高维RSA算法研究--一种基于格和代数数论的公开密钥密码体制
On the High Dimensional RSA Algorithm -- A Public Key Cryptosystem Based on Lattice and Algebraic Number Theory
摘要：
最著名的公开密钥密码系统是由Rivest，Shamir和Adleman于1978年提出的，现在以他们的名字命名为RSA公开密钥密码系统。后来，一些作者给出了RSA在代数数上的简单推广（见[20]- [22]），但他们要求代数整数的代数整数环是欧氏环，这个要求比类数一个条件强得多。本文利用代数数域的代数整数环和格理论，给出了RSA的一种高维形式。我们给出了一个可实现的算法（见下面的算法I），这是有意义的理论和实践的观点。本文的主要目的是证明高维RSA确实是一个基于公开密钥密码体制的格，它将被认为是后量子密码家族中的一个新数（见[17]和[18]）。另一方面，我们给出了任意代数数域的矩阵表示（见下面的定理2. 7），这即使在经典代数数论的意义上也是一个新的结果。
英文摘要：
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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