Cryptography is the study of securing communication over a public network against eavesdropping and tampering by untrusted third-parties.
Since the beginning of the digital age, the study of cryptography has become increasingly important. It is central to the way in which we use the internet today. For example, without cryptography:
- We would not have been able to do things such as online shopping and internet banking.
- E-mails and private messages on social networks would be readable by any nodes in the network that lie between the sender and the recipient.
- There would be no way to verify that we are truly visiting a financial institution’s website, and not some impostor.
Mathematics has always been fundamental to cryptography – most cryptographic methods rely on the conjectured difficulty of some mathematical problem. Elliptic curves are algebraic objects which have been studied for more than a century, and which have been used with great success to solve many problems in mathematics, including Andrew Wiles’s acclaimed proof of Fermat’s Last Theorem. As for practical uses, elliptic curves have recently been applied successfully to cryptography. In 1985, Neal Koblitz and Victor S. Miller independently suggested the use of cryptographic algorithms based on the properties of elliptic curves. Since the 2000s they have gained wider acceptance in the cryptographic community, and are now seen as a viable alternative to cryptosystems based on the difficulty of integer factorization, such as RSA.
However, elliptic curve cryptography is still a relatively new field of study. It is therefore important that researchers in the cryptographic community should study it thoroughly, in order to ensure that it is indeed secure. The objectives of my research are to:
- Thoroughly study and explain the mathematics behind elliptic curve cryptography.
- Review and describe various implementations of elliptic curve cryptosystems.
- Investigate and discuss a selection of cryptanalytic attacks which have been used to break the security of elliptic curve cryptosystems.