Chaos-based stream ciphers form a prospective class of data encryption techniques. Usually, in chaos-based encryption schemes, the pseudo-random generators based on chaotic maps are used as a source of randomness. Despite the variety of proposed algorithms, nearly all of them possess many shortcomings. While sequences generated from single-parameter chaotic maps can be easily compromised using the phase space reconstruction method, the employment of multi-parametric maps requires a thorough analysis of the parameter space to establish the areas of chaotic behavior. This complicates the determination of the possible keys for the encryption scheme. Another problem is the degradation of chaotic dynamics in the implementation of the digital chaos generator with finite precision. To avoid the appearance of quasi-chaotic regimes, additional perturbations are usually introduced into the chaotic maps, making the generation scheme more complex and influencing the oscillations regime. In this study, we propose a novel technique utilizing the chaotic maps with adaptive symmetry to create chaos-based encryption schemes with larger parameter space. We compare pseudo-random generators based on the traditional Zaslavsky map and the new adaptive Zaslavsky web map through multi-parametric bifurcation analysis and investigate the parameter spaces of the maps. We explicitly show that pseudo-random sequences generated by the adaptive Zaslavsky map are random, have a weak correlation and possess a larger parameter space. We also present the technique of increasing the period of the chaotic sequence based on the variability of the symmetry coefficient. The speed analysis shows that the proposed encryption algorithm possesses a high encryption speed, being compatible with the best solutions in a field. The obtained results can improve the chaos-based cryptography and inspire further studies of chaotic maps as well as the synthesis of novel discrete models with desirable statistical properties.