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Some Problems and Algorithms Related to the Weight Order Relation on the $n$-dimensional Boolean Cube
arXiv - CS - Discrete Mathematics Pub Date : 2018-11-11 , DOI: arxiv-1811.04421
Valentin Bakoev

The problem "Given a Boolean function $f$ of $n$ variables by its truth table vector. Find (if exists) a vector $\alpha \in \{0,1\}^n$ of maximal (or minimal) weight, such that $f(\alpha)= 1$." is considered here. It is closely related to the problem of fast computing the algebraic degree of Boolean functions. It is an important cryptographic parameter used in the design of S-boxes in modern block ciphers, PRNGs in stream ciphers, at Reed-Muller codes, etc. To find effective solutions to this problem we explore the orders of the vectors of the $n$-dimensional Boolean cube $\{0,1\}^n$ in accordance with their weights. The notion of "$k$-th layer" of $\{0,1\}^n$ is involved in the definition and examination of the "weight order" relation. It is compared with the known relation "precedes". Several enumeration problems concerning these relations are solved and the corresponding comments were added to 3 sequences in the On-line Encyclopedia of Integer Sequences (OEIS). One special order (among the numerous weight orders) is defined and examined in detail. The lexicographic order is a second criterion for an ordinance of the vectors of equal weights. So a total order called Weight-Lexicographic Order (WLO) is obtained. Two algorithms for generating the WLO sequence and two algorithms for generating the characteristic vectors of the layers are proposed. Their results were used in creating 2 new sequences: A294648 and A305860 in the OEIS. Two algorithms for solving the problem considered are developed--the first one works in a byte-wise manner and uses the WLO sequence, and the second one works in a bitwise manner and uses the characteristic vector as masks. The experimental results after many tests confirm the efficiency of these algorithms. Some other applications of the obtained algorithms are also discussed--for example, when representing, generating and ranking other combinatorial objects.

中文翻译:

$n$维布尔立方体上权阶关系的一些问题和算法

问题“给定一个包含 $n$ 个变量的布尔函数 $f$,通过其真值表向量。找到(如果存在)一个向量 $\alpha\in\{0,1\}^n$ 的最大(或最小)权重,使得 $f(\alpha)= 1$。" 这里考虑。它与快速计算布尔函数代数次数的问题密切相关。它是现代分组密码中 S-box 设计、流密码中 PRNG、Reed-Muller 码等设计中使用的重要密码参数。为了找到解决这个问题的有效方法,我们探索了 $n 向量的阶数$维布尔立方体 $\{0,1\}^n$ 按照它们的权重。$\{0,1\}^n$的“$k$-th layer”的概念涉及到“权重顺序”关系的定义和检验。它与已知的关系“先于”进行比较。解决了有关这些关系的几个枚举问题,并在整数序列在线百科全书 (OEIS) 中的 3 个序列中添加了相应的注释。定义并详细检查了一个特殊订单(在众多重量订单中)。字典顺序是相等权重向量排列的第二个标准。这样就得到了一个称为权重字典序(WLO)的总序。提出了两种生成 WLO 序列的算法和两种生成层特征向量的算法。他们的结果用于创建 2 个新序列:OEIS 中的 A294648 和 A305860。开发了两种用于解决所考虑问题的算法——第一种算法以字节方式工作并使用 WLO 序列,第二个以按位方式工作,并使用特征向量作为掩码。经过多次测试的实验结果证实了这些算法的有效性。还讨论了所获得算法的一些其他应用——例如,在表示、生成和排列其他组合对象时。
更新日期:2020-02-17
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