The study of elliptic curves by algebraists, algebraic geometers and number theorists dates back to the middle of the nineteenth century. Much attention has recently been focused on the use of elliptic curves in public key cryptography, first proposed in the work of Neal Koblitz and Victor Miller in 1985. Elliptic curve cryptography (ECC) is an exciting technology because for the same level of security as public-key cryptosystems such as RSA, it offers the benefits of smaller key sizes and hence of smaller memory and processor requirements. This makes them ideal for use in smart cards and other environments where resources such as storage, time, or power are at a premium. In the coming era of Internet of Things (IoT), the popularity of ECC is predicted to be significantly increased.
The Weil and Tate pairings on elliptic curves are used to construct protocols which cannot be implemented in another way. The most spectacular example of this is the identity-based encryption algorithm of Boneh and Franklin. Not only these protocols but also how these pairings can be efficiently implemented will be introduced.
The following topics will be brought to this course:
* Elliptic curve arithmetic
* Elliptic curves over finite fields
* Elliptic curve discrete logarithm problem
* Elliptic curve cryptography
* Elliptic curve factorization algorithm
* Efficient implementation of elliptic curves
* Determining the group order
* Bilinear pairings on elliptic curves
* Pairing-based cryptography
* Provable security of ECDSA
* Side-channel attacks and countermeasures
* Hyperelliptic curve cryptography