Homomorphic Rank Sort Using Surrogate Polynomials

In this paper we propose a rank based algorithm for sorting encrypted data using monomials. Greedy Sort is a sorting technique that achieves to minimize the depth of the homomorphic evaluations. It is a costly algorithm due to excessive ciphertext multiplications and its implementation is cumbersome. Another method Direct Sort has a slightly deeper circuit than Greedy Sort, nevertheless it is simpler to implement and scales better with the size of the input array. Our proposed method minimizes both the circuit depth and the number of ciphertext multiplications. In addition to its performance, its simple design makes it more favorable compared to the alternative methods which are hard to parallelize, e.g. not suitable for fast GPU implementations. Furthermore, we improve the performance of homomorphic sorting algorithm by adapting the SIMD operations alongside message slot rotation techniques. This method allow us to pack $N$ integers into a single ciphertext and compute $N$ comparisons at once, thus reducing $\mathcal{O}(N^2)$ comparisons to $\mathcal{O}(N)$

Integer Polynomial Recovery from Outputs and its Application to Cryptanalysis of a Protocol for Secure Sorting

{We investigate the problem of recovering integer inputs (up to an affine scaling) when given only the integer monotonic polynomial outputs. Given $n$ integer outputs of a degree-$d$ integer monotonic polynomial whose coefficients and inputs are integers within known bounds and $n \gg d$, we give an algorithm to recover the polynomial and the integer inputs (up to an affine scaling). A heuristic expected time complexity analysis of our method shows that it is exponential in the size of the degree of the polynomial but polynomial in the size of the polynomial coefficients. We conduct experiments with real-world data as well as randomly chosen parameters and demonstrate the effectiveness of our algorithm over a wide range of parameters. Using only the polynomial evaluations at specific integer points, the apparent hardness of recovering the input data served as the basis of security of a recent protocol proposed by Kesarwani et al. for secure $k$-nearest neighbour computation on encrypted data that involved secure sorting. The protocol uses the outputs of randomly chosen monotonic integer polynomial to hide its inputs except to only reveal the ordering of input data. Using our integer polynomial recovery algorithm, we show that we can recover the polynomial and the inputs within a few seconds, thereby demonstrating an attack on the protocol of Kesarwani et al

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

Recent Advances in Industrial and Applied Mathematics

This open access book contains review papers authored by thirteen plenary invited speakers to the 9th International Congress on Industrial and Applied Mathematics (Valencia, July 15-19, 2019). Written by top-level scientists recognized worldwide, the scientific contributions cover a wide range of cutting-edge topics of industrial and applied mathematics: mathematical modeling, industrial and environmental mathematics, mathematical biology and medicine, reduced-order modeling and cryptography. The book also includes an introductory chapter summarizing the main features of the congress. This is the first volume of a thematic series dedicated to research results presented at ICIAM 2019-Valencia Congress

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Knowledge, Representation, and the Physical World

This dissertation answers how mathematical representations enable knowledge of physical systems. Contemporary responses rely on matching the properties of physical systems to properties in mathematical models, arguing that such matching allows scientists to successfully draw conclusions about physical systems through the inspection of their models. We argue that such “matching accounts” cannot adapt to the routine mismatching pervasive in physical theories. These mismatching problems arise both when idealized models match some “similar” but better behaved potential physical system, and in cases we classify as pathological idealization, where the models employed must satisfy constraints that could not possibly be matched by realistic physical systems (e.g. requiring an infinite particle number or infinite density). In the latter cases such pathological constraints can also lead to incompatibilities with the governing laws of the physical theory. Despite such pathologies, conclusions drawn with these representations seem to enable improved understanding and empirically confirmable knowledge of the studied physical systems. To address this dichotomy, we develop a novel condition of successful mathematical representation, called epsilon-fidelity, under which mismatched models may facilitate knowledge of realistic physical systems. Arguing against direct matching, we propose that representations can meet the conditions of epsilon-fidelity by establishing a manifold of associations between topological neighborhoods of mathematical models and clusters of relevantly similar physical systems. We then demonstrate that this shift in the scope of representation relationships explains how suitably similar models entail conclusions about the relevant systems while avoiding the problems of individual model to system mismatching. As a signature case study, we investigate Einstein’s canonical interpretation of the geodesic principle, originally proposed to govern how gravitating bodies travel according to general relativity theory. We argue that under the canonical interpretation models of bodies must either meet unrealistic assumptions or violate the theory’s fundamental field equations, marking them as pathological idealizations. To recover the principle, we reinterpret geodesic dynamics as a universality thesis about the collective behavior of certain classes of systems, explaining how this reinterpretation satisfies the epsilon-fidelity criteria and can be used to gain knowledge about the observable motion of actual classes of gravitating bodies

Dynamics under Uncertainty: Modeling Simulation and Complexity

The dynamics of systems have proven to be very powerful tools in understanding the behavior of different natural phenomena throughout the last two centuries. However, the attributes of natural systems are observed to deviate from their classical states due to the effect of different types of uncertainties. Actually, randomness and impreciseness are the two major sources of uncertainties in natural systems. Randomness is modeled by different stochastic processes and impreciseness could be modeled by fuzzy sets, rough sets, Dempster–Shafer theory, etc

An Investigation on Holomorphic vector Bundles and Krichever-Lax matrices over an Algebraic curve

The work by N. Hitchin in 1987 opened a good possibility of describing the cotangent bundle of the moduli space of stable vector bundles over a compact Riemann surface in an explicit way. He proved that the space can be foliated by a family of certain spaces, i.e., the Jacobi varieties of spectral curves. The main purpose of this dissertation is to make the realization of the Hitchin system in a concrete way in the method initiated by I. M. Krichever and to give the necessary and sufficient condition for the linearity of flows in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work by P. A. Griffiths

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions