[Paper Review] Privacy in Distributed Computations based on Real Number Secret Sharing

Katrine Tjell, Rafael Wisniewski|arXiv (Cornell University)|Jul 2, 2021

Cryptography and Data Security29 references4 citations

TL;DR

This paper proposes a real-number secret sharing scheme that enables privacy-preserving distributed computations directly on real numbers without converting to integers or using finite field arithmetic. The scheme supports efficient homomorphic operations like addition, multiplication, and division on shared data, achieving high accuracy and low communication overhead—demonstrated via a privacy-preserving Kalman filter with minimal output error compared to non-private counterparts.

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.

Motivation & Objective

• Address the gap in privacy-preserving computation for real-valued data by avoiding integer conversion and modular arithmetic.
• Overcome the precision loss and computational infeasibility of division in finite field-based secret sharing schemes.
• Develop a practical framework that balances privacy, accuracy, and efficiency for real-world applications like Kalman filtering.
• Demonstrate that information leakage in the proposed scheme is bounded and asymptotically negligible.
• Provide a secure, efficient alternative to existing methods that rely on homomorphic encryption or differential privacy, which degrade utility.

Proposed method

• Design a secret sharing scheme that operates directly on real numbers, eliminating the need for integer conversion or finite field arithmetic.
• Implement homomorphic operations—addition, multiplication, and division—directly on secret-shared real values using linear algebraic operations.
• Use a Gaussian distribution with zero mean and variance 1000 to generate random shares for input data, ensuring randomness and distributional properties.
• Apply the scheme to a distributed Kalman filter by sharing system matrices (A, B, H), control inputs (u_k), and noise covariances (Q_k, R_k) across multiple parties.
• Perform all filter updates (prediction, update, gain computation) using only shared values, with no reconstruction until final output.
• Use a threshold of t=1 and n=3 parties to ensure robustness and prevent information leakage from any single party.

Experimental results

Research questions

• RQ1 Can a secret sharing scheme be designed to operate directly on real numbers without integer conversion, enabling efficient and accurate homomorphic operations?
• RQ2 How does the privacy leakage of such a scheme compare to perfectly secure schemes, and can it be bounded mathematically?
• RQ3 To what extent does the proposed scheme preserve output accuracy in privacy-preserving Kalman filtering compared to non-private or existing private alternatives?
• RQ4 What is the communication and computational overhead of the proposed scheme relative to state-of-the-art methods like homomorphic encryption or differential privacy?
• RQ5 Can the scheme be practically deployed in real-world applications such as sensor networks or medical monitoring with minimal utility loss?

Key findings

• The proposed real-number secret sharing scheme achieves significantly higher accuracy than integer-based counterparts, with the RSE (Relative Squared Error) between the private and non-private Kalman filter results remaining below 1.5×10⁻³ after 45 time steps.
• The scheme reduces computational and communication overhead compared to [12], with Algorithm 5 requiring only 27 interactive operations regardless of matrix dimension, versus an estimated 10M + l + 1 operations in [12] (where M is matrix dimension and l ≥ 24 bits).
• Information leakage is upper bounded and asymptotically approaches zero, demonstrating that while not perfectly secure, the scheme provides strong privacy guarantees in practice.
• The Kalman filter implementation using the proposed scheme maintains high output utility, with estimation errors in the third decimal place, whereas the method in [12] shows errors before the decimal point.
• The scheme enables direct computation on shared real numbers, including division, which is infeasible or highly inefficient in finite field-based secret sharing.
• Simulations confirm that the scheme is practical and scalable, with low error and consistent performance across time steps in a dynamic system estimation task.