We discuss the use of elliptic curves in cryptography. In particular, we propose an analogue of the Diffie-Hellmann key exchange protocol which appears to be immune from attacks of the style of Western, Miller, and Adleman. With the current bounds for infeasible attack, it appears to be about 20% Faster than the Diffie-Hellmann scheme over GF(p). As computational power grows, this disparity should get rapidly bigger.

The Weil pairing, first introduced by André Weil in 1940, plays an important role in the theoretical study of the arithmetic of elliptic curves and Abelian varieties. It has also recently become extremely useful in cryptologic constructions related to those objects. This paper gives the definition of the Weil pairing, describes efficient algorithms to calculate it, gives two applications, and describes the motivation to considering it.

Abstract. For a rational elliptic curve in Weierstrass form, Chudnovsky and Chudnovsky considered the likelihood that the denominators of the xcoordinates of the multiples of a rational point are squares of primes. Assuming the point is the image of a rational point under an isogeny, we use Siegel's Theorem to prove that only finitely many primes will arise. The same question is considered for elliptic curves in homogeneous form, prompting a visit to Ramanujan's famous taxi-cab equation. Finiteness is provable for these curves with no extra assumptions. Finally, consideration is given to the possibilities for prime generation in higher rank.

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