As a promising technique, the spatial information of an object can be acquired by employing active illumination of sinusoidal patterns in the Fourier single-pixel imaging. However, the major challenge in this field is that a large number of illumination patterns should be generated to record measurements in order to avoid the loss of object details. In this paper, an optical multiple-image authentication method is proposed based on sparse sampling and multiple logistic maps. To improve the measurement efficiency, object images to be authenticated are randomly sampled based on the spatial frequency distribution with smaller size, and the Fourier sinusoid patterns generated for each frequency are converted into binarized illumination patterns using the Floyd-Steinberg error diffusion dithering algorithm. In the generation process of the ciphertext, two chaotic sequences are used to randomly select spatial frequency for each object image and scramble all measurements, respectively. Considering initial values and bifurcation parameters of logistic maps as secret keys, the security of the cryptosystem can be greatly enhanced. For the first time to our knowledge, how to authenticate the reconstructed object image is implemented using a significantly low number of measurements (i.e., at a very low sampling ratio less than 5% of Nyquist limit) in the Fourier single-pixel imaging. The experimental results as well as simulations illustrate the feasibility of the proposed multiple-image authentication mechanism, which can provide an effective alternative for the related research.