Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.

翻译：安全计算常受益于使用关联随机性以实现快速、非加密的在线协议。Boyle等人（CCS 2018, Crypto 2019）近期提出的一种范式表明，伪随机关联生成器（PCG）可用于生成大量有用形式的关联（伪）随机性。该过程仅需最少的交互，随后完全由本地计算完成，从而产生静默安全的两方计算协议（即预处理阶段几乎无需通信的协议）。此外，一种称为可编程性的属性可将其扩展为构建多方协议。然而，现有可编程PCG的构建方式仅能生成大域上的OLE，并且依赖较新的可分裂环LPN假设。本研究克服了这两项局限性。为此，我们引入了准阿贝尔综合征解码问题（QA-SD），这是一类推广了已被充分研究的准循环综合征解码假设的假设族。基于QA-SD，我们构建了适用于任意域$\mathbb{F}_q$（其中$q>2$）上OLE的新型可编程PCG。我们的分析也揭示了Boyle等人（Crypto 2020）所使用的环LPN假设的安全性。利用这些新型PCG，我们首次得到了高效的多方静默安全计算协议，可用于计算任意$q>2$的域$\mathbb{F}_q$上的通用算术电路。