  The most known of public key cryptosystem was introduced in 1978 by Rivest,
Shamir and Adleman [19] and now called the RSA public key cryptosystem in their
honor. Later, a few authors gave a simply extension of RSA over algebraic
numbers field( see [20]- [22]), but they require that the ring of algebraic
integers is Euclidean ring, this requirement is much more stronger than the
class number one condition. In this paper, we introduce a high dimensional form
of RSA by making use of the ring of algebraic integers of an algebraic number
field and the lattice theory. We give an attainable algorithm (see Algorithm I
below) of which is significant both from the theoretical and practical point of
view. Our main purpose in this paper is to show that the high dimensional RSA
is a lattice based on public key cryptosystem indeed, of which would be
considered as a new number in the family of post-quantum cryptography(see [17]
and [18]). On the other hand, we give a matrix expression for any algebraic
number fields (see Theorem 2.7 below), which is a new result even in the sense
of classical algebraic number theory.
