单片段取证编码：基于多维循环可置换码
Single-Fragment Forensic Coding via Multidimensional Cyclically Permutable Codes
- Xidian University（西安电子科技大学）
- University of Science and Technology of China（中国科学技术大学）
- Capital Normal University（首都师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对3D打印物品的防伪追踪，提出基于多维循环可置换码的单片段取证编码，实现从任意含最小体积盒子的片段中恢复标识符，并给出最优冗余构造，在厚片段下速率趋近于一。
AI中文摘要：
3D打印技术的普及带来了新的安全和取证挑战，包括未经授权制造无法追踪的枪支和其他受管制物品的风险。为了实现可追溯性，我们考虑单片段取证编码，其中将唯一标识符嵌入打印对象中，并且必须能从任何包含指定最小体积和坐标方向厚度的轴平行盒子的片段中恢复该标识符，即使在存在替换错误的情况下也是如此。为了解决这个问题，我们引入并研究了多维循环可置换码（CPCs），对于这些码，每个循环平移唯一地确定原始码字和所应用的平移。通过对CPC基数组应用周期性提升，片段中包含的每个完整周期对应于码字的未知循环平移，从而将取证对齐问题简化为多维循环同步。我们在无噪声和替换错误设置下建立了理论界限并开发了显式构造。对于固定维度$d$和字母表大小$q$，我们获得了边长为$k$的无噪声$d$维CPC的最优冗余$d\log_q k+o(1)$。对于固定的$t$，$t$替换纠正$d$维CPC的最优冗余介于$(t+1)d\log_q k+O(1)$和$(2t+1)d\log_q k+O(1)$之间。对于二进制字母表和$d\geq2$，一个显式的鲁棒行锚构造实现了$(t+1)d\log_2 k+O(\log\log k)$冗余，匹配最优首项。在厚片段区域$h=cM^{1/d}$中，其中$M$和$h$分别下界于片段中包含的盒子的体积和边长，周期性提升产生了速率为$c^d-o(1)$的单片段取证码。特别是，当$c=1-o(1)$时，速率趋近于一。
英文摘要：
The proliferation of 3D printing raises new security and forensic challenges, including the risk of unauthorized fabrication of untraceable firearms and other regulated items. To enable traceability, we consider \emph{single-fragment forensic coding}, in which a unique identifier is embedded into a printed object and must be recoverable from any fragment containing an axis-parallel box of prescribed minimum volume and coordinatewise thickness, even in the presence of substitution errors. To address this problem, we introduce and study multidimensional cyclically permutable codes (CPCs), for which every cyclic translate uniquely determines both the original codeword and the applied translation. By applying periodic lifting to a CPC base array, every complete period contained in a fragment corresponds to an unknown cyclic translate of a codeword, thereby reducing the forensic alignment problem to multidimensional cyclic synchronization. We establish theoretical bounds and develop explicit constructions in both the noiseless and substitution-error settings. For fixed dimension $d$ and alphabet size $q$, we obtain the optimal redundancy $d\log_q k+o(1)$ for noiseless $d$-dimensional CPCs of side length $k$. For fixed $t$, the optimal redundancy of $t$-substitution-correcting $d$-dimensional CPCs lies between $(t+1)d\log_q k+O(1)$ and $(2t+1)d\log_q k+O(1)$. For binary alphabets and $d\geq2$, an explicit robust row-anchor construction achieves $(t+1)d\log_2 k+O(\log\log k)$ redundancy, matching the optimal leading term. In the thick-fragment regime $h=cM^{1/d}$, where $M$ and $h$ lower-bound the volume and the side lengths of a box contained in the fragment, periodic lifting yields single-fragment forensic codes of rate $c^d-o(1)$. In particular, the rate approaches one when $c=1-o(1)$.
