We present three results on the period-index problem for genus-one curves over global fields. Our first result implies that for every pair of positive integers (P, I ) such that I is divisible by P and divides P 2 , there exists a number field K and a genus-one curve C /K with period P and index I . Second, let E /K be any elliptic curve over a global field K , and let P > 1 be any integer indivisible by the characteristic of K . We construct infinitely many genus-one curves C /K with period P, index P 2 , and Jacobian E. Our third result, on the structure of ShafarevichTate groups under field extension, follows as a corollary. Our main tools are Lichtenbaum-Tate duality and the functorial properties of O'Neil's period-index obstruction map under change of period.
AbstractWe describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety.Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.
The index of a curve is the smallest positive degree of divisors which are rational over a fixed base field. The period is the smallest positive degree of divisor classes rational over the base field. Lichtenbaum proved certain divisibility conditions relating the period, index, and genus of a curve over a local field. We prove that his conditions are sharp by finding a curve of every admissible period, index, and genus. Our proof constructs degree two covers of elliptic curves with split multiplicative reduction.
Abstract. The period of a curve is the smallest positive degree of Galoisinvariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.
We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We perform some basic analysis of this black box system and show that it is resistant to black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. We explain why we believe this instantiation is likely to be secure.
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