: Group theory is one of the important branches of mathematics that has various applications in several fields, including number theory, geometry and cryptography .In this study, first, some facts of number theory were proven based on concepts in group theory, using new strategies such as the inverse of Wilson’s theorem and some other issues for distinguishing between prime numbers. Second, a practical application of these two theories (groups and numbers) in cryptography was presented. Two modifications of the well-known asymmetric encryption algorithms, RSA and ElGamal, were proposed. These modified encryption systems were named Power RSA (P-RSA) and Power ElGamal (P-ElGamal). These systems used powers of integers instead of using prime numbers directly. These powers were also kept secret by both the sender and receiver. Group theory has been used to prove that these cryptosystems work properly.

simulations were conducted to test the performance of the two proposed algorithms and compare each proposed algorithm with the standard algorithm using Python code.

The simulation results show that the execution time complexity of P-RSA encryption is approximately similar to that of RSA encryption, while the execution time for the encryption process of P-ElGamal is higher than that of both ElGamal and P-RSA. Furthermore, the decryption time complexity of P-RSA is higher than that of RSA and P-ElGamal, which in turn is higher than that of the ElGamal decryption process. The complexity of the ciphertext size of P-RSA and P-ElGamal encryption is larger than that of RSA and ElGamal, respectively. Furthermore, the complexity of the private key and ciphertext in P-RSA is higher than that of other comparable algorithms. This demonstrates that integrating group theory with number theory not only enhances theoretical understanding but also offers practical benefits in modern cryptography. The proposed improvements demonstrate the ability of sports structures to meet real-world security challenges.