4,343 research outputs found
Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory
The present survey reports on the state of the art of the different
cryptographic functionalities built upon the ring learning with errors problem
and its interplay with several classical problems in algebraic number theory.
The survey is based to a certain extent on an invited course given by the
author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other
authors/ comment of the author: quotation has been added to Theorem 5.
Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule
A driving question in (quantum) cohomology of flag varieties is to find
non-recursive, positive combinatorial formulas for expressing the product of
two classes in a particularly nice basis, called the Schubert basis. Bertram,
Ciocan-Fontanine and Fulton provided a way to compute quantum products of
Schubert classes in the Grassmannian of k-planes in complex n-space by doing
classical multiplication and then applying a combinatorial rim hook rule which
yields the quantum parameter. In this paper, we provide a generalization of
this rim hook rule to the setting in which there is also an action of the
complex torus. Combining this result with Knutson and Tao's puzzle rule then
gives an effective algorithm for computing all equivariant quantum
Littlewood-Richardson coefficients. Interestingly, this rule requires a
specialization of torus weights modulo n, suggesting a direct connection to the
Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in
Algebraic Combinatoric
- …