This work solves an open problem regarding the rate of time-bounded Kolmogorov complexity and polynomial-time dimension, conditioned on a hardness assumption. Hitchcock and Vinodchandran (CCC 2004) show that the polynomial-time dimension of infinite sequences (denoted ${\mathrm{cdim}}_\mathrm{P}$) defined using betting algorithms called gales, is lower bounded by the asymptotic lower rate of polynomial-time Kolmogorov complexity (denoted $\mathcal{K}_\text{poly}$). Hitchcock and Vindochandran and Stull asked whether the converse relationship also holds. This question has thus far resisted resolution. The corresponding unbounded notions, namely, the constructive dimension and the asymptotic lower rate of unbounded Kolmogorov complexity are equal for every sequence. Analogous notions are equal even at finite-state level. In view of these results, it is reasonable to conjecture that the polynomial-time quantities are identical for every sequence and set of sequences. However, under a plausible assumption which underlies modern cryptography - namely the existence of one-way functions, we refute the conjecture thereby giving a negative answer to the open question posed by Hitchcock and Vinodchandran and Stull . We show the following, conditioned on the existence of one-way functions: There are sets $\mathcal{F}$ of infinite sequences whose polytime dimension strictly exceeds $\mathcal{K}_\text{poly}(\mathcal{F})$, that is ${\mathrm{cdim}}_\mathrm{P}(\mathcal{F}) > \mathcal{K}_\text{poly}(\mathcal{F})$. We establish a stronger version of this result, that there are individual sequences $X$ whose poly-time dimension strictly exceeds $\mathcal{K}_\text{poly}(X)$, that is ${\mathrm{cdim}}_\mathrm{P}(X) > \mathcal{K}_\text{poly}(X)$. Further, we show that the gap between these quantities can be made arbitrarily close to 1. We also establish similar bounds for strong poly-time dimension

翻译：本研究在硬度假设条件下，解决了关于时间有界柯尔莫哥洛夫复杂度与多项式时间维数比率的一个开放性问题。Hitchcock与Vinodchandran（CCC 2004）证明，使用称为赌金的投注算法定义的无穷序列多项式时间维数（记为${\mathrm{cdim}}_\mathrm{P}$）存在下界，该下界由多项式时间柯尔莫哥洛夫复杂度的渐进下确界比率（记为$\mathcal{K}_\text{poly}$）给出。Hitchcock、Vinodchandran及Stull曾提出逆关系是否成立的问题，该问题至今悬而未决。对于无界情形，构造性维数与无界柯尔莫哥洛夫复杂度的渐进下确界比率对所有序列均相等。类似概念在有限状态层面亦保持相等。鉴于这些结果，有理由推测多项式时间量对所有序列及序列集合均具有同一性。然而，在现代密码学基础——即单向函数存在的合理假设下，我们证伪了该猜想，从而对Hitchcock、Vinodchandran及Stull提出的开放性问题给出了否定答案。我们在单向函数存在性条件下证明：存在无穷序列集合$\mathcal{F}$，其多项式时间维数严格超越$\mathcal{K}_\text{poly}(\mathcal{F})$，即${\mathrm{cdim}}_\mathrm{P}(\mathcal{F}) > \mathcal{K}_\text{poly}(\mathcal{F})$。我们建立了该结果的强化版本：存在个体序列$X$，其多项式时间维数严格超越$\mathcal{K}_\text{poly}(X)$，即${\mathrm{cdim}}_\mathrm{P}(X) > \mathcal{K}_\text{poly}(X)$。进一步证明这两个量之间的差距可无限趋近于1。同时，我们对强多项式时间维数也建立了类似界限。