固定字母多比特伪随机码的依赖公钥的对抗删除上限

中文摘要 AI 辅助

伪随机码（PRC）是一种带密钥的纠错码，其码字在计算上与均匀字符串不可区分。我们研究固定字母表上的公钥PRC对抗对抗删除的问题，其中删除信道可同时依赖于公钥编码密钥和传输的码字。令γ_q^LCS表示两个独立均匀q元字符串的渐近归一化最长公共子序列长度。我们证明，对于每个固定的q≥2，任何具有单输出解码器的多消息公钥PRC，对于任意δ>1-γ_q^LCS，都无法对所有此类δ删除信道具有鲁棒性。对于q=2，当前严格边界γ_2≥0.792665992排除了所有常数δ>0.207334008。该证明仅使用伪随机性将LCS事件从均匀字符串转移到独立采样的码字，因此所得的碰撞论证是信息论的，不需要密钥。我们还将该论证扩展到列表解码：对于每个固定常数L，只要消息空间包含至少L+1个消息，任何此类PRC的输出列表大小最多为L时，对于δ>1-γ_2^(L+1)都不具有鲁棒性。由于γ_2^(m)=1/2+Θ(1/√m)，这些阈值趋近于1/2。我们的边界特定于依赖公钥的对抗信道，不适用于无意识编辑信道。

英文摘要

A pseudorandom code (PRC) is a keyed error-correcting code whose codewords are computationally indistinguishable from uniform strings. We study public-key PRCs over fixed alphabets against adversarial deletions, where the deletion channel may both depend on the public encoding key and the transmitted codeword. Let $γ_q^{\mathrm{LCS}}$ denote the asymptotic normalised longest-common-subsequence length of two independent uniform $q$-ary strings. We prove that for every fixed $q\ge2$ no multi-message public-key PRC with a single-output decoder is robust against all such $δ$-deletion channels for any $δ>1-γ_q^{\mathrm{LCS}}$. For $q=2$, the current rigorous bound $γ_2\ge0.792665992$ rules out every constant $δ>0.207334008$. The proof uses pseudorandomness only to transfer an LCS event from uniform strings to independently sampled codewords and therefore also the resulting collision argument is information-theoretic and requires no secret key. We also extend the argument to list decoding: for every fixed constant $L$, provided the message space contains at least $L+1$ messages, no such PRC with output lists of size at most $L$ is robust for $δ>1-γ_2^{(L+1)}$. Since $γ_2^{(m)}=1/2+Θ(1/\sqrt m)$, these thresholds approach $1/2$. Our bounds are specific to public-key-dependent adversarial channels and do not apply to oblivious edit channels.

发表机构

• UC Berkeley（加州大学伯克利分校）

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