Security and Efficiency of Linear Feedback Shift Registers in <i>GF</i>(2^<i>n</i>) Using <i>n</i>-Bit Grouped Operations

Abstract

Many stream ciphers employ linear feedback shift registers (LFSRs) to generate pseudorandom sequences. Many recent LFSRs are defined in GF(2n) 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(2n) are used as feedback polynomials in real ciphers. In this article, we present an efficient implementation of the LFSRs defined in GF(2n). 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(2n).

Keywords

• LFSR
• stream cipher
• <i>m</i>-sequence
• primitive polynomial
• extended Galois field
• symmetric encryption