A binary de Bruijn sequence is a sequence of period $2^n$2n in which every binary $n$n-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 $2^{n+k}$2n+k by an NLFSR from a de Bruijn sequence of period $2^n$2n through a composition operation repeatedly applying $k$k 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 $2^k$2k, 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 $2^{160}$2160 to $2^{1056}$21056. For instance, a class of de Bruijn sequences of period $2^{160}$2160 (resp. $2^{288}$2288) 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.

中文翻译：

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一类复合De Bruijn序列的分析与有效实现

二进制 de Bruijn 序列是一个周期序列 $2^n$2n 其中每个二进制 $n$n-tuple 在每个时期只出现一次。de Bruijn 序列具有良好的随机性，例如周期长、元组分布理想、线性复杂度高，可以由非线性反馈移位寄存器（NLFSR）生成。寻找可以生成长周期 de Bruijn 序列的高效 NLFSR 是一项重大挑战。“复合构造”是一种构造 de Bruijn 周期序列的技术$2^{n+k}$2n+克 通过来自 de Bruijn 周期序列的 NLFSR $2^n$2n 通过组合操作反复应用 $千$克次。本文的目标是进一步研究 de Bruijn 序列与高效硬件实现的复合构造，并确定随机性属性，例如线性复杂度。我们在本文中的贡献如下。首先，我们提出了复合 de Bruijn 序列的广义构造，它是通过在复合构造的反馈函数中添加不同长度的共轭对组合来构造的，从而生成一类大小为 de Bruijn 序列$2^k$2克，而原始的复合结构只能生成两个序列。其次，我们研究了新类 de Bruijn 序列的线性复杂度和相关性。我们从理论上证明了这类 de Bruijn 序列的线性复杂度是最优的或接近最优的。有趣的是，我们还证明了此类所有序列的线性复杂度相等，这加强了 Etzion 关于具有相等线性复杂度的 de Bruijn 序列数量的猜想 (JCTA 1985, IEEE-IT 1999)。这是第一个已知的任意长周期的 de Bruijn 序列构造，其线性复杂度在理论上是确定的。最后，我们在硬件中实现我们的构造以证明其实用性。$2^{160}$2160 至 $2^{1056}$21056. 例如，一类 de Bruijn 周期序列$2^{160}$2160 （分别 $2^{288}$2288) 可以用 65 nm ASIC 中的 3.43 (分别为 6.71) kGE 和 Spartan6 FPGA 中的 83 (分别为 229) 个切片来实现。