Theory of Computing Systems ( IF 0.5 ) Pub Date : 2023-10-19 , DOI: 10.1007/s00224-023-10147-7 David E. Brown , David Skidmore
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A method is given to reduce the problem of finding a nontrivial factorization of a positive integer \(\alpha \), greater than one, to the problem of finding a solution to a system of Boolean equations, that is, a system of equations such that each equation is of the form \(f=g\) where f and g are Boolean functions, meaning \(\{0,1\}\)-valued functions in zero or more Boolean (\(\{0,1\}\)-valued) variables. Our system is obtained by applying a sequence of reductions to an initial system of equations of the form
$$\begin{aligned} \{f_i(\vec {x},\vec {y}) = \alpha _i \mid i \in \{0, \dots , 2n\} \} \end{aligned}$$where for each i \(f_i(\vec {x},\vec {y})=f_i(x_0,\dots ,x_n,y_0,\dots ,y_n)\) gives the coefficient of \(2^i\) in the binary expansion of
$$\begin{aligned} (x_0+2x_1+\dots +2^{n}x_n)(y_0+2y_1+\dots +2^{n}y_n), \end{aligned}$$\(\alpha _i\) gives the coefficient of \(2^i\) in the binary expansion of \(\alpha \), and \(x_i\) and \(y_i\) are \(\{0,1\}\)-valued variables. That is, the initial system represents a binary multiplier whose output bits have been set equal to the bits of \(\alpha \). It is shown that each Boolean function in our reduced system, that is, each Boolean function
$$\begin{aligned} (g-h)~ {\text {mod}} 2 = {\left\{ \begin{array}{ll} 0 &{} \iff g=h \\ 1 &{} \iff g \ne h \end{array}\right. } \end{aligned}$$such that \(g=h\) is an equation in the reduced system, can be represented by a type of graph called an ordered binary decision diagram (OBDD) with an upper bound on its number of vertices of \(\mathcal {O} \left( \log _2(\alpha /\log _2(\alpha )))^3 \right) \). Previous work has shown that the initial system has at least one Boolean function with an OBDD representation with number of vertices exponential in \(\log _2(\alpha )\).
中文翻译：
使用有序二元决策图表示整数分解问题
给出了一种方法，可以将寻找大于 1 的正整数\(\alpha \)的非平凡因式分解的问题简化为寻找布尔方程组（即，这样的方程组）的解的问题每个方程的形式为\(f=g\)，其中f和g是布尔函数，表示零个或多个布尔值的\(\{0,1\}\)值函数 ( \(\{0,1 \}\) -值）变量。我们的系统是通过对以下形式的初始方程组应用一系列归约而获得的
$$\begin{对齐} \{f_i(\vec {x},\vec {y}) = \alpha _i \mid i \in \{0, \dots , 2n\} \} \end{对齐}$ $其中对于每个i \(f_i(\vec {x},\vec {y})=f_i(x_0,\dots ,x_n,y_0,\dots ,y_n)\) 给出 \(2^i\)的系数在二进制展开式中
$$\begin{对齐} (x_0+2x_1+\dots +2^{n}x_n)(y_0+2y_1+\dots +2^{n}y_n), \end{对齐}$$\(\alpha _i\)给出 \ (\alpha \)的二进制展开中\(2^i\)的系数，而\(x_i\)和\(y_i\)为\(\{0,1 \}\) -值变量。也就是说，初始系统表示一个二进制乘法器，其输出位已设置为等于\(\alpha \)的位。表明我们的简化系统中的每个布尔函数，即每个布尔函数
$$\begin{对齐} (gh)~ {\text {mod}} 2 = {\left\{ \begin{array}{ll} 0 &{} \iff g=h \\ 1 &{} \iff g \ne h \end{数组}\right。} \end{对齐}$$使得\(g=h\)是简化系统中的方程，可以用一种称为有序二元决策图 (OBDD) 的图来表示，其顶点数上限为 \(\mathcal { O } \left( \log _2(\alpha /\log _2(\alpha )))^3 \right) \)。先前的工作表明，初始系统至少有一个具有 OBDD 表示形式的布尔函数，其顶点数量以\(\log _2(\alpha )\)为指数。
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