  The Shapley value (SV) is a fair and principled metric for contribution
evaluation in cross-silo federated learning (cross-silo FL), wherein
organizations, i.e., clients, collaboratively train prediction models with the
coordination of a parameter server. However, existing SV calculation methods
for FL assume that the server can access the raw FL models and public test
data. This may not be a valid assumption in practice considering the emerging
privacy attacks on FL models and the fact that test data might be clients'
private assets. Hence, we investigate the problem of secure SV calculation for
cross-silo FL. We first propose HESV, a one-server solution based solely on
homomorphic encryption (HE) for privacy protection, which has limitations in
efficiency. To overcome these limitations, we propose SecSV, an efficient
two-server protocol with the following novel features. First, SecSV utilizes a
hybrid privacy protection scheme to avoid ciphertext--ciphertext
multiplications between test data and models, which are extremely expensive
under HE. Second, an efficient secure matrix multiplication method is proposed
for SecSV. Third, SecSV strategically identifies and skips some test samples
without significantly affecting the evaluation accuracy. Our experiments
demonstrate that SecSV is 7.2-36.6 times as fast as HESV, with a limited loss
in the accuracy of calculated SVs.
