Nonsingularity of Trivium-like cascade FSRs over finite fields via semi-tensor product

Abstract

In stream cipher designing, nonsingularity is a crucial requirement to ensure that the feedback shift registers (FSRs) do not produce keys that are equivalent to one another. This study uses a semi-tensor product to examine the nonsingularity of Trivium-like cascade FSRs over a finite field. The Trivium-like cascade FSRs are expressed algebraically using the semi-tensor product, allowing them to be viewed as logical networks and introducing a novel state transition matrix. Several necessary and sufficient conditions for the nonsingularity of FSRs and Trivium-like cascade FSRs over a finite field are established by dividing the structural matrices of feedback functions into different parts. These findings are also applicable to binary FSRs and binary Trivium-like cascade FSRs.

Keywords:

• Logical networks
• nonsingularity
• semi-tensor product
• Trivium-like cascade FSRs

Additional information

Funding

This document is the results of the research project funded by the National Natural Science Foundation (NNSF) of China [grant number 62273201] and Taishan Scholar Project of Shandong Province.