Isogenies of Elliptic Curves: A Computational Approach

The study of elliptic curves has historically been a subject of almost purely mathematical interest. However, Koblitz and Miller independently showed that elliptic curves can be used to implement cryptographic primitives [13], [17]. This thrust elliptic curves from the abstract realm of pure mathematics to the preeminently applied world of communications security. Public key cryptography in general advanced the development of the Internet and was in turn further advanced by this new use. Elliptic curve cryptography (ECC) was also developed and advanced along with the general field of public key cryptography. Elliptic curves provide benefits over the groups previously proposed for use in cryptography. Unlike finite fields, elliptic curves do not have a ring structure (the two related group operations of addition and multiplication), and hence are not vulnerable to index calculus like attacks [12]. The direct effect of this is that using elliptic curves over smaller finite fields yields the same security as using discrete log or factoring based public key crypto systems of Diffie-Hellman and RSA with larger moduli. This makes ECC ideally suited to small embedded and low power devices such as cell phones. So it is unsurprising that as these type of small devices have increased in popularity in recent years, ECC has as well. As elliptic curves are now used in cryptography, the computational aspects of them have real world applications. The underlying theory is very deep and touches on many different branches of mathematics. Elliptic curves have a very rich mathematical structure and the subject of ECC is about determining how to best apply and efficiently compute with this deep structure. The maps defined on any mathematical object are a key part of the underlying structure. In the case of elliptic curves, the principal maps of interest are the isogenies. An isogeny is a non-constant function, defined on an elliptic curve, that takes values on another elliptic curve and preserves point addition. In short, isogenies are functions that preserve the elliptic curve structure. As