In this paper we examine the underlying hard problems in asymmetric pairings, their precise relationships and how they affect a number of existing protocols. Furthermore, we present a new model for the elliptic curve groups used in asymmetric pairings, which allows both an efficient pairing and an efficient computable isomorphism.

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and ĥ be the canonical height on E. Bounds for the difference h − ĥ are of tremendous theoretical and practical importance. It is possible to decompose h − ĥ as a weighted sum of continuous bounded functions Ψυ : E(Kυ) → R over the set of places υ of K. A standard method for bounding h− ĥ, (due to L...

Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 12℘ ′(u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the functions ℘(u) and σ(u) are attached. Here the complex number u and the coordinate (x(u), y(u)) correspond by the equality

Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 12℘ ′(u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the functions ℘(u) and σ(u) are attached. Here the complex number u and the coordinate (x(u), y(u)) correspond by the equality

We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the p-adic setting. We prove, for primes p > 3, that the trace of Frobenius of any elliptic curve over Fp, whose jinvariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of ...

Suppose M ≡ N . Recently, there has been much interest in the classification of groups. We show that ‖π‖ > ∅. In [31], the authors address the structure of manifolds under the additional assumption that xΩ = πV . Therefore in [31], the main result was the classification of non-elliptic monoids.

1 SL2(Z) and elliptic curves 2 1.1 SL2(Z) and the moduli of complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Fundamental region and a system of generators . . . . . . . . . . . . . . . . . . . . . 3 1.3 The Weierstrass ℘ function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Nonsingular cubics and the invariant j . . . . . . . . . . . ...