The potential of discrete memristors to improve chaotic systems for safe communication has been demonstrated by recent advancements. This paper presents a novel four-dimensional (4D) memristive hyperchaotic map that combines a sine map with a two-dimensional logistic-Gaussian hyperchaotic (2D LGH) map through a sinusoidal discrete memristor. The proposed system exhibits high-dimensional chaotic behavior, including hyperchaos and strong randomness, as illustrated through bifurcation diagrams, Lyapunov exponent (LE) analysis, and complexity metrics. We present a high-security image encryption approach based on these features, incorporating the proposed 4D memristive hyperchaotic system with a non-associative LA-group structure. This method generates algebraic-chaotic sequences by mixing non-associative algebraic operations with chaotic sequences, significantly broadening the key space. The encryption framework provides great efficiency, low computing cost, and resilience to cryptographic attacks. Comprehensive security assessments demonstrate the scheme’s efficacy, such as key sensitivity analysis, entropy evaluation, correlation testing, and differential attack resistance. The simulation results show that the suggested technique provides better encryption quality, high unpredictability, and resistance to statistical and differential attacks, making it an appealing choice for secure image transmission and data security in current communication systems.