Modular Multiplication Research Articles

Subdivision algorithms with modular arithmetic

We study the de Casteljau subdivision algorithm for Bezier curves and the Lane-Riesenfeld algorithm for uniform B-spline curves over the integers mod m, where m>2 is an odd integer. We place the integers mod m evenly spaced around a unit circle so that the integer k mod m is located at the position on the unit circle ate2πki/m=cos(2kπ/m)+isin(2kπ/m)↔(cos(2kπ/m),sin(2kπ/m)).Given a sequence of integers (s0,…,sm)mod m, we connect consecutive values sjsj+1 on the unit circle with straight line segments to form a control polygon. We show that if we start these subdivision procedures with the sequence (0,1,…,m)mod m, then the sequences generated by these recursive subdivision algorithms spawn control polygons consisting of the regular m-sided polygon and regular m-pointed stars that repeat with a period equal to the minimal integer k such that 2k=±1modm. Moreover, these control polygons represent the eigenvectors of the associated subdivision matrices corresponding to the eigenvalue 2−1modm. We go on to study the effects of these subdivision procedures on more general initial control polygons, and we show in particular that certain control polygons, including the orbits of regular m-sided polygons and the complete graphs of m-sided polygons, are fixed points of these subdivision procedures.

A new approach to implement stream cipher using memristor-based Chua’s circuit for secure communication

The advancement in cryptographic algorithms is the primary need for modern secure communication systems. In secure communication system, the stream cipher serves the basis for cryptographic algorithms. The properties of cryptographic algorithms are often analyzed on the basis of randomness involved in implementing cipher texts while complexity of the hardware is also taken care of. Therefore, the main objective of this paper is to propose a new stream cipher scheme having higher complexity (high degree of randomness) while keeping the hardware as simple as possible. The proposed stream cipher algorithm gives the new direction to develop the highly secure and simple models in communication field. The proposed scheme offers the benefit of adjusting variable-length key(s) according to the variable length of plaintext using modular arithmetic and memristor—based chaotic behavioral Chua’s circuit. To explain the encryption–decryption process at transmitter and receiver ends, the Devanagari script has been taken. Although, any other script or data type is supported by the proposed algorithm. In addition, the synchronization is also achieved without having any requirement of additional process.

Knots with Composite Colors

Summary The technique of distinguishing one knot from another by coloring arcs and applying some basic modular arithmetic is part of most standard undergraduate knot theory classes. When we study n-colorability, we are usually only interested when n is a prime number. But what if n is composite? What can we say then?

Extended Diffie Hellman Protocol Based on Exponential and Multiplication Operation Over Modular Arithmetic

Extended Diffie Hellman Protocol Based on Exponential and Multiplication Operation Over Modular Arithmetic

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>

Extended Diffie-Hellman protocol based on exponential and multiplication operations over modular arithmetic

Extended Diffie-Hellman protocol based on exponential and multiplication operations over modular arithmetic

Structures and applications of graphs arising from congruences over moduli

<p>For any positive integer $ \mathrm{n} $, let $ \mathrm{M_{p}} $ contain the prime numbers less than $ \mathrm{n} $. Assuming $ \mathrm{M_{p}} $ as the set of moduli, we draw a graph with the vertex set $ \{1, 2, 3, \cdots, \mathrm{n}\} $, and an edge will be built between the vertices $ \mathrm{p} $ and $ \mathrm{q} $ if and only if $ \mathrm{p}\equiv (\mathrm{q}\; mod\; \mathrm{m}) $ for some $ \mathrm{m}\in $ $ \mathrm{M_{p}}. $ We call this graph a prime congruence simple graph and label this graph as $ \mathrm{G}(\mathrm{n}, \mathrm{M}_{p}) $. The objective of this work is to characterize prime congruence simple graphs, and afterwards, by utilizing these graphs, solutions to the system of linear congruences are suggested and demonstrated by applying modular arithmetic. It is shown that this graph is always a connected graph. The generalized formulae for vertex degrees, size, chromatic number, domination number, clique number, and eccentricity of the prime congruence simple graphs are proposed and proved. Also, independence numbers as well as a covering number for the proposed graph through vertices and edges are evaluated. Lastly, as an application of prime congruence simple graphs, the solution of a system of linear congruences is discussed in terms of the degrees of the vertices.</p>

Integer Modular Multiplication With Barrett Reduction and Its Variants for Homomorphic Encryption Applications: A Comprehensive Review and an Empirical Study

Integer Modular Multiplication With Barrett Reduction and Its Variants for Homomorphic Encryption Applications: A Comprehensive Review and an Empirical Study

PaReNTT: Low-Latency Parallel Residue Number System and NTT-Based Long Polynomial Modular Multiplication for Homomorphic Encryption

PaReNTT: Low-Latency Parallel Residue Number System and NTT-Based Long Polynomial Modular Multiplication for Homomorphic Encryption

Analysis of the Utility of Chinese Traditional Culture in Contemporary Digital Media Communication Based on a Big Data Platform

Abstract Chinese traditional culture is profound and has important historical and cultural values, and in the digital information era, the dissemination of Chinese traditional culture is facing new challenges and opportunities. This paper begins with the utility mechanism of Chinese traditional culture dissemination and develops a path for Chinese traditional culture dissemination using digital media. On this basis, a digital platform for Chinese traditional culture is established through the modular division of digital platforms and joint visual and interaction design. Then, the hierarchical analysis method (AHP) is combined with (PCA) to complete the weighting. Then, the subjective and objective weights are balanced by the information entropy method (IE). Finally, the comprehensive evaluation model of the utility of digital communication in Chinese traditional culture is constructed by combining with the method of the distance between the superiority and inferiority solutions (TOPSIS), and the evaluation index system of the utility of cultural communication is established. Among them, the benefit value has the highest weight of 0.2007. The other indicators in order of weight are cultural, emotional experience (0.1735), cultural content experience (0.1687), interactive experience (0.1533), social impact (0.1531), and digital function experience (0.1507). Finally, the communication effect of traditional Chinese culture is analyzed using the digital platform of this paper. The average score is 8.75, with excellent communication effects. Based on the utility analysis of digital media communication, this paper provides a digital path for the communication of Chinese traditional culture and establishes a comprehensive evaluation method of communication utility, which provides an effective research method for the utility of digital media communication of traditional culture.

On the Exponential Diophantine Equation 27^x – 11^y = z^2

In this article, we study and establish the theorem of the exponential Diophantine equation 27^x -11^y = z^2 , which has exactly two solutions where x, y and z are non-negative integers using Catalan’s conjecture, modular arithmetic, factorial method and elementary mathematical concepts.

High-Performance Implementation of a 1024-bit Full-Word Montgomery Modular Multiplier for High-Speed Encryption Hardware Circuit

High-Performance Implementation of a 1024-bit Full-Word Montgomery Modular Multiplier for High-Speed Encryption Hardware Circuit

High-performance montgomery modular multiplier with NTT and negative wrapped convolution

Modular multiplication plays a crucial role in modern cryptography. Montgomery modular multiplication(MMM), one of the most classic and practical modular multiplication algorithms, has been widely used in cryptographic algorithms such as RSA, Diffie–Hellman algorithm, and Elliptic Curve Cryptography. In this paper, we incorporate negative wrapped convolution (NWC) into the FFT-based Montgomery modular multiplication to avoid the issue of zero-padding and use carry-save arithmetics for parallel computation. By utilizing coefficient pairs (pos_part and neg_part), we reconstruct the final result and eliminate the restrictions imposed by nega-cyclic parts. Moreover, Karatsuba-like algorithm is introduced for building fine-grained large integer multipliers. We have modified the parameter specifications for our design to meet requirements from diverse application scenarios. We implement the design on Xilinx Virtex-7 FPGA under different conditions and compare the results with the state-of-the-art MMM designs. The comparisons confirm that our design has the following characteristics: low latency for process, competitive area-latency-product(ALP), efficient DSP usage, and constant delay, which enhances security against timing attacks.

High-performance unified modular multiplication algorithm and hardware architecture over G(2m)

The finite field Modular Multiplier (MM) over GF (2m) based on the National Institute of Standards and Technology (NIST) recommended polynomials can be used as a critical component in many cryptosystems. But much of the research suffers from the problems of only supporting a single curve or being very inefficient. In this paper, we present a KA-based unified algorithm (hybrid field size) and its corresponding architecture for high-performance implementation of the systolic modular multiplier over GF (2m), which can support all the NIST recommended polynomials. A number of efficient techniques have been explored and used to realize a high-performance implementation of the multiplier. First, we propose a three-term KA based MM approach and make it fit for systolic implementation. Second, we propose a configurable, unified irreducible polynomial that can support all five curves recommended by NIST and the corresponding modular reduction method. Third, the corresponding systolic structure is also designed in accordance with (1) and (2) to improve efficiency, and we adopt register sharing and computation unit sharing methods to reduce the space complexity of the proposed structure. Additionally, parallel computation of two163-width MMs is employed to improve resource utilization further. From the synthesis results on ASIC, it is shown that the proposed multipliers have significantly lower latency and higher performance than the existing designs. e.g., the proposed structure could achieve up to a 73 % reduction in area-delay product (ADP) over the existing structures.

Minimizing CNOT-count in quantum circuit of the extended Shor’s algorithm for ECDLP

The elliptic curve discrete logarithm problem (ECDLP) is a popular choice for cryptosystems due to its high level of security. However, with the advent of the extended Shor’s algorithm, there is concern that ECDLP may soon be vulnerable. While the algorithm does offer hope in solving ECDLP, it is still uncertain whether it can pose a real threat in practice. From the perspective of the quantum circuits of the algorithm, this paper analyzes the feasibility of cracking ECDLP using an ion trap quantum computer with improved quantum circuits for the extended Shor’s algorithm. We give precise quantum circuits for extended Shor’s algorithm to calculate discrete logarithms on elliptic curves over prime fields, including modular subtraction, three different modular multiplication, and modular inverse. Additionally, we incorporate and improve upon windowed arithmetic in the circuits to reduce the CNOT-counts. Whereas previous studies mostly focused on minimizing the number of qubits or the depth of the circuit, we focus on minimizing the number of CNOT gates in the circuit, which greatly affects the running time of the algorithm on an ion trap quantum computer. Specifically, we begin by presenting implementations of basic arithmetic operations with the lowest known CNOT-counts, along with improved constructions for modular inverse, point addition, and windowed arithmetic. Next, we precisely estimate that, to execute the extended Shor’s algorithm with the improved circuits to factor an n-bit integer, the CNOT-count required is 1237n3/logn+2n2+n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1237n^3/\\log n+2n^2+n$$\\end{document}. Finally, we analyze the running time and feasibility of the extended Shor’s algorithm on an ion trap quantum computer.

Korselt Numbers Through Computational Algorithms

The Korselt numbers and sets were discussed for the first time in 2007. The problem can be considered as a new one with limited literature making it as a new field of research. Let N be a positive integer and α a non-zero integer. If N ≠ α and p divides N for each prime divisor p of N, then N is called an α−Korselt number (Kα-number). In this thesis, many concepts such as Korselt numbers that are related to Carmichael numbers have been studied. Korselt determined Korselt numbers by studying the converse of Fermat’s Little theorem and it can be noticed that all these numbers depend on number theory, prime numbers, divisibility and modular arithmetic. To validate the concerned theorems, an illustrated proofs were followed through detailed steps in addition to many examples are solved in order to support the correctness of these theories. It is important to say that some errors in literature were addressed by us. Consequently, we introduced proper corrections for them. Finally, many notes have been taken and directed us to build and develop a number of complicated algorithms, some of them in order to find Korselt sets for relatively large numbers in an effective way in a short time which may require a great time and need tedious effort if it is to be calculated manually

Abacus Algorithms: A Pure Mathematical Approach to Ancient Calculation Tools

The abacus is one of the oldest calculating tools still in use today. Despite its simplicity, the bead-based interface allows users to conduct complex mathematical operations through a system of sliding beads along wires or rods. While the physical abacus itself provides an intuitive and visual approach to calculation, the underlying operations rely on fundamental mathematical principles. This paper provides a comprehensive mathematical framework that formally describes the algorithms behind abacus calculations. Beginning with basic abacus configuration, we define key components like rods, beads, and bead values required to model abacus states. We then characterize the core abacus algorithms for addition, subtraction, multiplication, and division through set notation, recurrence relations, and state transition diagrams. Our formalized abacus algorithms leverage concepts from number theory, modular arithmetic, combinatorics, and algebra. In addition to offering new mathematical insights into ancient technologies, our work helps bridge connections between the tangible abacus interface and the abstract algorithms powering it. Through examples and proofs, we show how bead manipulations precisely correspond to mathematical transformations. This level of formalization not only helps explain the effectiveness of the abacus, but also illustrates how even rudimentary calculation tools utilize profound mathematical ideas. Our mathematical abacus framework lays the foundation for further analysis as well as modifications and extensions of the classic abacus approach.

Software Support for Arbitrary Precision Arithmetic in Programming Languages

Arbitrary precision arithmetic, also known as bignum arithmetic, is a computational technique that allows programmers to perform arithmetic operations on numbers with significantly higher precision and magnitude than what is typically supported by the built-in numerical data types in programming languages. This technique is especially useful when working with extremely large or extremely precise numbers, such as in cryptography or scientific computations. Arbitrary precision arithmetic has many applications in the areas of Cryptography, Numerical Computation, Statistical Analysis, and High Precision measurements. For example, the calculation of the modulus in the RSA algorithm involves numbers with 1024-bit numbers and higher. An arithmetic calculation involving Multiplication and exponentiation of such numbers using Modulo arithmetic cannot be easily carried out in the existing programming Languages unless special software is provided. We can calculate the mathematical constant Pi to many thousand decimal places using the support provided in Programming Languages. Many programming languages provide built-in support for libraries for arbitrary precision arithmetic. We discuss the support provided in C/C++, Java and Python Languages with examples. Besides Programming Languages, Toolkits like Matlab and Sagemath also are used for scientific computation and special software support provided in these toolkits can enable arbitrary precision arithmetic. Most Programming Languages have support for floating-point arithmetic. We also discuss how arbitrary precision floating point arithmetic. can be supported in C/C++, Java, and Python. In addition, we discuss support for arbitrary precision integer arithmetic in Ruby, Javascript and Matlab and support for arbitrary precision floating point arithmetic in the Perl Language. Finally, we provide an example of computing the constant Pi to many decimal places using the sagemath tool.

Hardware Implementations of Elliptic Curve Cryptography Using Shift-Sub Based Modular Multiplication Algorithms

Elliptic curve cryptography (ECC) over prime fields relies on scalar point multiplication realized by point addition and point doubling. Point addition and point doubling operations consist of many modular multiplications of large operands (256 bits for example), especially in projective and Jacobian coordinates which eliminate the modular inversion required in affine coordinates for every point addition or point doubling operation. Accelerating modular multiplication is therefore important for high-performance ECC. This paper presents the hardware implementations of modular multiplication algorithms, including (1) interleaved modular multiplication (IMM), (2) Montgomery modular multiplication (MMM), (3) shift-sub modular multiplication (SSMM), (4) SSMM with advance preparation (SSMMPRE), and (5) SSMM with CSAs and sign detection (SSMMCSA) algorithms, and evaluates their execution time (the number of clock cycles and clock frequency) and required hardware resources (ALMs and registers). Experimental results show that SSMM is 1.80 times faster than IMM, and SSMMCSA is 3.27 times faster than IMM. We also present the ECC hardware implementations based on the Secp256k1 protocol in affine, projective, and Jacobian coordinates using the IMM, SSMM, SSMMPRE, and SSMMCSA algorithms, and investigate their cost and performance. Our ECC implementations can be applied to the design of hardware security module systems.

Efficient Hardware Implementation of Elliptic-Curve Diffie–Hellman Ephemeral on Curve25519

Hardware architecture optimized for implementing the elliptic-curve Diffie–Hellman ephemeral (ECDHE) on 256-bit Montgomery elliptic curves presents unique challenges, particularly for resource-constrained IoT and mobile devices. This work aims to provide an efficient hardware implementation of ECDHE on Curve25519, including a dedicated finite state machine (FSM) designed to handle point multiplication and ECDHE operations, utilizing constant-time algorithms and a unified memory block for resource management. Additionally, we introduce an optimized modular computation unit that covers modular addition, subtraction, multiplication, and inversion. Our proposed hardware architecture enhances the efficiency of ECDHE operations while maintaining low resource utilization, considerably reduced latency, and low power consumption. Synthesized on the Xilinx Artix-7 platform, our design boasts 64,000 Slices and a clock speed of 102 MHz, and it computes an ECDHE scalar multiplication operation in 1.1 ms, consuming 117 mW. The proposed hardware design can be applied to various platforms, including mobile devices and IoT systems.