Abstract

In this paper, we consider the finite groups $$\begin{aligned} H_{(t,l,m)}=\langle a,b,c | a^t=b^l=c^m=1, [a,b]=c, [a,c]=[b,c]=1\rangle . \end{aligned}$$ We obtain the generalized order-k Pell sequences and study their periods. We prove the period of the order-k Pell sequence divided the period of the generalized order-k Pell sequence in the Heisenberg group. Then, the generalized order-k Pell sequence in Heisenberg group are used to define new cyclic groups. As an application, these groups are used in encryption algorithms.

中文翻译：

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基于海森堡群中广义k阶佩尔序列的新循环群及其在密码学中的应用

摘要

在本文中，我们考虑有限群$$\begin{aligned} H_{(t,l,m)}=\langle a,b,c | a^t=b^l=c^m=1, [a,b]=c, [a,c]=[b,c]=1\rangle 。 \end{aligned}$$我们获得了广义的k阶Pell 序列并研究了它们的周期。我们证明了k阶Pell序列的周期除以海森堡群中广义k阶Pell序列的周期。然后，利用海森堡群中的广义k阶Pell序列来定义新的循环群。作为应用程序，这些组用于加密算法。