A private compression design problem is studied, where an encoder observes useful data $Y$, wishes to compress it using variable length code and communicates it through an unsecured channel. Since $Y$ is correlated with private data $X$, the encoder uses a private compression mechanism to design encoded message $\cal C$ and sends it over the channel. An adversary is assumed to have access to the output of the encoder, i.e., $\cal C$, and tries to estimate $X$. Furthermore, it is assumed that both encoder and decoder have access to a shared secret key $W$. In this work, we generalize the perfect privacy (secrecy) assumption and consider a non-zero leakage between the private data $X$ and encoded message $\cal C$. The design goal is to encode message $\cal C$ with minimum possible average length that satisfies non-perfect privacy constraints. We find upper and lower bounds on the average length of the encoded message using different privacy metrics and study them in special cases. For the achievability we use two-part construction coding and extended versions of Functional Representation Lemma. Lastly, in an example we show that the bounds can be asymptotically tight.

翻译：本文研究了一种私有压缩设计问题，其中编码器观测到有用数据$Y$，希望使用变长码对其进行压缩，并通过不安全的信道传输。由于$Y$与私有数据$X$相关，编码器采用私有压缩机制设计编码消息$\cal C$并将其通过信道发送。假设攻击者能够获取编码器的输出（即$\cal C$），并试图估计$X$。此外，假设编码器和解码器共享一个密钥$W$。本研究将完美隐私（保密）假设推广至非完美情形，考虑私有数据$X$与编码消息$\cal C$之间存在非零泄露。设计目标是在满足非完美隐私约束的前提下，以最小可能平均长度对消息$\cal C$进行编码。我们采用不同隐私度量给出了编码消息平均长度的上界和下界，并在特殊情形下对其进行了研究。在可达性证明中，我们采用了两部分构造编码和函数表示引理的扩展版本。最后，通过一个算例展示了这些界可以是渐近紧致的。