New versions Elliptic Curve Group Ring Public Key Cryptosystems

In this work, a group ring (GR), which is defined on an elliptic curve over a prime field, is used to create new versions of public key cryptosystems. First version is elliptic curve group ring ElGamal public key (ECGR-EPKC). On the ECGR- EPKC, a public key, ciphertext and decrypting a plaintext are computing in according to the scalar multiplication that defined on the ECGR over a prime field. Two cases have been considered on the proposed ECGR-EPKC. First case, with an order of ring is less than or equal to the elliptic curve group (ECG). Whereas, on second case, an order of a ring is greater than the order of ECG. Also, the hybrid algorithm of RSA and EPKC based on the ECGR is proposed. The security of the proposed versions, ECGR-EPKE and RSA-EPKC based on ECGR are determined. The proposed RSA-EPKC based on ECGR considers as a more secure for communication schemes in compare with the proposed ECGR-EPKC in two cases and with original algorithms of EPKC and RSA alone.