Item-Based Privacy-Preserving Recommender System with Offline Users and Reduced Trust Requirements

Abstract

Safeguarding privacy of ratings assigned by users is an important issue for recommender systems. There are several existing protocols that allow a server to generate recommendations from homomorphically encrypted ratings, thereby ensuring privacy of rating data. After collecting the encrypted ratings, the server may require further interaction with each user, which is problematic in case some users were to go offline. To solve the offline user problem previous solutions use additional semi-honest third parties. In this paper, we propose a privacy-preserving recommender system that does not suffer from the offline user problem. Unlike previous works, our proposal does not require any additional third party. We demonstrate with the help of experiments that the time required to generate recommendations is efficient for practical applications.

Appendices

A Similarity Computation

Cosine similarity is used here to show the similarity between two items. It computes similarity between rating vectors of two items using cosine of angle between the vectors, smaller angle represents more similarity.

Algorithm 3 shows the similarity computation phase. Before executing these phases, the users would have been divided into groups and generated group keys for respective groups. The common key Y is available to all users. The server has computed the average rating of all items.

The similarity between two items $\theta$ and $\theta '$ is computed as:

$$\begin{aligned} s\left( \theta ,\theta '\right) = \frac{\sum _{i=1}^{n}r_{i,\theta }r_{i,\theta '}}{\sqrt{\sum _{i=1}^{n}r^{2}_{i,\theta }} \sqrt{ \sum _{i=1}^{n}r^{2}_{i,\theta '}}} \end{aligned}$$

B CBF-Based Recommendation

To generate recommendations, the server computes predicted ratings for all items and send them to the user. At the user end items with the highest ratings are shown to the user as recommended items list. The server sends predicted ratings encrypted under the target user’s public key so these predictions can be seen by the intended user only. In CBF-Based recommendations, for a user u the rating for item $\theta$ is computed as:

$$\begin{aligned} RP_{u,\theta } = \frac{\sum _{j=1}^{m}r_{u,\theta } s\left( p_j, p_\theta \right) }{\sum _{j=1}^{m}s \left( p_j, p_\theta \right) } \end{aligned}$$

The user sends its encrypted ratings to the server, and the server computes numerator and denominator values of rating predictions for all items, encrypts them using the target user’s public key and sends back to the user. The user decrypts all predicted ratings and receives the recommendations. Algorithm 4 shows the detailed steps.

C CF-Based Recommendation

To generate recommendations using CF-based technique, the data flow is similar to that of CBF-based technique with following equation for prediction:

$$\begin{aligned} RP_{u,\theta } = \frac{\bar{p}_\theta \sum _{j=1}^{m}s\left( p_\theta , p_j\right) + \sum _{j=1}^{m} \left( r_{u,j} - \bar{p}_j\right) s\left( p_\theta , p_j\right) }{\sum _{j=1}^{m}s\left( p_\theta , p_j\right) } \end{aligned}$$

Detailed steps are shown in Algorithm 5.