A random local function defined by a $d$-ary predicate $P$ is one where each output bit is computed by applying $P$ to $d$ randomly chosen bits of its input. These represent natural distributions of instances for constraint satisfaction problems. They were put forward by Goldreich as candidates for low-complexity one-way functions, and have subsequently been widely studied also as potential pseudo-random generators. We present a new search-to-decision reduction for random local functions defined by any predicate of constant arity. Given any efficient algorithm that can distinguish, with advantage $ε$, the output of a random local function with $m$ outputs and $n$ inputs from random, our reduction produces an efficient algorithm that can invert such functions with $\tilde{O}(m(n/ε)^2)$ outputs, succeeding with probability $Ω(ε)$. This implies that if a family of local functions is one-way, then a related family with shorter output length is family of pseudo-random generators. Prior to our work, all such reductions that were known required the predicate to have additional sensitivity properties, whereas our reduction works for any predicate. Our results also generalise to some super-constant values of the arity $d$, and to noisy predicates.

翻译：由 $d$ 元谓词 $P$ 定义的随机局部函数，其每个输出位通过对输入中随机选择的 $d$ 个位应用 $P$ 计算得出。这类函数代表了约束满足问题实例的自然分布。Goldreich 提出将其作为低复杂度单向函数的候选方案，随后它们也被广泛研究作为潜在的伪随机生成器。针对任何常数元数谓词定义的随机局部函数，我们提出了一种新的搜索到决策归约。给定任何能够以优势 $ε$ 区分具有 $m$ 个输出和 $n$ 个输入的随机局部函数输出与随机输出的高效算法，我们的归约可产生一个高效算法，该算法能以 $\tilde{O}(m(n/ε)^2)$ 的输出量反转此类函数，并以 $Ω(ε)$ 的概率成功。这意味着，如果一个局部函数族是单向的，那么具有更短输出长度的相关函数族就是伪随机生成器族。在我们的工作之前，所有已知的此类归约都要求谓词具备额外的敏感性性质，而我们的归约适用于任何谓词。我们的结果还可推广到某些超常数元数值 $d$ 以及带噪声的谓词。