Security and Efficiency of Linear Feedback Shift Registers in \\({GF(2^n)}\\) Using \\({n}\\)-Bit Grouped Operations

Many stream ciphers employ linear feedback shift registers (LFSRs) to generate pseudorandom sequences. Many recent LFSRs are defined in \\({GF(2^n)}\\) to take advantage of the \\({n}\\)-bit processors, instead of using the classic binary field. In this way, the bit generation rate increases at the expense of a higher complexity in computations. For this reason, only certain primitive polynomials in \\({GF(2^n)}\\) are used as feedback polynomials in real ciphers. In this article, we present an efficient implementation of the LFSRs defined in \\({GF(2^n)}\\). The efficiency is achieved by using equivalent binary LFSRs in combination with binary \\({n}\\)-bit grouped operations, \\({n}\\) being the processor word's length. This improvement affects the general considerations about the security of cryptographic systems that uses LFSR. The model also allows the development of a faster method to test the primitiveness of polynomials in \\({GF(2^n)}\\).