<p>The construction of memristive hyperchaotic maps has long been an important topic in chaos research. Over the past few years, second-order memristors (SOMs) have become a growing focus of research; however, the development of their corresponding discrete maps remains limited. To address this gap, a novel discrete second-order memristor (SOM) model is proposed and coupled with a sine map, resulting in the construction of a three-dimensional second-order memristive sine hyperchaotic map (SOM-SHM). Numerical investigations using phase portraits, Lyapunov exponent spectra (LEs) and bifurcation diagrams (BDs) show that the proposed system exhibits rich nonlinear dynamics in representative parameter regions, including periodic, chaotic, and hyperchaotic behaviors, together with initial-offset boosting. Numerical simulations and NIST test results indicate that the sequences derived from the proposed map exhibit good statistical randomness under the tested conditions. The map is further implemented on an STM32 hardware platform to validate its digital implementability. Finally, an image encryption example based on the SOM-SHM is constructed, and standard diagnostic evaluations are conducted to illustrate its potential application in information processing.</p>