Do you have Qualtrics Directory enabled? That would allow you to load the the lists in the directory (aka list) and use samples to send the emails on a daily basis. Your survey flow could then have branching to determine if the test is their DOB or contact number. If they have a DOB listed, then the authenticator that points to the directory list and autofills the externaldatareference variable (their participant ID) and asks for DOB. If the DOB is empty it does similar but instead asks for their contact number.

bstrahin thank you for your response. The problems is I create a new list everyday and the way I understand the authentication works, it has to be set up for each list separately, is there a way of automating it so any future list would be checked the exact same way?

Simin that's why I am asking you if you have Directory. With Directory, you can continue to add contacts to a list. You then take samples from the list to distribute. So the "lists" you generate every day, you would just keep adding them to the same list in the directory not creating new lists in the directory. Samples of the list replace the new lists you are currently making.
You can then use the flow I gave you. And because it points to a list and not a sample, it always stays updated even with adding new contacts every day.