Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Quantitative spectral gap for thin groups of hyperbolic isometries

Abstract: Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1 , 1)$. The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002)

On the Analysis of Cryptographic Assumptions in the Generic Ring Model

Abstract: At Eurocrypt 2009 Aggarwal and Maurer proved that breaking RSA is equivalent to factoring in the generic ring model . This model captures algorithms that may exploit the full algebraic structure of the ring of integers modulo n , but no properties of the given representation of ring elements. This interesting result raises the question how to interpret proofs in the generic ring model. For instance, one may be tempted to deduce that a proof in the generic model gives some evidence that solving the considered problem is also hard in a general model of computation. But is this reasonable? We prove that computing the Jacobi symbol is equivalent to factoring in the generic ring model. Since there are simple and efficient non-generic algorithms computing the Jacobi symbol, we show that the generic model cannot give any evidence towards the hardness of a computational problem. Despite this negative result, we also argue why proofs in the generic ring model are still interesting, and show that solving the quadratic residuosity and subgroup decision problems is generically equivalent to factoring.