Integer Inputs Research Articles

The Multiple Millionaires' Problem: New Algorithmic Approaches and Protocols

We study a fundamental problem in Multi-Party Computation, which we call the Multiple Millionaires Problem (MMP). Given a set of private integer inputs, the problem is to identify the subset of inputs that equal the maximum (or minimum) of that set, without revealing any further information on the inputs beyond what is implied by the desired output. Such a problem is a natural extension of the Millionaires Problem, which is the very first Multi-Party Computation problem that was presented in Andrew Yaos seminal work (FOCS 1982). A closely related problem is MaxP, in which the value of the maximum is sought. We study these fundamental problems and describe several algorithmic approaches and protocols for their solution. In addition, we compare the performance of the protocols under several selected settings. As applications of privacy-preserving computation are more and more commonly implemented in industrial systems, MMP and MaxP become important building blocks in privacy-preserving statistics, machine learning, auctions and other domains. One of the prominent advantages of the protocols that we present here is their simplicity. As they solve fundamental problems that are essential building blocks in various application scenarios, we believe that the presented solutions to those problems, and the comparison between them, will serve well future researchers and practitioners of secure distributed computing.

Multi-LRA: Multi logical residual architecture for spiking neural networks

Recently, it has been noticed that certain current state-of-the-art (SOTA) spiking neural networks incorporated non-spike data by use of residual connections and count coding. However, these architectures may not be well-suited for the neuromorphic chips optimized primarily for processing binary inputs and floating-point weights. On the other hand, some pure-spike structures, such as SEW-AND and SEW-IAND, have been found to exhibit spike vanishing and limited capacity. To overcome these shortcomings, we analyze the energy consumption and expressive ability of binary and integer inputs on neuromorphic chips, and then propose a new Multi Logical Residual Architecture (Multi-LRA), which eliminates non-spike computation on neuromorphic chips and addresses the issues of spike vanishing and limited capacity in SEW-AND and SEW-IAND. Furthermore, we prove that the upper bound of conditional entropy of Multi-LRA is higher than that of SEW-AND and logical-OR, which means that Multi-LRA has better capacity and may avoid the spike vanishing. Finally, experiments on CIFAR-10 have shown that Multi-LRA can significantly reduce energy consumption because of its pure-spike computation and logical operation, where Multi-LRA-ResNet50 can reach the SOTA accuracy 96.27% in just 4 time steps. In addition, the effectiveness of Multi-LRA has been validated on ImageNet, CIFAR10-DVS and DVS128 Gesture.

Encrypted Model Predictive Control of a Nonlinear Chemical Process Network

This work focuses on developing and applying Encrypted Lyapunov-based Model Predictive Control (LMPC) in a nonlinear chemical process network for Ethylbenzene production. The network, governed by a nonlinear dynamic model, comprises two continuously stirred tank reactors that are connected in series and is simulated using Aspen Plus Dynamics. For enhancing system cybersecurity, the Paillier cryptosystem is employed for encryption–decryption operations in the communication channels between the sensor–controller and controller–actuator, establishing a secure network infrastructure. Cryptosystems generally require integer inputs, necessitating a quantization parameter d, for quantization of real-valued signals. We utilize the quantization parameter to quantize process measurements and control inputs before encryption. Through closed-loop simulations under the encrypted LMPC scheme, where the LMPC uses a first-principles nonlinear dynamical model, we examine the effect of the quantization parameter on the performance of the controller and the overall encryption to control the input calculation time. We illustrate that the impact of quantization can outweigh those of plant/model mismatch, showcasing this phenomenon through the implementation of a first-principles-based LMPC on an Aspen Plus Dynamics process model. Based on the findings, we propose a strategy to mitigate the quantization effect on controller performance while maintaining a manageable computational burden on the control input calculation time.

Integer polynomial recovery from outputs and its application to cryptanalysis of a protocol for secure sorting

AbstractWe investigate the problem of recovering integer inputs (up to an affine scaling) when given only the integer monotonic polynomial outputs. Givennninteger outputs of a degree-ddinteger monotonic polynomial whose coefficients and inputs are integers within known bounds andn≫dn\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 securekk-nearest neighbor 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. By 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.

Theory of binary-valued data envelopment analysis: an application in assessing the sustainability of suppliers

PurposeThe objective of this study is to present a binary-valued data envelopment analysis (DEA) theory. The authors’ proposed approach, for the first time, combines binary-valued and integer-valued theories concurrently in the DEA context. To do so, new production possibility sets (PPSs) with some distinguished features are developed.Design/methodology/approachThe authors address integer inputs and outputs in the proposed approach by introducing a new PPS.FindingsTo take into account the binary data, the authors develop axiomatic DEA principles. The binary production principles guarantee any combination of convexity and feasibility. Furthermore, the authors develop a new DEA model to consider integer and real data. A case study is presented to show the usefulness of the developed models. Using the proposed models, the authors obtained benchmarks to solve the sustainable supplier selection problems.Originality/value(1) For the first time, binary-valued and integer-valued theories are presented in an integrated DEA model. (2) To deal with the pure binary data, a new PPS is proposed. (3) To consider real, integer and binary data, a new PPS is introduced. (4) New technologies are developed to propose feasible solutions. (5) The proposed models can project inefficient decision-making units (DMUs) on efficiency frontier given binary, integer and real data. (6) A case study is given for the performance evaluation of sustainable suppliers.

Random Kernel Forests

Random forests of axis-parallel decision trees still show competitive accuracy in various tasks; however, they have drawbacks that limit their applicability. Namely, they perform poorly for multidimensional sparse data that is frequently input in many text and image processing tasks. A straightforward solution is to create forests of decision trees with oblique splits; however, most training approaches have low performance or are hard to tune. Besides, those ensembles appear unstable and easily overfit, so they should be regularized to find the trade-off between complexity and generalization. This paper proposes an algorithm to train kernel decision trees and random forests. At the top level, this algorithm follows a common greedy procedure to train binary decision trees; however, it produces quasi-optimal linear and kernel splits at the decision stump level. At each stump, the algorithm finds a quasi-optimal distribution of classes to subtrees and trains this stump via optimization of an SVM-like loss function with a margin re-scaling approach, which helps simultaneously optimize the margin between subtree data and arbitrary impurity criteria. We also try to reveal uniform stability-based generalization bounds for those forests and apply them to select the regularization technique. The obtained bounds explicitly consider primary hyperparameters of forests, trees, and decision stumps. The bounds also show a relationship between outliers in decision tree structure, stability and generalization, and how forests smooth these outliers. We performed an experimental evaluation on several tasks, such as studying the reaction of social media users, image recognition, and bank scoring. The experimental results show that the proposed algorithm trains ensembles, which outperform other oblique or kernel forests in many commonly-recognized datasets. Namely, the proposed algorithm shows 99.1% accuracy on MNIST and 58.1% on CIFAR-10. It has been confirmed that the selected regularization technique helps reduce overfitting on several datasets. Therefore, the proposed algorithm may be considered a small step toward customized and regularized ensembles of oblique or kernel trees that keep reasonable training performance on large datasets.We believe that the proposed algorithmcan be utilized as a construction element of approaches to training context-aware models on complex problems with integer inputs and outputs, such as text mining and image recognition.

Integer-Wise Functional Bootstrapping on TFHE: Applications in Secure Integer Arithmetics

TFHE is a fast fully homomorphic encryption scheme proposed by Chillotti et al. in Asiacrypt’ 2018. Integer-wise TFHE is a generalized version of TFHE that can encrypt the plaintext of an integer that was implicitly presented by Chillotti et al., and Bourse et al. presented the actual form of the scheme in CRYPTO’ 2018. However, Bourse et al.’s scheme provides only homomorphic integer additions and homomorphic evaluations of a sign function. In this paper, we construct a technique for operating any 1-variable function in only one bootstrapping of the integer-wise TFHE. For applications of the scheme, we also construct a useful homomorphic evaluation of several integer arithmetics: division, equality test, and multiplication between integer and binary numbers. Our implementation results show that our homomorphic division is approximately 3.4 times faster than any existing work and that its run time is less than 1 second for 4-bit integer inputs.

A Bicriteria Approach for Saving a Path Maximizing Dynamic Contraflow

The maximum dynamic contraflow problem in transportation networks seeks to maximize the flow from a source to a sink within a given time horizon with a possibility of arc reversals. This may result into blockage of paths of desired length from some node of the network towards the source. In some cases such as the evacuation planning, we may require a path towards the source to move some facilities, for example, emergency vehicles. In this work, we model the problem of saving such a path as a bicriteria optimization problem which minimizes the length of the path and maximizes the dynamic flow with arc reversals. We use the [Formula: see text]-constraint approach to solve the problem and propose a procedure that gives the set of all Pareto optimal solutions in a single-source-single-sink network with integer inputs. We also present computational performance of the algorithm on a road network of Kathmandu city, and on randomly generated networks. The results are of both theoretical and practical importance.

Using submodularity within column generation to solve the flight-to-gate assignment problem

In this paper, we provide a column generation-based approach for solving the airport flight-to-gate assignment problem, where the goal is to minimize the on-ground portion of arrival delays by optimally assigning each scheduled flight to a compatible gate. Specifically, we use a set covering formulation for the master problem and decompose the pricing problem such that each gate is the basis for an independent pricing problem to be solved for assignment patterns with negative reduced costs. We use a combination of an approximation algorithm based on the submodularity of the underlying set and dynamic programming algorithms to solve the independent pricing problems. To the best of our knowledge, this is the first use of submodularity property to efficiently solve pricing problems and improve the performance of column generation algorithm. We show that the dynamic programming algorithm is pseudo-polynomial when there are integer inputs. We also design and employ a rolling horizon method and block decomposition algorithm to solve large-sized instances. Finally, we perform extensive computational experiments to validate the performance of our approach.

Reinforcement Learning for mixed-integer problems based on MPC

Model Predictive Control has been recently proposed as policy approximation for Reinforcement Learning, offering a path towards safe and explainable Reinforcement Learning. This approach has been investigated for Q-learning and actor-critic methods, both in the context of nominal economic MPC and Robust (N)MPC, showing very promising results. In that context, actor-critic methods seem to be the most reliable approach. Many applications include a mixture of continuous and integer inputs, for which the classical actor-critic methods need to be adapted. In this paper, we present a policy approximation based on mixed-integer MPC schemes, and propose a computationally inexpensive technique to generate exploration in the mixed-integer input space that ensures a satisfaction of the constraints. We then propose a simple compatible advantage function approximation for the proposed policy, that allows one to build the gradient of the mixed-integer MPC-based policy.

Fast constant-time gcd computation and modular inversion

This paper introduces streamlined constant-time variants of Euclid’s algorithm, both for polynomial inputs and for integer inputs. As concrete applications, this paper saves time in (1) modular inversion for Curve25519, which was previously believed to be handled much more efficiently by Fermat’s method, and (2) key generation for the ntruhrss701 and sntrup4591761 lattice-based cryptosystems.

Threesomes, Degenerates, and Love Triangles

The 3SUM problem is to decide, given a set of n real numbers, whether any three sum to zero. It is widely conjectured that a trivial O ( n 2 )-time algorithm is optimal on the Real RAM, and optimal even in the nonuniform linear decision tree model. Over the years the consequences of this conjecture have been revealed. This 3SUM conjecture implies Ω ( n 2 ) lower bounds on numerous problems in computational geometry, and a variant of the conjecture for integer inputs implies strong lower bounds on triangle enumeration, dynamic graph algorithms, and string matching data structures. In this article, we refute the conjecture that 3SUM requires Ω ( n 2 ) in the Real RAM and refute more forcefully the conjecture that its complexity is Ω ( n 2 ) in the linear decision tree model. In particular, we prove that the decision tree complexity of 3SUM is O ( n 3/2 √ log n ) and give two subquadratic 3SUM algorithms, a deterministic one running in O ( n 2 / (log n / log log n ) 2/3 ) time and a randomized one running in O ( n 2 (log log n ) 2 / log n ) time with high probability. Our results lead directly to improved bounds on the decision tree complexity of k -variate linear degeneracy testing for all odd k ≥ 3. Finally, we give a subcubic algorithm for a generalization of the (min ,+)-product over real-valued matrices and apply it to the problem of finding zero-weight triangles in edge-weighted graphs. We give a depth- O ( n 5/2 √ log n ) decision tree for this problem, as well as a deterministic algorithm running in time O ( n 3 (log log n ) 2 /log n ).

Mitigating Silent Data Corruptions in Integer Matrix Products: Toward Reliable Multimedia Computing on Unreliable Hardware

The generic matrix multiply (GEMM) routine comprises the compute and memory-intensive parts of many information retrieval, machine learning, and object recognition systems that process integer inputs. Therefore, it is of paramount importance to ensure that integer GEMM computations remain robust to silent data corruptions (SDCs), which stem from accidental voltage or frequency overscaling, or other hardware nonidealities. In this paper, we introduce a new method for SDC mitigation based on the concept of numerical packing. The key difference between our approach and all existing methods is the production of redundant results within the numerical representation of the outputs, rather than as a separate set of checksums. Importantly, unlike well-known algorithm-based fault tolerance approaches for GEMM, the proposed approach can reliably detect the locations of the vast majority of all possible SDCs in the results of GEMM computations. An experimental investigation of voltage-scaled integer GEMM computations for visual descriptor matching within state-of-the-art image and video retrieval algorithms running on an Intel i7-4578U 3 GHz processor shows that SDC mitigation based on numerical packing leads to comparable or lower execution and energy-consumption overhead in comparison with all other alternatives.

A Computationally Efficient Model Predictive Control Strategy for Linear Systems With Integer Inputs

For linear systems with integer inputs, the model predictive control problem with output reference tracking is formulated as an integer least-squares (ILS) problem. The ILS problem is solved using a modified sphere decoding algorithm, which is a particular branch-and-bound method. To reduce the computational complexity of the sphere decoder, a reduction algorithm is added as a preprocessing stage to reshape the search space in which the integer solution lies. The computational complexity of the proposed algorithm is modest, enabling its implementation in a real-time system even when considering long prediction horizons. A variable-speed drive system with a three-level voltage source inverter serves as an illustrative example to demonstrate the effectiveness of the proposed algorithm.

Lower bounds for monotone counting circuits

A monotone arithmetic circuit computes a given multivariate polynomial f if its values on all nonnegative integer inputs are the same as those of f. The circuit countsf if this holds for 0–1 inputs; on other inputs, the circuit may output arbitrary values. The circuit decidesf if it has the same 0–1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and that some polynomials can be exponentially easier to decide than to count them. Our main results are general lower bounds on the size of counting circuits.

A Novel Approach of Equivalence Class Partitioning for Numerical Input

Software testing is an indispensible phase of software development that supports not only productivity but also quality assurance from small to large scale software products. Software Systems essentially need an effective way to develop a strategy for testing various aspects of the system. When we are testing software the first and potentially most critical step is to design test cases. There are many methods which assist test case design. This paper will document the approach known as Equivalence class partitioning. We have partitioned the input into two classes Even and Odd. The proposed approach works for integer inputs only. Partitioning is done on basis of assessment of input conditions that serves as a set of valid as well as invalid set of input values for a test case

A Note on the Complexity of Comparing Succinctly Represented Integers, with an Application to Maximum Probability Parsing

The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs. Input instance: Four lists of positive integers: a 1,..., an ∈N+ n ; b 1,..., bn ∈N+ n ; c 1,..., cm ∈N+ m ; d 1, ..., dm ∈N+ m ; where each of the integers is represented in binary. Problem 1 (equality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn = c 1 d 1 c 2 d 2⋯ cmdm . Problem 2 (inequality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn ≥ c 1 d 1 c 2 d 2⋯ cmdm . Problem 1 is easily decidable in polynomial time using a simple iterative algorithm. Problem 2 is much harder. We observe that the complexity of Problem 2 is intimately connected to deep conjectures and results in number theory. In particular, if a refined form of the ABC conjecture formulated by Baker in 1998 holds, or if the older Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in logarithms holds, then Problem 2 is decidable in P-time (in the standard Turing model of computation). Moreover, it follows from the best available quantitative bounds on linear forms in logarithms, namely, by Baker and Wüstholz [1993] or Matveev [2000], that if m and n are fixed universal constants then Problem 2 is decidable in P-time (without relying on any conjectures). This latter fact was observed earlier by Shub [1993]. We describe one application: P-time maximum probability parsing for arbitrary stochastic context-free grammars (where ε -rules are allowed).

A new robust mixed integer-valued model in DEA

The mixed integer linear programming (MILP) models are proposed to estimate the performance of decision making units (DMUs) including both integer and real values in data envelopment analysis (DEA). There are several studies to propose MILPs in the literature of DEA; however, they have some major shortcomings unfortunately. This study firstly mentioned the shortcomings in the previous researches and secondly suggests a robust MILP based on the Kourosh and Arash Method (KAM). The proposed linear model, integer-KAM (IKAM), has almost all capabilities of the linear KAM and significantly removes the shortcomings in the current MILPs. For instance, IKAM benchmarks and ranks all technically efficient and inefficient DMUs at the same time. It detects outliers, and estimates the production frontier significantly. A numerical example of 39 Spanish airports with four integer inputs and three outputs including two integer values and a real value also represents the validity of the statements.

A mixed-integer model predictive control formulation for linear systems

Since their inception in the early 1980s industrial model predictive controllers (MPC) rely on continuous quadratic programming (QP) formulations to derive their optimal solutions. More recent advances in mixed-integer programming (MIP) algorithms show that MIP formulations have the potential of being advantageously applied to the MPC problem. In this paper, we present an MIP formulation that can overcome difficulties faced in the practical implementation of MPCs. In particular, it is possible to set explicit priorities for inputs and outputs, define minimum moves to overcome hysteresis, and deal with digital or integer inputs. The proposed formulation is applied to simulated process systems and the results compared with those achieved by a traditional continuous MPC. The solutions of the resulting mixed-integer quadratic programming (MIQP) problems are derived by a computer implementation of the Outer Approximation method (OA) also developed as part of this work.

An FPGA realization of simplified turbo decoder architecture

The key issue of applying Turbo codes is to find an efficient implementation of turbo decoder. This paper addresses the implementation of a simplified and efficient turbo decoder in field programmable gate array (FPGA) technology. A simplified and efficient implementation of a Turbo decoder with minor performance loss has been proposed. An integer Turbo decoder based on the standard 2’s complement number system after considering the issues of dynamic range, truncation effect and other algorithm related subjects has been introduced. The efficient implementation comes from algorithm modification, integer arithmetic and compact hardware management. Based on the Max-Log-MAP decoding algorithm, the branch metric is modified by weighting a priori value, resulting in a significant BER improvement. The Turbo decoder takes in 8-level integer inputs generates 7-bit soft-decisions and calculates all metrics on integers, avoiding complex floating point or fixed-point arithmetic. By manipulating memory address, delay associated with interleaving and de-interleaving is eliminated, resulting in much higher throughput. Also, by taking advantage of identical decoder function, Turbo decoder is implemented in a single-decoder structure, making efficient use of memory and logic cells. Key words: Turbo, Max-Log-MAP, field programmable gate array, bit error ratio.