In the current study we introduce an improved identity-based cryptographic (IBC) scheme that aims at addressing the weaknesses of existing frameworks. In further extension of the approach of Meshram, Meshram, and Zhang [21], who formulated an IBC method based on integer-factorization problems and generalized discrete-logarithm without reference to bilinear pairings, this paper determines and removes the stalemate of their construction. Though the scheme proposed by Meshram, Meshram, and Zhang [21] offers protection of data and adversarial resistance, the confidentiality of the encryption mechanism is still on a condition of secret data available in the key authentication center, which is the resource that cannot be accessed by the end users. To overcome this drawback, we present an enhanced subtree-based cryptographic protocol of secure group communication which uses partial discrete logarithms in finite fields. This plan not only removes the problem of the deadlock but also provides security improvement to the solution of Pang, Li, and Wang [31] whose work was undermined by the lack of full security analysis. The scheme proposed incorporates a strong key-distribution design and is more robust to any attack that might occur and thus enhances the security and trustworthiness of identity-based cryptographic schemes.