Local Differential Privacy (LDP) is the predominant privacy model for safeguarding individual data privacy. Existing perturbation mechanisms typically require perturbing the original values to ensure acceptable privacy, which inevitably results in value distortion and utility deterioration. In this work, we propose an alternative approach -- instead of perturbing values, we apply randomization to indexes of values while ensuring rigorous LDP guarantees. Inspired by the deniability of randomized indexes, we present CRIAD for answering subset counting queries on set-value data. By integrating a multi-dummy, multi-sample, and multi-group strategy, CRIAD serves as a fully scalable solution that offers flexibility across various privacy requirements and domain sizes, and achieves more accurate query results than any existing methods. Through comprehensive theoretical analysis and extensive experimental evaluations, we validate the effectiveness of CRIAD and demonstrate its superiority over traditional value-perturbation mechanisms.