This note is an attempt to unconditionally prove the existence of weak one way functions (OWF). Starting from a provably intractable decision problem $L_D$ (whose existence is nonconstructively assured from the well-known discrete time-hierarchy theorem from complexity theory), we construct another intractable decision problem $L\subseteq \{0,1\}^*$ that has its words scattered across $\{0,1\}^\ell$ at a relative frequency $p(\ell)$, for which upper and lower bounds can be worked out. The value $p(\ell)$ is computed from the density of the language within $\{0,1\}^\ell$ divided by the total word count $2^\ell$. It corresponds to the probability of retrieving a yes-instance of a decision problem upon a uniformly random draw from $\{0,1\}^\ell$. The trick to find a language with known bounds on $p(\ell)$ relies on switching from $L_D$ to $L_0:=L_D\cap L'$, where $L'$ is an easy-to-decide language with a known density across $\{0,1\}^*$. In defining $L'$ properly (and upon a suitable G\"odel numbering), the hardness of deciding $L_D\cap L'$ is inherited from $L_D$, while its density is controlled by that of $L'$. The lower and upper approximation of $p(\ell)$ then let us construct an explicit threshold function (as in random graph theory) that can be used to efficiently and intentionally sample yes- or no-instances of the decision problem (language) $L_0$ (however, without any auxiliary information that could ease the decision like a polynomial witness). In turn, this allows to construct a weak OWF that encodes a bit string $w\in\{0,1\}^*$ by efficiently (in polynomial time) emitting a sequence of randomly constructed intractable decision problems, whose answers correspond to the preimage $w$.

翻译：本文档尝试无条件证明弱单向函数（OWF）的存在性。从可证明难解的判定问题$L_D$（其存在性由复杂性理论中著名的离散时间谱定理非构造性地保证）出发，我们构造另一个难解的判定问题$L\subseteq \{0,1\}^*$，其词语以相对频率$p(\ell)$散布在$\{0,1\}^\ell$中，且该频率的上下界可被计算。值$p(\ell)$由语言在$\{0,1\}^\ell$中的密度除以总词数$2^\ell$得到，它对应于从$\{0,1\}^\ell$中均匀随机抽取时获得判定问题肯定实例的概率。寻找具有已知$p(\ell)$边界的语言的关键在于将$L_D$转换为$L_0:=L_D\cap L'$，其中$L'$是已知在$\{0,1\}^*$上具有密度的易解语言。通过适当定义$L'$（并采用合适的哥德尔编号），判定$L_D\cap L'$的困难性由$L_D$继承，而其密度则由$L'$的密度控制。$p(\ell)$的下界和上界近似使我们能构造显式阈值函数（如随机图理论中），该函数可高效且有目的地采样判定问题（语言）$L_0$的肯定或否定实例（但无需任何能简化判定的辅助信息，如多项式见证）。从而，这允许构造一个弱OWF，通过高效（多项式时间内）发射一系列随机构造的难解判定问题对比特串$w\in\{0,1\}^*$进行编码，这些问题的答案对应于原像$w$。