We mainly consider the existence of a random exponential attractor (positive invariant compact measurable set with finite fractal dimension and attracting orbits exponentially) for stochastic discrete long wave-short wave resonance equation driven by multiplicative white noise. Firstly, we prove the existence of a random attractor of the considered equation by proving the existence of a uniformly tempered pullback absorbing set and making an estimate on the "tails" of solutions. Secondly, we show the Lipschitz property of the solution process generated by the considered equation. Finally, we prove the existence of a random exponential attractor of the considered equation, which implies the finiteness of fractal dimension of random attractor.2010 Mathematics Subject Classification. 37L55; 35B40; 60H15. Key words and phrases. Random exponential attractor, fractal dimension, multiplicative white noise, discrete long wave-short wave resonance equation.

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