Scalar Multiplication Research Articles

Generating key-dependent involutory MDS matrices through permutations, direct exponentiation, and scalar multiplication

Generating key-dependent involutory MDS matrices through permutations, direct exponentiation, and scalar multiplication

What is the variant of fractal dimension under addition of functions with same dimension and related discussions

<abstract><p>This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let $ g(x) $ and $ h(x) $ have the same box dimension $ t\; (1 < t\leq 2) $, the box dimension of $ g(x)+h(x) $ may or may not exist. If it exists, it can take an arbitrary real number $ \gamma $ satisfying $ 1 < \gamma\leq t $. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers $ t_1\; {\rm and}\; t_2 $ that satisfy $ 1 < t_1 < t_2 < t\leq 2 $. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function $ W_t(x) $, we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals.</p></abstract>

Methods for using elliptic curves in cryptography

Elliptic Curve Cryptography (ECC), a significant modern cryptography, is more secure and robust than most others due to its construction using an elliptic curve and the application of mathematical operations for encryption and key generation. Furthermore, ECC based on elliptic curves can continually enhance speed and intensity through improvements in accelerators, scalar multiplication, and order processing speed.

CHARACTERISATION OF EQUIVALENT NORMS ON A LINEAR SPACE USING EXPONENTIAL VECTOR SPACE

In this paper we have found a necessary and sufficient condition for equivalence of two norms on a linear space using the theory of exponential vector space. Exponential vector space (‘evs’ in short) is an ordered algebraic structure which can be considered as an algebraic ordered extension of vector space. This structure is axiomatised on the basis of the intrinsic properties of the hyperspace C(X) comprising all nonempty compact subsets of a Hausdorff topological vector space X. Exponential vector space is a conglomeration of a semigroup structure, a scalar multiplication and a compatible partial order. We have shown that the collection of all norms defined on a linear space, together with the constant function zero, forms a topological exponential vector space. Then using the concept of comparing function (a concept defined on a topological exponential vector space) we have proved the aforesaid necessary and sufficient condition; also we have proved using comparing function that in an infinite dimensional linear space there are uncountably many non-equivalent norms.

Practical Certificate-Less Infrastructure with Application in TLS

We propose highly efficient certificate-less (CL) protocols for the infrastructure used by authenticated key exchange (AKE). The construction is based on elliptic curves (EC) without pairing, which means it can be easily supported by most industrial cryptography libraries on constrained devices. Compared with other pairing-free CL solutions, the new CL-AKE protocol enjoys the least number of scalar multiplications over EC groups. We use a unified game-based model to formalize the security of each protocol, while most previous works only assess the security against a list of attacks, provide informal theorems without proper modeling, or use separate models for protocols in different stages. We also present an efficient integration of the core protocols into the TLS cipher suites and a stand-alone implementation for constrained devices. The performance is evaluated on constrained devices in real-world settings, which further confirms the efficiency of our proposal.

Near-linear algebra

In this paper, we demonstrate that the realm of near-vector spaces enables us to address nonlinear problems while also providing access to most of the tools that linear algebra offers. We establish fundamental results for near-vector spaces, which serve to extend classical linear algebra into the realm of near-linear algebra. Within this paper, we finalize the algebraic proof that for a given scalar group [Formula: see text], any nonempty [Formula: see text]-subspace that remains stable under addition and scalar multiplication constitutes an [Formula: see text]-subspace. We prove that any quotient of a near-vector space by an [Formula: see text]-subspace is itself a near-vector space, along with presenting the First Isomorphism Theorem for near-vector spaces. In doing so, we obtain comprehensive descriptions of the span. By defining linear independence outside the quasi-kernel, we introduce a new concept of basis. We also establish that near-vector spaces are characterized based on the presence of a scalar basis.

Fast and Clean: Auditable high-performance assembly via constraint solving

Handwritten assembly is a widely used tool in the development of highperformance cryptography: By providing full control over instruction selection, instruction scheduling, and register allocation, highest performance can be unlocked. On the flip side, developing handwritten assembly is not only time-consuming, but the artifacts produced also tend to be difficult to review and maintain – threatening their suitability for use in practice.In this work, we present SLOTHY (Super (Lazy) Optimization of Tricky Handwritten assemblY), a framework for the automated superoptimization of assembly with respect to instruction scheduling, register allocation, and loop optimization (software pipelining): With SLOTHY, the developer controls and focuses on algorithm and instruction selection, providing a readable “base” implementation in assembly, while SLOTHY automatically finds optimal and traceable instruction scheduling and register allocation strategies with respect to a model of the target (micro)architecture.We demonstrate the flexibility of SLOTHY by instantiating it with models of the Cortex-M55, Cortex-M85, Cortex-A55 and Cortex-A72microarchitectures, implementing the Armv8.1-M+Helium and AArch64+Neon architectures. We use the resulting tools to optimize three workloads: First, for Cortex-M55 and Cortex-M85, a radix-4 complex Fast Fourier Transform (FFT) in fixed-point and floating-point arithmetic, fundamental in Digital Signal Processing. Second, on Cortex-M55, Cortex-M85, Cortex-A55 and Cortex-A72, the instances of the Number Theoretic Transform (NTT) underlying CRYSTALS-Kyber and CRYSTALS-Dilithium, two recently announced winners of the NIST Post-Quantum Cryptography standardization project. Third, for Cortex-A55, the scalar multiplication for the elliptic curve key exchange X25519. The SLOTHY-optimized code matches or beats the performance of prior art in all cases, while maintaining compactness and readability.

A special scalar multiplication algorithm on Jacobi quartic curves

A special scalar multiplication algorithm on Jacobi quartic curves

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.

On Modules over G-sets

Let R be a commutative ring with unity, M a module over R and let S be a G–set for a finite group G. We define a set MS to be the set of elements expressed as the formal finite sum of the form ∑s∈Smss where ms∈M. The set MS is a module over the group ring RG under the addition and the scalar multiplication similar to the RG–module MG. With this notion, we not only generalize but also unify the theories of both of the group algebra and the group module, and we also establish some significant properties of (MS)RG. In particular, we describe a method for decomposing a given RG–module MS as a direct sum of RG–submodules. Furthermore, we prove the semisimplicity problem of (MS)RG with regard to the properties of MR, S and G.

Enhanced BaTiO3/Si3N4 integrated photonic platform with VO2 technology for large-scale neuromorphic computing [Invited

The hybrid barium titanate (BaTiO3 or BTO) – silicon nitride (Si3N4 or SiN) platform integrated on silicon has been established as a promising candidate for implementing photonic integrated circuits with unique features in terms of high modulation speeds and low transmission loss. However, despite the high Pockels effect in BTO, switching device footprints are relatively large, which could compromise their use in applications with large scalability requirements, such as neuromorphic computing hardware. To address this limitation, we propose the integration of vanadium dioxide (VO2) in the SiN/BTO platform to enable ultra-compact amplitude switching devices offering a scalar multiplication functionality with multilevel operation. More concretely, an electronically reprogrammable switching device with a 5-bit amplitude encoding capability and an insertion loss of only 0.5 dB is developed. The proposed device is built with a 9-µm-long VO2/SiN/BTO waveguide structure integrated with an efficient microheater using a transparent conducting oxide. Such a vital building block would offer significant potential for developing more complex photonic integrated circuits, including dot-product or matrix-vector multiplication engines. The combination of high speed, low loss, and reduced footprint makes the proposed enhanced platform an attractive solution for application in scalable and energy-efficient neuromorphic computing hardware.

Normed Space of Fuzzy Intervals and Its Topological Structure

The space, Fcc(R), of all fuzzy intervals in R cannot form a vector space. However, the space Fcc(R) maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of Fcc(R) is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in Fcc(R). Although Fcc(R) cannot form a vector space, we still can endow a norm on the space Fcc(R) by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets.

A tight upper bound on the number of non-zero weights of a constacyclic code

For a simple-root λ-constacyclic code C over Fq, let 〈ρ〉 and 〈ρ,M〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ, and by the cyclic shift ρ and the scalar multiplication group M, respectively. Let NG(C⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C⁎=C﹨{0}. In this paper, we establish explicit formulas for N〈ρ〉(C⁎) and N〈ρ,M〉(C⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N〈ρ〉(C⁎)=N〈ρ,M〉(C⁎) is presented.

A novel homomorphic polynomial public key encapsulation algorithm

Background: One of the primary drivers in development of novel quantum-safe cryptography techniques is the ongoing National Institute of Standards and Technology (NIST) Post-Quantum Cryptography (PQC) competition, which aims to identify quantum-safe algorithms for standardization. Although NIST has recently announced candidates to be standardized, the development of novel PQC algorithms remains desirable to address the challenges of quantum computing. Furthermore, to enhance security and improve performance. Methods: This paper introduces a novel public key encapsulation algorithm that incorporates an additional layer of encryption during key construction procedure, through a hidden ring. This encryption involves modular multiplication over the hidden ring using a homomorphism operator that is closed under addition and scalar multiplication. The homomorphic encryption key is comprised of two values - one used to create the hidden ring and the other to form an encryption operator. This homomorphic encryption can be applied to any polynomials during key construction over a finite field with their coefficients considered private. Particularly, the proposed homomorphic encryption operator can be applied to the public key of the Multivariate Public Key Cryptography schemes (MPKC) to hide the structure of its central map construction. Results: This paper presents a new variant of the MPKC with its public key encrypted using the proposed homomorphic operator. This novel scheme is called the Homomorphic Polynomial Public Key (HPPK) algorithm, which simplifies MPKC central map to two multivariate polynomials constructed from polynomial multiplications. The HPPK algorithm employs a single polynomial vector for the plaintext and a multi-variate noise vector associated with the central map. In contrast, in MPKC, a single multivariate vector is created by segmenting the secret plaintext over a small finite field. The HPPK algorithm is Indistinguishability Under Chosen-Plaintext Attack (IND-CPA) secure, and its classical complexity for cracking is exponential in the size of the prime field GF(p).

Frugal and decentralised resolvent splittings defined by nonexpansive operators

Frugal resolvent splittings are a class of fixed point algorithms for finding a zero in the sum of the sum of finitely many set-valued monotone operators, where the fixed point operator uses only vector addition, scalar multiplication and the resolvent of each monotone operator once per iteration. In the literature, the convergence analyses of these schemes are performed in an inefficient, algorithm-by-algorithm basis. In this work, we address this by developing a general framework for frugal resolvent splitting which simultaneously covers and extends several important schemes in the literature. The framework also yields a new resolvent splitting algorithm which is suitable for decentralised implementation on regular networks.

An area-efficient and low-latency elliptic curve scalar multiplication accelerator over prime field

For the elliptic curve cryptography (ECC) over prime field, this paper presents an area-efficient and low-latency elliptic curve scalar multiplication (ECSM) accelerator, which supports to process any prime number p ≤ 256 bits. The proposed accelerator is based on a parallel hardware implementation of iterative digit-digit Montgomery multiplication and an optimized data flow architecture for elliptic curve point operations. The proposed design is implemented in Xilinx Virtex-7 FPGA. The experimental results show that the proposed architecture occupies 23.7k look-up tables (LUTs) and performs single 256-bit scalar multiplication in 0.56 ms. The area-time product of the proposed architecture is only 13.36. Compared with other state-of-the-art ECSM implementations, the proposed architecture has obvious advantage in terms of the area-time product.

Lattice attacks on pairing-based signatures

ABSTRACT Practical implementations of cryptosystems often suffer from critical information leakage through side-channels (such as their power consumption or their electromagnetic emanations). For public-key cryptography on embedded systems, the core operation is usually group exponentiation – or scalar multiplication on elliptic curves – which is a sequence of group operations derived from the private-key that may reveal secret bits to an attacker (on an unprotected implementation). We present lattice-based polynomial-time (heuristic) algorithms that recover the signer’s secret in popular pairing-based signatures when used to sign several messages under the assumption that blocks of consecutive bits of the corresponding exponents are known by the attacker. Our techniques rely upon Coppersmith's method and apply to many signatures in the so-called exponent-inversion framework in the standard security model (i.e. Boneh-Boyen, Gentry and Pontcheval-Sanders signatures) as well as in the random oracle model (i.e. Sakai-Kasahara signatures).

Elliptic Curve Implementation and its Applications: A Review

Encryption is regarded as essential within the safety measures used to protect data via unsafe transport methods. Elliptic Curve encryption (ECC), one of the several asymmetric encryption algorithms currently in use, has become popular because of its high level of security and small key sizes. In contrast to Rivest, Shamir and Adleman (RSA), for instance, ECC can keep security levels at a certain level with a smaller key. The primary purpose of ECC is elliptic curve point multiply (ECPM), which also has the largest hardware cost. Numerous hardware applications have been made to accelerate the calculus of the ECPM. This paper discusses the idea behind the cryptography of elliptic curves (ECC's) and the benefits it quicker, moreover believed way of encoding as compared to the RSA public_key encryption techniques that are currently used as standards. ECC's specifications encompass all relevant asymmetry cryptography elements, including electronic signatures & key negotiation processes. The scalar multiplication Key Point (K.P) function, which is the fundamental operation of ECCs, is utilized to accomplish this objective. Where P is a value and k represents an integer and is exists on an elliptic curve. This article explains how ECC contributes to system safety and applications of it in the field of security

1D convolutional neural networks-based soil fertility classification and fertilizer prescription

Sustainable agriculture is essential to meet the demands of the global population. An adequate application of fertilizers is essential for sustainable agricultural productivity. This research aims to determine soil fertility and provide precise fertilizer to improve crop yield. Many researchers have proposed soil fertility classification using deep learning-based approaches, such as extreme learning machines (ELMs) and multilayer perceptrons (MLPs). Although both ELM and MLP have the highest performance, insufficient training data prevent them from being useful. To address this limitation, this research proposes a 1D convolution neural networks (1D-CNN)-based soil fertility classification method that is straightforward, compact, and supports scalar additions and multiplications. To classify soil fertility, the classifier employs laboratory-measured soil data that encompasses electrical conductivity, pH, organic carbon, potassium, phosphorus, sulfur, boron, copper, iron, manganese, and zinc. The proposed approach employs MinMax normalization and the synthetic minority oversampling technique (SMOTE) to improve the classifier performance. The results of soil classification are used to recommend fertilizers. An experimental study using a laboratory-measured soil dataset showed that the proposed technique outperformed ELM and MLP classifiers. The proposed approach outperformed ELM and MLP with a classification accuracy of 97.9%, while ELM and MLP achieved classification accuracies of 69.80% and 87.06%, respectively. The proposed method can help farmers manage soil fertility sustainably to increase crop production.