You are right, in this case, part of the rent may be illegal.
The reason is that Housing Authority(HA) decides what type of housing this lady can have. For example, HA may decide that she needs a house of three bedrooms and set her allowance to be $1500/month (e.g. market rent for three bedroom house is $1500). In this $1500, HA will calculate how much the lady need to pay out of pocket based on her salary. For example, HA may say she need to pay $300, and HA will take care of the rest ($1200). By law, she is NOT allow to rent a house that is higher then $1500.
Now if the lady wants to rent a house of four bedrooms with rent $1800, she can only tell HA that the rent is $1500, and the difference of $300 will have to be under the table with the landlord.
