# Current Issue** [Vol. 08, No. 09]** [September 2022]

- Citation
- Abstract
- Reference
- PDF Download

Paper Title | :: | On the High Dimensional RSA Algorithm--- A Public Key Cryptosystem Based on Lattice and Algebraic Number Theory |

Author Name | :: | Zhiyong Zhenga || Fengxia Liub || Man Chen |

Country | :: | China |

Page Number | :: | 01-16 |

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

**Key Words:**RSA, The Ring of Algebraic Integers, Ideal Matrix, Ideal Lattice, HNF Basis[1]. M. Ajtai, C. Dwork. A Public-Key Cryptosystem with Worst-Case/Avarage -Case Equivalence. 29th ACM Symposium on Theory of Computing, 1997, 284-293.

[2]. D. Bonech. Twenty Years of Attacks on the RSA Cryptosystem. Notices of the Ams, 2002, 46(2): 203-213.

[3]. D. Coppersmith. Finding Small Solutions to Small Degree polynomials. Lecture Notes in Computer Science, 2001, 2146: 20-31.

[4]. H. Cohen. A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics. Springer-Verlag, 1993.

[5]. P. J. Davis. Circulant Matrices. 2nd Edition, Chelseea Publishing, New York, 1994.

[2]. D. Bonech. Twenty Years of Attacks on the RSA Cryptosystem. Notices of the Ams, 2002, 46(2): 203-213.

[3]. D. Coppersmith. Finding Small Solutions to Small Degree polynomials. Lecture Notes in Computer Science, 2001, 2146: 20-31.

[4]. H. Cohen. A Course in Computational Algebraic Number Theory, Graduate Texts in Mathematics. Springer-Verlag, 1993.

[5]. P. J. Davis. Circulant Matrices. 2nd Edition, Chelseea Publishing, New York, 1994.