In this paper, we propose ByzSecAgg, an efficient secure aggregation scheme for federated learning that is resistant to Byzantine attacks and privacy leakages. Processing individual updates to manage adversarial behavior, while preserving the privacy of the data against colluding nodes, requires some sort of secure secret sharing. However, the communication load for secret sharing of long vectors of updates can be very high. In federated settings, where users are often edge devices with potential bandwidth constraints, excessive communication overhead is undesirable. ByzSecAgg solves this problem by partitioning local updates into smaller sub-vectors and sharing them using ramp secret sharing. However, this sharing method does not admit bilinear computations, such as pairwise distances calculations, which are needed for distance-based outlier-detection algorithms, and effective methods for mitigating Byzantine attacks. To overcome this issue, each user runs another round of ramp sharing, with a different embedding of the data in the sharing polynomial. This technique, motivated by ideas from coded computing, enables secure computation of pairwise distance. In addition, to maintain the integrity and privacy of the local update, ByzSecAgg also uses a vector commitment method, in which the commitment size remains constant (i.e., does not increase with the length of the local update), while simultaneously allowing verification of the secret sharing process. In terms of communication load, ByzSecAgg significantly outperforms the related baseline scheme, known as BREA.