Relationship between special RSA modulus and quadratic residue in CL (Camenisch-Lysyanskaya) signature
The standard RSA ($c=m^{e}$ $mod(n)$) is not semantically secure, since it reveals one bit information about the message. This information is jacobi symbol of the message. You can find many resources about jacobi symbol but basically it states that a value is in $QR_n$ or not. In CL signature, we can handle with this situation easilly, since we determine the base values for exponentiation operations. If we choose the base values from $QR_n$, any exponentiation operation using them outputs another value from $QR_n$. Thus, there is no revealed information. Ofcourse, it comes with a disadvantage that any output of exponentiation is in $QR_n$ we decrase the number of possible outputs that we can get.