The rise of generative AI (GenAI) has impacted many aspects of human life. As these systems become embedded in everyday practices, understanding public trust in them is also essential for responsible adoption and governance. Prior work on trust in AI has largely drawn from psychology and human-computer interaction, but there is a lack of computational, large-scale, and longitudinal approaches to measuring trust and distrust in GenAI and large language models (LLMs). This paper presents the first computational study of trust and distrust in GenAI, using a multi-year Reddit dataset (2022--2025) spanning 39 subreddits and 230,576 posts. Crowd-sourced annotations of a representative sample were combined with classification models to scale analysis. We find that trust and distrust are nearly balanced over time, although trust modestly outweighs distrust, with shifts around major model releases. Technical performance and usability dominate as dimensions, while personal experience is the most frequent reason shaping attitudes. Distinct patterns also emerge across trustors (e.g., experts, ethicists, and general users). Our results provide a methodological framework for large-scale trust analysis and insights into evolving public perceptions of GenAI.