On the Optimal Source Key Size of Secure Gradient Coding

With gradient coding, a user node can efficiently aggregate gradients from server nodes processing local datasets, achieving low communication costs and maintaining resilience against straggling servers. This paper considers a secure gradient coding problem, where a user aims to compute the sum of the gradients from $K$ datasets with the assistance of $N$ distributed servers. The user should recover the sum of gradients by receiving transmissions from any $N_r$ servers, and each dataset is assigned to $N - N_r + m$ servers. The security constraint guarantees that even if the user receives transmissions from all servers, it cannot obtain any additional information about the datasets beyond the sum of gradients. It has been shown in the literature that this security constraint does not increase the optimal communication cost of the gradient coding problem, provided enough source keys are shared among the servers. However, the minimum required source key size that ensures security while maintaining this optimal communication cost has only been studied for the special case $m = 1$. In this paper, we focus on the more general case $m \geq 1$ and aim to determine the minimum required source key size for this purpose. We propose a new information-theoretic converse bound on the source key size, as well as a new achievable scheme with carefully designed data assignments. Our scheme outperforms the existing optimal scheme based on the widely used cyclic data assignment and coincides with the converse bound under certain system parameters.