be the child's father, is not privileged (other than under the 5th Amendment right against self-incrimination where applicable), and so, a woman can generally be legally compelled to disclose this information by subpoena or in response to discovery in litigation.
For example, a woman under the age of consent at the time her child was conceived can be incarcerated for contempt of court for failing to disclose the name of the father in connection with a criminal prosecution of a suspected father of the child for statutory rape.