We are concerned with an underlying primitive element common to the
key-privacy encryption and the ring signature schemes, that is, a family of
trap-door RSA permutations with a common domain. For a standard family of
trap-door RSA permutations, even if all of the functions in a family use RSA
moduli of the same size (the same number of bits), it will have domains with
different sizes. In this paper, we construct a new family of trap-door RSA
permutations with a common domain, and propose the applications of our
construction to the key-privacy encryption and ring signature schemes, which
have some advantage to the previous schemes.