Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBilinear pairings on elliptic curves
216
0
0.0
(
0
)
Authors
Andreas Enge
Ask ChatGPT about the research
No Arabic abstract
We give an elementary and self-contained introduction to pairings on elliptic curves over finite fields. For the first time in the literature, the three different definitions of the Weil pairing are stated correctly and proved to be equivalent using Weil reciprocity. Pairings with shorter loops, such as the ate, ate$_i$, R-ate and optimal pairings, together with their twisted variants, are presented with proofs of their bilinearity and non-degeneracy. Finally, we review different types of pairings in a cryptographic context. This article can be seen as an update chapter to A. Enge, Elliptic Curves and Their Applications to Cryptography - An Introduction, Kluwer Academic Publishers 1999.
rate research
Read More
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every curve is pairing-friendly. These have recently found notable applications in pairing-based cryptography, for instance in improving the scalability of distributed ledger technologies. We construct a new cycle of length 4 consisting of MNT curves, and characterize all the possibilities for cycles consisting of MNT curves. We rule out cycles of length 2 for particular choices of small embedding degrees. We show that long cycles cannot be constructed from families of curves with the same complex multiplication discriminant, and that cycles of composite order elliptic curves cannot exist. We show that there are no cycles consisting of curves from only the Freeman or Barreto--Naehrig families.
In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.
Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $ell$-part of the relative class group $G_{L/K} := textrm{Cl}(L/K)$ when $K$ contains $ell^n$th roots of unity. While the moments of a conjectural distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $psi_{L/K}$ and $omega_{L/K},$ and study the distribution of $(G_{L/K}, psi_{L/K}, omega_{L/K})$ using a linear random matrix model. Using this linear model, we calculate the distribution (including our new invariants) in the function field case, and then make local adjustments at the primes lying over $ell$ and $infty$ to make a conjecture in the number field case, which agrees with some numerical experiments.
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $operatorname{char} k mid N$. For $P in C$, let $s_P$ be a rational function with divisor $N cdot P - N cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
