This paper presents an enhanced variant of the RSA cryptosystem that integrates chaotic exponent selection and ciphertext blinding to address security limitations inherent in classical RSA. Traditional RSA relies on a fixed public exponent, which generates predictable encryption patterns and increases exposure to exponent-based attacks. In the proposed scheme, the encryption exponent is dynamically derived from a logistic-map–based chaotic sequence, introducing high sensitivity to initial conditions and producing session-dependent exponent values. This chaotic exponentiation increases unpredictability without modifying the established RSA framework. Additionally, a ciphertext blinding factor is incorporated to prevent deterministic outputs and strengthen resistance against chosen-ciphertext and side-channel attacks. The paper outlines the mathematical background of the logistic map, details the complete encryption and decryption procedures, and demonstrates the correctness of the method through a numerical example using small primes. A theoretical security analysis shows that the combined effects of chaotic exponent selection and blinding significantly improve resistance to key-related attacks while maintaining compatibility with the original RSA structure. These enhancements offer a lightweight and practical improvement to RSA for environments requiring increased confidentiality and unpredictability in exponent selection.