This work studies differential privacy in the context of the recently proposed shuffle model. Unlike in the local model, where the server collecting privatized data from users can track back an input to a specific user, in the shuffle model users submit their privatized inputs to a server anonymously. This setup yields a trust model which sits in between the classical curator and local models for differential privacy. The shuffle model is the core idea in the Encode, Shuffle, Analyze (ESA) model introduced by Bittau et al. (SOPS 2017). Recent work by Cheu et al. (EUROCRYPT 2019) analyzes the differential privacy properties of the shuffle model and shows that in some cases shuffled protocols provide strictly better accuracy than local protocols. Additionally, Erlingsson et al. (SODA 2019) provide a privacy amplification bound quantifying the level of curator differential privacy achieved by the shuffle model in terms of the local differential privacy of the randomizer used by each user.In this context, we make three contributions. First, we provide an optimal single message protocol for summation of real numbers in the shuffle model. Our protocol is very simple and has better accuracy and communication than the protocols for this same problem proposed by Cheu et al. Optimality of this protocol follows from our second contribution, a new lower bound for the accuracy of private protocols for summation of real numbers in the shuffle model. The third contribution is a new amplification bound for analyzing the privacy of protocols in the shuffle model in terms of the privacy provided by the corresponding local randomizer. Our amplification bound generalizes the results by Erlingsson et al. to a wider range of parameters, and provides a whole family of methods to analyze privacy amplification in the shuffle model. * The Alan Turing Institute. jbell@posteo.net.

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Despite recent improvements, men still have worse HIV outcomes than women in South Africa. This study describes how young men form distinct behavioural and attitudinal subgroups, and is intended to inform the design of targeted interventions to encourage HIV testing and initiation on antiretroviral therapy. Data were collected using a cross-sectional survey with questions on men’s attitudes, beliefs and behaviours around HIV/AIDS. A total of 2,019 men were randomly sampled from eight district municipalities in KwaZulu-Natal and Mpumalanga provinces between October 2018 and January 2019. Men were eligible to participate if they were aged 20–34, Black African, had an education level below university graduation, were aware of HIV and were willing to disclose whether they had tested for HIV. Each participant responded to a questionnaire asking about their demographic characteristics, reported sexual behaviour, engagement with HIV testing and treatment services, alcohol consumption, HIV knowledge, attitudes to gender equity and reported level of depressive symptoms. Data were analysed using canonical correlation, hierarchical clustering and factor analysis techniques to produce five groups of men. The results were synthesised using Human Centred Design principles to suggests areas for potential intervention for each segment. The results showed that men vary based on their attitudes to gender and masculinity, use of alcohol, testing and treatment behaviour, HIV-related fears and preferences for testing modalities. Segment 1 (21%) avoids the topic of HIV, perhaps fearful of the impact on his life. Segment 2 (23%) is well connected to his community and has social concerns about HIV. Segment 3 (15%) struggles with more distal determinants of HIV acquisition such as unemployment and poor mental health. Segment 4 (25%) has concerns about the lifestyle changes that would be required if he were HIV positive. Segment 5 (16%) has a strong traditional mindset and is fearful of the ramifications of HIV in his community. The results will be used to design targeted interventions to increase HIV testing and treatment rates among young men in South Africa. Further research is required to understand the impact of interventions designed in this way.