BetterTimes

Abstract

We present a privacy-assured multiplication protocol using which an arbitrary arithmetic formula with inputs from two parties over a finite field $\mathbb {F}_p$ can be jointly computed on encrypted data using an additively homomorphic encryption scheme. Our protocol is secure against malicious adversaries. To motivate and illustrate applications of this technique, we demonstrate an attack on a class of known protocols showing how to compromise location privacy of honest users by manipulating messages in protocols with additively homomorphic encryption. We evaluate our approach using a prototypical implementation. The results show that the added overhead of our approach is small compared to insecure outsourced multiplication.

Appendices

A A Concrete Instantiation to Secure Hallgren et al.

To make the protocol from Hallgren et al., and other afflicted solutions, secure against format attacks from Alice, the distance can be computed directly on the coordinates instead of using several correlated values. The secured algorithm could be modeled as follows:

B Visualization of Privacy-Preserving Arithmetic Formula

Figure 7 depicts the system for privacy-preserving arithmetic formulas presented in this paper, during an execution where Alice is honest. Alice is the initiating party, and starts by sending her inputs to Bob. Bob then hardwires both is and Alice’s inputs into a instruction of nested operations, forming a tree like in Fig. 3. Depending on g, Bob computes any local operations and executes BetterTimes as necessary, with as many iterations as necessary. Finally, he computes the ciphertext $[\![result ]\!]$. Since Alice by assumption is honest, $[\![result ]\!]$ will hold the output of g (and would hold the encryption of a random element in $\mathbb {F}_p$ if Alice was dishonest). BetterTimes is simplified here, for a more complete visualization see Fig. 4.

Fig. 7.

Visualization of actions by each principal, where R and O means repeatable and optional, respectively.