| Summary: | Many stream ciphers employ linear feedback shift registers (LFSRs) to generate pseudorandom sequences. Many recent LFSRs are defined in <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> to take advantage of the <i>n</i>-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 <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula> are used as feedback polynomials in real ciphers. In this article, we present an efficient implementation of the LFSRs defined in <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula>. The efficiency is achieved by using equivalent binary LFSRs in combination with binary <i>n</i>-bit grouped operations, <i>n</i> 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 <inline-formula><math display="inline"><semantics><mrow><mi>G</mi><mi>F</mi><mo>(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>)</mo></mrow></semantics></math></inline-formula>.
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