# On the High Dimensional RSA Algorithm – A Public Key Cryptosystem Based on Lattice and Algebraic Number Theory

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.

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