In the DP$-3$T protocol for proximity tracing against Covid$19,$ each client device Y compares her stored EphIDs with the EphIDs in each new infection report published by the server.

$($a$)$ If the number of infection reports published per day is constant, and assuming the number of EphIDs stored by client devices grows linearly with the number d of elapsed days, how does the number of comparisons carried out by Y grow with d$?$

$($b$)$ Regarding privacy, what do the server and the user holding Y learn on the places visited by an infected user X or on the people X approached?