  While machine learning has become pervasive in as diversified fields as
industry, healthcare, social networks, privacy concerns regarding the training
data have gained a critical importance. In settings where several parties wish
to collaboratively train a common model without jeopardizing their sensitive
data, the need for a private training protocol is particularly stringent and
implies to protect the data against both the model's end-users and the actors
of the training phase. Differential privacy (DP) and cryptographic primitives
are complementary popular countermeasures against privacy attacks. Among these
cryptographic primitives, fully homomorphic encryption (FHE) offers ciphertext
malleability at the cost of time-consuming operations in the homomorphic
domain. In this paper, we design SHIELD, a probabilistic approximation
algorithm for the argmax operator which is both fast when homomorphically
executed and whose inaccuracy is used as a feature to ensure DP guarantees.
Even if SHIELD could have other applications, we here focus on one setting and
seamlessly integrate it in the SPEED collaborative training framework from
"SPEED: Secure, PrivatE, and Efficient Deep learning" (Grivet S\'ebert et al.,
2021) to improve its computational efficiency. After thoroughly describing the
FHE implementation of our algorithm and its DP analysis, we present
experimental results. To the best of our knowledge, it is the first work in
which relaxing the accuracy of an homomorphic calculation is constructively
usable as a degree of freedom to achieve better FHE performances.
