论文标题
在高维RSA算法上 - 基于晶格和代数理论的公共密钥加密系统
On the High Dimensional RSA Algorithm -- A Public Key Cryptosystem Based on Lattice and Algebraic Number Theory
论文作者
论文摘要
最著名的公共密钥密码系统是由Rivest，Shamir和Adleman [19]于1978年推出的，现在以其荣誉称为RSA公共密钥密码系统。后来，一些作者简单地将RSA扩展到代数数字字段（见[20] - [22]），但他们要求代数整数的环是欧几里得环，此要求比一类第一条件强得多。在本文中，我们通过利用代数数字字段和晶格理论的代数整数环和晶格理论引入了高维形式的RSA。从理论和实际的角度来看，我们提供了可实现的算法（请参见下面的算法I）。本文我们的主要目的是表明，高维RSA确实是基于公共密钥密码系统的晶格，在后量词后加密家族中，它将被视为新数字（请参阅[17]和[18]）。另一方面，我们为任何代数数字字段提供了矩阵表达式（请参见下面的定理2.7），即使在经典代数数理论的意义上，这也是一个新的结果。
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.
