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The first published solution to key distribution problem is due to Diffie–Hellman, which allows two parties that have never communicated earlier, to jointly establish a shared secret key over an insecure channel. In this paper, we propose a new key exchange protocol in a non-commutative semigroup over group ring whose security relies on the hardness of Factorization with Discrete Logarithm Problem (FDLP). We have also provided its security and complexity analysis. We then propose a ElGamal cryptosystem based on FDLP using the group of invertible matrices over group rings.

ElGamal cryptosystem has emerged as one of the most important construction in Public Key Cryptography (PKC) since Diffie-Hellman key exchange protocol was proposed. However, public key schemes which are based on number theoretic problems such as discrete logarithm problem (DLP) are at risk because of the evolution of quantum computers. As a result, other non-number theoretic alternatives are a dire need of entire cryptographic community.In 2016, Saba Inam and Rashid Ali proposed a ElGamal-like cryptosystem based on matrices over group rings in ‘Neural Computing & Applications’. Using linear algebra approach, Jia et al. provided a cryptanalysis for the cryptosystem in 2019 and claimed that their attack could recover all the equivalent keys. However, this is not the case and we have improved their cryptanalysis approach and derived all equivalent key pairs that can be used to totally break the ElGamal-like cryptosystem proposed by Saba and Rashid. Using the decomposition of matrices over group rings to larger size matrices over rings, we have made the cryptanalysing algorithm more practical and efficient. We have also proved that the ElGamal cryptosystem proposed by Saba and Rashid does not achieve the security of IND-CPA and IND-CCA.

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Group Key Distribution (GKD) protocols are designed to distribute a group key to several users for establishing a secure communication over a public network. The central trusted authority, called the key distribution center (KDC) is in charge of distributing the group keys. For securing the communication, all the users share a common secret key in advance with KDC. In this paper, we propose a secure and efficient Group Authenticated Key Distribution (GAKD) protocol based on the simple idea of encryption in matrix rings. In this protocol, each user registers in private with the KDC, while all the other information can be transferred publicly. The scheme also supports authentication of group keys without assuming computational hard problems such as Integer Factorization Problem (IFP).The analysis of our GAKD protocol shows that the proposed protocol is resistant to reply, passive and impersonation attacks. Our construction leads to a secure, cost and computation- effective GAKD protocol.