Privacy in Distributed Computations based on Real Number Secret Sharing

Abstract

Privacy preservation in distributed computations is an important subject as digitization and new technologies enable collection and storage of vast amounts of data, including private data belonging to individuals. To this end, there is a need for a privacy preserving computation framework that minimises the leak of private information during computations while being efficient enough for practical usage. This paper presents a step towards such a framework with the proposal of a real number secret sharing scheme that works directly on real numbers without the need for conversion to integers which is the case in related schemes. The scheme offers computations like addition, multiplication, and division to be performed directly on secret shared data (the cipher text version of the data). Simulations show that the scheme is much more efficient in terms of accuracy than its counterpart version based on integers and finite field arithmetic. The drawback with the proposed scheme is that it is not perfectly secure. However, we provide a privacy analysis of the scheme, where we show that the leaked information can be upper bounded and asymptotically goes to zero. To demonstrate the scheme, we use it to perform Kalman filtering directly on secret shared data.

Analog Secure Distributed Matrix Multiplication over Complex Numbers

“…The problem of secure coded computation over the analog domain was considered in [25]- [27]. Secret sharing over real numbers is considered in [25] using Lagrange coded secret sharing.…”

Secure PAC Bayesian Regression via Real Shamir Secret Sharing

“…Instead of using secret sharing over a finite field we base our secret sharing directly on the real numbers. This idea has been introduced in Tjell and Wisniewski (2021). This comes with the small drawback that a share might reveal a small amount of information.…”

“…In this section, we will give a brief introduction to the ideas from Tjell and Wisniewski (2021) and some of the obstacles they have to circumvent in order to apply Shamir's scheme over the real numbers. One of the obstacles is that if the shares are constructed from the polynomial in (1) the information the shares carry about the secret is dependent on how close the evaluation point is to 0.…”

“…We end this section with a description of how to compute on the shares. In the paper Tjell and Wisniewski (2021) protocols for securely computing sums, multiplications, and divisions over the real numbers are presented which are generalizations of protocols and techniques for finite fields described in Cramer, Damgård, and Nielsen (2015).…”

“…For this we propose to use techniques from secure multiparty computation (MPC). Originally, MPC is carried out over a finite field, but recent research consider MPC over real numbers Tjell and Wisniewski (2021) and we will make use of this approach.…”