Cryptography is the technique of employing codes and ciphers to protect communication and information, ensuring that only the intended recipients can decode and handle it. This technique is essential for ensuring secure communication, where a message or data shared between two parties is protected against unwanted access by attackers. Data confidentiality, Data integrity, Authentication and Non-repudiation are core principles of modern cryptography. In the field of cryptography, an adversary refers to a malevolent entity that aims to undermine the security of a system by intercepting, modifying, or counterfeiting data. Opponents challenge the efficiency of cryptographic protocols by attempting to bypass these fundamental principles, highlighting the need for the creation of strong cryptographic techniques to ensure information security. In this chapter we have discussed recent developments in mathematical cryptography that are important for a number of reasons, primary among them being the necessity to improve digital system security, counter new threats, and accommodate the increasing complexity of international communications and transactions.