Theorem 15.1. Let E/Fq be an elliptic curve over a finite field, and let πE be the Frobenius endomorphism of E. Then E is supersingular if and only if trπE ≡ 0 mod p. Proof. Let q = pn and let π be the p-power Frobenius map π(x, y) = (xp, yp) (note that π is an isogeny, but not necessarily an endomorphism, since E need not be defined over Fp). We have π̂π = [p], where [p] denotes the multiplicat...

We study the supersingular curves on Picard modular surfaces modulo a prime p which is inert in the underlying quadratic imaginary ...eld. We analyze the automorphic vector bundles in characteristic p; and as an application derive a formula relating the number of irreducible components in the supersingular locus to the second Chern class of the surface. The characteristic p ...bers of Shimura v...

Let q be a prime and K = Q( √ −D) be an imaginary quadratic field such that q is inert in K. If q is a prime above q in the Hilbert class field of K, there is a reduction map rq : E``(OK) −→ E``(Fq2) from the set of elliptic curves over Q with complex multiplication by the ring of integers OK to the set of supersingular elliptic curves over Fq2 . We prove a uniform asymptotic formula for the nu...

We focus on the implementation and security aspects of cryptographic protocols that use Type 1 and Type 4 pairings. On the implementation front, we report improved timings for Type 1 pairings derived from supersingular elliptic curves in characteristic 2 and 3 and the first timings for supersingular genus-2 curves in characteristic 2 at the 128-bit security level. In the case of Type 4 pairings...

In their seminal paper, Miyaji, Nakabayashi and Takano [12] describe a simple method for the creation of elliptic curves of prime order with embedding degree 3, 4, or 6. Such curves are important for the realisation of pairing-based cryptosystems on ordinary (non-supersingular) elliptic curves. We provide an alternative derivation of their results, and extend them to allow for the generation of...

Abstract L. Moret-Bailly constructed families $${\mathfrak {C}}\rightarrow {\mathbb {P}}^1$$ C → P 1 of genus 2 curves with supersingular Jacobian. In this paper we first classify the reducible fibers a family usi...

In implementation of elliptic curve cryptography, three kinds of finite fields have been widely studied, i.e. prime field, binary field and optimal extension field. In pairing-based cryptography, however, pairingfriendly curves are usually chosen among ordinary curves over prime fields and supersingular curves over extension fields with small characteristics. In this paper, we study pairings on...

In 1965 Hayashida and Nishi initiated the study of genus 2 curves C whose Jacobian JC is isomorphic to a product A = E1×E2 of two elliptic curves. In their papers [12], [14] and [13], they determined the number of curves C with JC ' A for a fixed A in many cases, thereby exhibiting the existence of such curves. A similar count was done for supersingular curves by Ibukiyama, Katsura and Oort[16]...