Abstract

Cryptography has existed since the time humans found it necessary to keep secrets from each other. In the past, it was simply enough to shift characters in a message by a fixed amount to hide a message but as the digital age dawned, ubiquitous computing has made it necessary to rely on complex mathematics to aid in secure transmission of data. Reliance on probability theory which studies events and their likelihood plays a central role in cryptography as it helps answer questions such as “What is the likelihood that two hash values are identical?”. Following this, combinatorics which is the study of counting allows a cryptographer to understand the design, capabilities, and limits of a cryptographic system. Number Theory enables the understanding of prime numbers, integer factorization, and finding the greatest common divisor which are the key ingredients of Diffie-Hellman key exchange protocols, the RSA algorithm, digital signatures, and encryption protocols. Efficient integer arithmetic lays the foundation for Montgomery multiplication and Karatsuba multiplication which enable efficient RSA and ECC implementations. Abstract algebra is focused on defining, describing, and manipulating mathematical objects and their structure. The study of abstract algebra has led to many new algorithms and ideas relating to cryptography, including some of the most famous and widely used such as AES. Abstract algebra defines the infrastructure for both information theory and cryptography algorithms. Elliptic curves have been the study of mathematicians going back to the 3rd century AD and are now becoming the new wave in public key cryptography. We review this material also in Chapter 2. In the current age of a proliferation of computing devices, the need for robust cryptographic solutions that protect these devices from malicious attacks has become mandatory. Thankfully, mathematics allows us to lay the foundation for these solutions.