<p>Random matrices are fundamental in photonic computing because of their ability to model and enhance complex light interactions and signal processing capabilities. In manipulating classical light, random operations are utilized for random projections and dimensionality reduction, which are important for analog signal processing, computing, and imaging. In quantum information processing, random unitary operations are essential to boson sampling algorithms for multiphoton states in linear photonic circuits. Random operations are typically realized in photonic circuits through fixed disordered structures or through large meshes of interferometers with reconfigurable phase shifters, requiring a large number of active components. In this article, we introduce a compact photonic circuit for generating random matrices by utilizing programmable phase modulation layers interlaced with a fixed mixing operator. We show that using only two random phase layers is sufficient for producing output optical signals with a white-noise profile, even for highly sparse input optical signals. We experimentally demonstrate these results using a silicon-based photonic circuit with tunable thermal phase shifters and waveguide lattices as mixing layers. The proposed circuit offers a practical method for generating random matrices for photonic information processing and for applications in data encryption.</p>