基于基域表示的SM9高次扩域运算优化OA

The domestic cryptographic algorithm SM9,which is an independently innovated identity-based crypto-graphic system in China,was limited by performance bottlenecks in its underlying high-degree extension field opera-tions.In the traditional tower extension method,operations such as multiplication,squaring,inversion,and Frobenius mapping on high-degree extension fields involved redundant computational overhead due to multi-level hierarchical processing,which constrained the overall efficiency improvement of the algorithm.To address this issue,a base field representation method was proposed for optimizing high-degree extension field operations.Through theoretical deri-vation,base field expressions for various operations on high-degree extension fields were established,transforming these operations directly into large integer arithmetic on the base field and thereby avoiding the substantial intermedi-ate layer overhead inherent in the tower extension process.Theoretical analysis demonstrated that,through distribu-tive law optimization and combination of like terms,modular operations were reduced by over two-thirds,with other operations simplified accordingly.Experimental results show that the proposed base field representation achieves a speedup of 2～3 times for various high-degree extension field operations compared to the tower extension method,with peak memory consumption reduced by 13.30%～63.60%.Furthermore,performance improvements of 95.39%～158.36%were observed in seven SM9 algorithms including digital signature,key exchange,and encryption.The base field representation method significantly enhances the execution efficiency and resource utilization of SM9 and its derived cryptographic schemes while preserving mathematical equivalence and security.Moreover,this method exhibits generality and can be extended to other high-degree extension field scenarios beyond SM9.