Activity-tracking applications, where people record and upload information about their location-based activities (e.g., the routes of their activities), are increasingly popular. Such applications enable users to share information and compete with their friends on activity-based social networks but also, in some cases, to obtain discounts on their health insurance premiums by proving they conduct regular fitness activities. However, they raise privacy and security issues: the service providers know the exact locations of their users; the users can report fake location information, for example, to unduly brag about their performance. In this paper, we present SecureRun, a secure privacy-preserving system for reporting location-based activity summaries (e.g., the total distance covered and the elevation gain). SecureRun is based on a combination of cryptographic techniques and geometric algorithms, and it relies on existing Wi-Fi access-point networks deployed in urban areas. We evaluate SecureRun by using real data-sets from the FON hotspot community networks and from the Garmin Connect activity-based social network, and we show that it can achieve tight (up to a median accuracy of more than 80%) verifiable lower-bounds of the distance covered and of the elevation gain, while protecting the location privacy of the users with respect to both the social network operator and the access point network operator(s). The results of our online survey, targeted at RunKeeper users recruited through the Amazon Mechanical Turk platform, highlight the lack of awareness and significant concerns of the participants about the privacy and security issues of activity-tracking applications. They also show a good level of satisfaction regarding SecureRun and its performance.