Decrypting the future : a mathematical review of error-correcting codes and cryptography

Data encryption plays a crucial role in postmodern society. As technology continues to advance, the current predominate system, RSA cryptography, will fail to maintain the safety of information. Thus, corporations, governments, and individuals alike must turn towards a different alternative. I discuss how the security of data is related to the mathematical properties of finite fields. Throughout the discussion, mathematical examples are demonstrated using programs in Mathematica code. First, error-correcting codes are presented and analyzed to demonstrate how miscommunications can effectively and efficiently be prevented. Second, the mathematical structure of ElGamal and elliptic curve cryptography is explained. Lastly, all presented cryptosystems are analyzed for their level of security and implementation.