General Polynomial Research Articles

Parallel Accelerating Number Theoretic Transform for Bootstrapping on a Graphics Processing Unit

The bootstrapping procedure has become the main bottleneck affecting the efficiency of all known fully homomorphic encryption (FHE) schemes. The state-of-the-art scheme for efficient bootstrapping, which is called fully homomorphic encryption over the torus (TFHE), accelerates polynomial multiplication by leveraging number theoretic transform (NTT) and implementing NTT in parallel on a GPU. Unfortunately, almost none of the recent advancements in NTT take full advantage of a GPU, leading to the need for more time. With this in mind, in this work, a novel faster number theoretic transform based on a GPU is proposed, in which matrix multiplication is used to implement a decomposed small-point NTT. When implementing matrix multiplication, we introduce a merging preprocessing method to merge multiple inputs of the small-point NTT, aiming to effectively minimize the count of modulo operations. Subsequently, when the merged result is multiplied by rotation factors, we use logical left shift rather than arithmetic multiplication to improve the computational efficiency. Our scheme can easily be used to realize a 1024-point NTT and the results of the experiments show that the speedup ratio of our method over the butterfly algorithm is about 2.49.

Hilbert's 13th problem for algebraic groups

The algebraic form of Hilbert's 13th problem asks for the resolvent degree \operatorname{RD}(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + \cdots + a_n of degree n , where a_1, \ldots, a_n are independent variables. The resolvent degree is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with \mathbb{C}(a_1, \ldots, a_n) and adjoining algebraic functions in \!{}\leqslant\nobreak d variables at each step. Recently Farb and Wolfson defined the resolvent degree \operatorname{RD}_k(G) for every finite group G and any base field k of characteristic 0 . In this setting \operatorname{RD}(n) = \operatorname{RD}_{\mathbb{C}}(\operatorname{S}_n) , where \operatorname{S}_n denotes the symmetric group. In this paper we extend their definition of \operatorname{RD}_k(G) to an arbitrary algebraic {group} G over an arbitrary field k . We investigate the dependency of this quantity on k and show that \operatorname{RD}_k(G) \leqslant 5 for any field k and any connected group G . The question whether \operatorname{RD}_k(G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.

Data-driven prediction of the surface layer state in hard-turning for optimization of component quality

Data-driven approaches are an effective solution for modeling problems in machining. To increase the service life of hard-turned components, it is important to quantify the correlation between the cutting parameters such as feed rate, cutting speed and depth of cut and the near-surface properties. For obtaining high-quality models with small data sets, different data-driven approaches are investigated in this contribution. Additionally, models that enable uncertainty quantification are crucial for effective decision-making and the adjustment of cutting parameters. Therefore, parametric multiple polynomial regression and Takagi–Sugeno models, as well as non-parametric Gaussian process regression as a Bayesian approach are considered and compared regarding their capability to predict residual stress and surface roughness values of 51CrV4 specimens after hard-turning. Moreover, a novel method based on optimization of data driven non-linear models is proposed that allows for identification of cutting parameter combinations, which at the same time lead to satisfactory surface roughness and residual stress states.

Power-series solutions of fractional-order compartmental models

Compartmental models based on coupled differential equations of fractional order have been widely employed in the literature for modeling. An abstraction of these models is given by a system with polynomial vector field. We investigate the use of power series for solving generic polynomial differential equations in any dimension, with Caputo fractional derivative. As is well known, power series convert a continuous formulation into a discrete system of difference equations, which are easily solved by recursion. The novelty of this paper is that we rigorously prove that the series converge on a neighborhood of the initial instant, which is an analogue of the Cauchy–Kovalevskaya theorem. Besides, these series are proved to be continuous with respect to the fractional index. For applications, a general-purpose symbolic implementation of truncated power series is developed, and its execution is illustrated for the fractional SIR epidemiological model.

Numerical evaluation of oscillatory integrals via automated steepest descent contour deformation

Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.

On Generalized Class of Bell Polynomials Associated with Geometric Applications

In this paper, we introduce a new class of special polynomials called the generalized Bell polynomials, constructed by combining two-variable general polynomials with two-variable Bell polynomials. The concept of the monomiality principle was employed to establish the generating function and obtain various results for these polynomials. We explore certain related identities, properties, as well as differential and integral formulas. Further, specific members within the generalized Bell family—such as the Gould-Hopper-Bell polynomials, Laguerre-Bell polynomials, truncated-exponential-Bell polynomials, Hermite-Appell-Bell polynomials, and Fubini-Bell polynomials—were examined, unveiling analogous outcomes for each. Finally, Mathematica was utilized to investigate the zero distributions of the Gould-Hopper-Bell polynomials.

Performance and Efficiency Exploration of Hardware Polynomial Multipliers for Post-Quantum Lattice-Based Cryptosystems

The significant effort in the research and design of large-scale quantum computers has spurred a transition to post-quantum cryptographic primitives worldwide. The post-quantum cryptographic primitive standardization effort led by the US NIST has recently selected the asymmetric encryption primitive Kyber as its candidate for standardization and indicated NTRU, as a valid alternative if intellectual property issues are not solved. Finally, a more conservative alternative to NTRU, NTRUPrime was also considered as an alternate candidate, due to its design choices that remove the possibility for a large set of attacks preemptively. All the aforementioned asymmetric primitives provide good performances, and are prime choices to provide IoT devices with post-quantum confidentiality services. In this work, we present a comprehensive exploration of hardware designs for the computation of polynomial multiplications, the workhorse operation in all the aforementioned cryptosystems, with a thorough analysis of performance, compactness and efficiency. The presented designs cope with the differences in the arithmetics of polynomial rings employed by distinct cryptosystems, benefiting from configurations and optimizations that are applicable at synthesis time and/or run time. In this context, we target a use case scenario where long-term key pairs are used, such as the ones for VPNs (e.g., over IPSec), secure shell protocols and instant messaging applications. Our high-performance design variants exhibit figures of latency comparable to the ones needed for the execution of the symmetric cryptographic primitives also included in the Post-Quantum schemes. Notably, the performance figures of the designs proposed for NTRU and NTRU Prime surpass the ones described in the related literature.

Norm of the general polynomial differentiation composition operator from the space of Cauchy transforms to the mth weighted‐type space on the unit disk

We introduce the general polynomial differentiation composition operator where , , , are holomorphic functions on the open unit disk and , , are holomorphic self‐maps of , calculate norm of the operator acting from the space of Cauchy transforms to the th weighted‐type space, and characterize its boundedness, as well as the boundedness of the operator acting from the space of Cauchy transforms to the little th weighted‐type space.

Linear integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document} spaces representing polynomial covariance type commutation relations

In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$AB=BF(A)$$\\end{document} by linear integral operators in Banach spaces Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_p$$\\end{document}. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.

A fast NTRU software implementation based on 5-way TMVP

The fast and efficient operation of post-quantum cryptographic algorithms has become an important research topic, especially with the recent developments and the process of the PQC Standardization Project by NIST. Various algorithms based on lattices are chosen to be finalists in this project. Number Theoretic Transform (NTT), Toom–Cook, Karatsuba, and Toeplitz matrix–vector product (TMVP) are considered in order to efficiently perform multiplications in quotient polynomial rings required by the cryptographic schemes based on lattices. In this paper, we propose a 5-way split TMVP-based algorithm for multiplication in polynomial quotient rings and determine its computational complexity. The new algorithm is derived from a 5-term polynomial multiplication algorithm using 13 multiplications with coefficients of only 1 or -1 that provide efficiency. We also implement it for all parameter sets of NTRU. The results show that we have up to 34%, 35%, and 157% speed-up against Toom4-Karatsuba implementation in key generation, encapsulation, and decapsulation, respectively.

О генерической сложности проблемы вычисления функции Эйлера

We study the generic complexity of the problem of the Euler function computation. This problem has important applications in modern cryptography. For example, the cryptographic strength of the famous public key encryption system RSA is based on the assumption of its hardness. We prove that under the condition of worst-case hardness and P = BPP there is no polynomial strongly generic algorithm for this problem. For a strongly generic polynomial algorithm, there is no efficient method for random generation of inputs on which the algorithm cannot solve the problem. Thus, this result justifies the application of the problem of computing the Euler function in public key cryptography. To prove this theorem, we use the method of generic amplification, which allows us to construct generically hard problems from the problems that are hard in the classical sense. The main feature of this method is the cloning technique, which combines the input data of a problem into sufficiently large sets of equivalent input data. Equivalence is understood in the sense that the problem is solved in a similar way for them.

On Polynomial Modular Number Systems over $ \mathbb{Z}/{p}\mathbb{Z} $

<p style='text-indent:20px;'>Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on particular binomials and trinomials. But these polynomials do not always provide systems with interesting characteristics such as small digits, fast reduction, etc.</p><p style='text-indent:20px;'>In this work, we study a larger family of polynomials that can be exploited to design a safe and efficient PMNS. To do so, we first state a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds. Then, we present classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.</p>

Генерическая NP-полнота проблем разрешимости систем уравнений над конечными группами, полугруппами и полями

In this paper we prove that problems of solvability of systems of equations over finite fields, non-Abelian finite groups and non-commutative finite monoids are complete with respect to generic polynomial reducibility in generic analogue of the NP class.

FPGA Implementation of Elliptic-Curve Point Multiplication Over GF(2233) Using Booth Polynomial Multiplier for Area-Sensitive Applications

FPGA Implementation of Elliptic-Curve Point Multiplication Over GF(2²³³) Using Booth Polynomial Multiplier for Area-Sensitive Applications

A High-Throughput Toom-Cook-4 Polynomial Multiplier for Lattice-Based Cryptography Using a Novel Winograd-Schoolbook Algorithm

A High-Throughput Toom-Cook-4 Polynomial Multiplier for Lattice-Based Cryptography Using a Novel Winograd-Schoolbook Algorithm

LTE: Lightweight and Timing-Efficient Unequal-Sized Polynomial Multiplication Accelerators

LTE: Lightweight and Timing-Efficient Unequal-Sized Polynomial Multiplication Accelerators

SCOPE: Schoolbook-Originated Novel Polynomial Multiplication Accelerators for NTRU-Based PQC

SCOPE: Schoolbook-Originated Novel Polynomial Multiplication Accelerators for NTRU-Based PQC

Quantum algorithm for lattice Boltzmann (QALB) simulation of incompressible fluids with a nonlinear collision term

We present a full quantum algorithm for the lattice Boltzmann method for simulating fluid flows, the only such algorithm to implement both the streaming and collision substeps as unitary operators using an efficient number of qubits. We use Hamiltonian simulation as the main route underlying the algorithm, and show that it uses qubits that scale logarithmically in the flow Reynolds number but gates that scale only polynomially. We motivate the discussion by a brief overview of existing attempts at simulating classical fluids on quantum computers and present a pedagogical discussion on assigning quantum operators to classical variables in both the streaming and the collision substeps, after highlighting the incompatibility of the latter with the implementations of the streaming step that exists in the literature. We use the Bhatnagar–Gross–Krook ansatz for the collision term, representing the relaxation toward an equilibrium distribution. For nonlinear collisions, we use Kowalski's framework that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the Fock space of the bosons. We present the qubit and gate complexities, in terms of the chosen accuracy and the Reynolds number. In the Appendix, we work out the details of implementing the operators of the truncated bosonic Fock space in terms of single-qubit gates as well as the error scaling for a general polynomial driving function.

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

Chance-Constrained Optimization Based PV Hosting Capacity Calculation Using General Polynomial Chaos

On double cyclic codes over $ \mathbb{Z}_2+u\mathbb{Z}_2 $

<abstract><p>In this paper, we introduced double cyclic codes over $ R^r\times R^s $, where $ R = \mathbb{Z}_{2}+u\mathbb{Z}_{2} = \{0, 1, u, 1+u\} $ is the ring with four elements and $ u^2 = 0 $. We first determined the generator polynomials of $ R $-double cyclic codes for odd integers $ r $ and $ s $, then gave the generators of duals of free double cyclic codes over $ R^r\times R^s $. By defining a linear Gray map, we looked at the binary images of $ R $-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from $ R $-double cyclic codes. Moreover, we studied self-dual $ R $-double cyclic codes and presented an example of a self-dual $ R $-double cyclic code.</p></abstract>