Analysis and Efficient Implementations of a Class of Composited de Bruijn Sequences

A binary de Bruijn sequence is a sequence of period <inline-formula><tex-math notation="LaTeX">$2^n$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq1-2979460.gif"/></alternatives></inline-formula> in which every binary <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives><mml:math><mml:mi>n</mml:mi></mml:math><inline-graphic xlink:href="mandal-ieq2-2979460.gif"/></alternatives></inline-formula>-tuple occurs exactly once in each period. A de Bruijn sequence has good randomness properties, such as long period, ideal tuple distribution, and high linear complexity, and can be generated by a nonlinear feedback shift register (NLFSR). Finding an efficient NLFSR that can generate a de Bruijn sequence with a long period is a significant challenge. “Composited construction” is a technique for constructing a de Bruijn sequence of period <inline-formula><tex-math notation="LaTeX">$2^{n+k}$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq3-2979460.gif"/></alternatives></inline-formula> by an NLFSR from a de Bruijn sequence of period <inline-formula><tex-math notation="LaTeX">$2^n$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mi>n</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq4-2979460.gif"/></alternatives></inline-formula> through a composition operation repeatedly applying <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="mandal-ieq5-2979460.gif"/></alternatives></inline-formula> times. The goal of this article is to further investigate the composited construction of de Bruijn sequences with efficient hardware implementations, and determine randomness properties such as linear complexity. Our contributions in this article are as follows. First, we present a generalized construction of composited de Bruijn sequences that is constructed by adding a combination of conjugate pairs of different lengths in the feedback function of the composited construction, which results in generating a class of de Bruijn sequences of size <inline-formula><tex-math notation="LaTeX">$2^k$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq6-2979460.gif"/></alternatives></inline-formula>, whereas the original composited construction can generate only two sequences. Second, we investigate the linear complexity and the correlation property of the new class of de Bruijn sequences. We prove theoretically that the linear complexity of this class of de Bruijn sequences is optimal or close to optimal. Interestingly, we also prove that the linear complexities of all the sequences of this class are equal, which strengthens Etzion's conjecture (JCTA 1985, IEEE-IT 1999) about the number of de Bruijn sequences with equal linear complexity. This is the first known construction of de Bruijn sequences of an arbitrarily long period whose linear complexities are determined theoretically. Finally, we implement our construction in hardware to demonstrate its practicality. We synthesize our implementations for a 65 nm ASIC and a Xilinx Spartan FPGA and present hardware areas, and performances of de Bruijn sequences of periods in the range of <inline-formula><tex-math notation="LaTeX">$2^{160}$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>160</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq7-2979460.gif"/></alternatives></inline-formula> to <inline-formula><tex-math notation="LaTeX">$2^{1056}$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>1056</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq8-2979460.gif"/></alternatives></inline-formula>. For instance, a class of de Bruijn sequences of period <inline-formula><tex-math notation="LaTeX">$2^{160}$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>160</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq9-2979460.gif"/></alternatives></inline-formula> (resp. <inline-formula><tex-math notation="LaTeX">$2^{288}$</tex-math><alternatives><mml:math><mml:msup><mml:mn>2</mml:mn><mml:mn>288</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="mandal-ieq10-2979460.gif"/></alternatives></inline-formula>) can be implemented with an area of 3.43 (resp. 6.71) kGEs in 65 nm ASIC, and 83 (resp. 229) slices in Spartan6 FPGA.

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