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 attribute $X$, 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 $W$. The design goal is to encode message $\cal C$ with minimum possible average length that satisfies a perfect privacy constraint. To do so we first consider two different privacy mechanism design problems and find upper bounds on the entropy of the optimizers by solving a linear program. We use the obtained optimizers to design $\cal C$. In two cases we strengthen the existing bounds: 1. $|\mathcal{X}|\geq |\mathcal{Y}|$; 2. The realization of $(X,Y)$ follows a specific joint distribution. In particular, considering the second case we use two-part construction coding to achieve the upper bounds. Furthermore, in a numerical example we study the obtained bounds and show that they can improve the existing results.

翻译：研究了一个私有压缩设计问题：编码器观察到有用数据$Y$，期望使用变长码对其进行压缩，并通过不安全信道传输。由于$Y$与私有属性$X$相关，编码器采用私有压缩机制设计编码消息$\cal C$并通过信道发送。假设攻击者能够获取编码器输出即$\cal C$，并试图估计$X$。此外，假设编码器和解码器共享同一密钥$W$。设计目标是在满足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。为此，我们首先考虑两种不同的隐私机制设计问题，通过求解线性规划得到优化器的熵上界，并利用所得优化器设计$\cal C$。在两种情形下我们强化了现有界：1. $|\mathcal{X}|\geq |\mathcal{Y}|$；2. $(X,Y)$的联合分布服从特定形式。特别地，针对第二种情形采用两部分构造编码以实现上界。最后通过数值算例研究所得界，表明其能改进现有结果。