We present a heuristic asymptotic formula as x → ∞ for the number of isogeny classes of pairing-friendly elliptic curves over prime fields with fixed embedding degree k ≥ 3, with fixed discriminant, with rho-value bounded by a fixed ρ0 such that 1 < ρ0 < 2, and with prime subgroup order at most x.

We give a classification of all principally polarized abelian surfaces that admit an (l, l)-isogeny to themselves, and show how to compute all the abelian surfaces that occur. We make the classification explicit in the simplest case l = 2. As part of our classification, we also show how to find all principally polarized abelian surfaces with multiplication by a given imaginary quadratic order.

We show that there are no non-zero semi-stable abelian varieties over Q( √ 5) with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X0(15).

There are two types of elliptic curves, ordinary curves and supersingular curves. In 2012, Sutherland proposed an efficient almost deterministic algorithm for determining whether a given curve is or supersingular. Sutherland's based on sequences isogenies started from the input curve, computation each isogeny requires square root computations, which dominant cost algorithm. this paper, we reduc...

We provide explicit bounds on the difference of heights isogenous Drinfeld modules. derive a finiteness result in isogeny classes. In rank 2 case, we also obtain an upper bound size coefficients modular polynomials atta

• Ω = ∫ E(R) |ω|, and ω is the global section of Ω 1 N(E)/Z corresponding to a choice of basis of the Z-line Cot0(N(E)) for the Néron model N(E) of E over Z, • cp is the number of connected components of N(E)Fp that are geometrically connected over Fp, or equivalently (by Lang’s theorem applied to N(E)0Fptorsors) have a rational point, so cp coincides with the number of Fp-points of the finite ...

Consider the problem of efficiently evaluating isogenies $$\phi : \mathcal {E}\rightarrow {E}/H$$ elliptic curves over a finite field $$\mathbb {F}_q$$ , where kernel $$H = \langle {G}\rangle $$ is cyclic group odd (prime) order: given $$\mathcal {E}$$ $$G$$ and point (or several points) P on we want to compute (P)$$ . This at heart efficient implementations group-action- isogeny-based post-qua...

Abstract In supersingular isogeny-based cryptography, the path-finding problem reduces to endomorphism ring problem. Can be reduced knowing just one endomorphism? It is known that a small degree enables polynomial-time and computation (in: Love Boneh, ANTS XIV-Proceedings of Fourteenth Algorithmic Number Theory Symposium, volume 4 Open Book Ser. Math. Sci. Publ., Berkeley, 2020). An gives an ex...