Abstract Computing a function of some private inputs while maintaining the confidentiality of those inputs is an important problem, to which Differential Privacy and Secure Multi-party Computation can offer solutions under specific assumptions. Research in randomised algorithms aims at improving the privacy of such inputs by randomising the output of a computation while ensuring that large distortions of outputs occur with low probability. But use cases such as e-voting or auctions will not tolerate large distortions at all. Thus, we develop a framework for randomising the output of a privacypreserving computation, while guaranteeing that output distortions stay within a specified bound. We analyse the privacy gains of our approach and characterise them more precisely for our notion of sparse functions. We build randomisation algorithms, running in linearithmic time in the number of possible input values, for this class of functions and we prove that the computed randomisations maximise the inputs’ privacy. Experimental work demonstrates significant privacy gains when compared with existing approaches that guarantee distortion bounds, also for non-sparse functions.

Elaborate protocols in Secure Multi-party Computation enable several participants to compute a public function of their own private inputs while ensuring that no undesired information leaks about the private inputs, and without resorting to any trusted third party. However, the public output of the computation inevitably leaks some information about the private inputs. Recent works have introduced a framework and proposed some techniques for quantifying such information flow. Yet, owing to their complexity, those methods do not scale to practical situations that may involve large input spaces. The main contribution of the work reported here is to formally investigate the information flow captured by the min-entropy in the particular case of secure three-party computations of affine functions in order to make its quantification scalable to realistic scenarios. To this end, we mathematically derive an explicit formula for this entropy under uniform prior beliefs about the inputs. We show that this closed-form expression can be computed in time constant in the inputs sizes and logarithmic in the coefficients of the affine function. Finally, we formulate some theoretical bounds for this privacy leak in the presence of non-uniform prior beliefs.

The purpose of secure multi-party computation is to enable protocol participants to compute a public function of their private inputs while keeping their inputs secret, without resorting to any trusted third party. However, opening the public output of such computations inevitably reveals some information about the private inputs. We propose a measure generalizing both Rényi entropy and g -entropy so as to quantify this information leakage. In order to control and restrain such information flows, we introduce the notion of function substitution, which replaces the computation of a function that reveals sensitive information with that of an approximate function. We exhibit theoretical bounds for the privacy gains that this approach provides and experimentally show that this enhances the confidentiality of the inputs while controlling the distortion of computed output values. Finally, we investigate the inherent compromise between accuracy of computation and privacy of inputs and we demonstrate how to realize such optimal trade-offs.