Ordinary Multiplication Research Articles

SM2-based low-cost and efficient parallel modular multiplication

Elliptic curve encryption (ECC) has been widely used in public key cryptography, and modular multiplication is one of the core operations of elliptic curve encryption. This paper presents a low-cost high-speed parallel modular multiplication implementation based on SM2. Using the characteristics of the prime (P256), the two-step multiplication and reduction of modular multiplication are performed in parallel. The 8-part karatsuba algorithm is used in multiplication. In the process of performing multiplication, in order to reduce the consumption of multiplier resources, the karatsuba algorithm is used to improve the ordinary multiplier. The control signal EN=0/1 is used to control the multiplier to perform ordinary multiplication or karatsuba algorithm multiplication. Then the multiplier is reused. 4 improved 32-bit multipliers are used in complete 256-bit modular multiplication. Experiments show that on the 100 MHz Artix-7 FPGA hardware platform, only 12K LUTs are needed, and a modular multiplication operation can be completed in 0.09μs. Comprehensive time and area, our design has certain advantages.

Impacts of climate change and meteo-solar parameters on photosynthetically active radiation prediction using hybrid machine learning with Physics-based models

The photosynthetically active radiation (PAR) ratio of 0.45 has been applied in numerous ecological studies to convert global solar radiation (H) to PAR by ordinary multiplication. The ratio is essentially useful and convenient, particularly when in situ measurement of climate data sets are not readily available due to inadequate instrumentation network and experience required. Up till now, there is no general agreement on whether the ratio is 0.45 and how the solar fluxes are distributed in Nigeria or around the world. Numerous empirical ratios have been reported in literature. This study proposed five (5) machine learning models (bagging, boosting, artificial neural network (MLP), automatic regression integrated moving average (ARIMA), controlled ARIMA), and 5 Physics-based models (Physics-based Gumbel model (PGM), popularly applied ratio (PCR), observed PAR ratio (POR), empirical, and PGM-CARIMA hybrid models. The 10 models were used to predict PAR using generalized datasets in Nigeria and the PGM-CARIMA model emerged as the best performing fit based on the R2, RMSE, MAPE, and BIC error metric indicators. PGM-CARIMA reported a better estimate for predicting PAR compared to POR and PCR with a relative error ranging from 0.00002 to 0.0503% compared to POR and PCR which recorded 4.657% and 6.178%. This implies that the use of PGM-CARIMA is more suitable for predicting the PAR ratio as it introduces a lower marginal error of approximately 0.00002–0.0503% when applied for the calculation of net primary productivity (NNP) and of climate change with a significant reduction from 6.17% and 4.65% was achieved in using the ratio of PCR and POR, respectively. Using the proposed PGM-CARIMA evolutionary hybrid model, we demonstrated that the annual mean of the ratio yielded a range of 0.4377 to 0.4712 in Nigeria. The authors also proposed the Physics-based classical climate change model (PCM) to assess the impact of climate change on PAR productivity in Nigeria. The findings revealed that changes in PAR productivity as a result of the impact of climate change may accelerate food production and crop yields in Southern Nigeria and depreciate overall food production and crop yields in Northern Nigeria in 2030, 2040, and 2021–2040. The results of the climate analysis also revealed that north-central Nigeria often referred to as the nation's food basket is most negatively impacted by the impact of climate change externalities over the periods studied. The authors recommended mitigation and adaptation plans such as delayed planting dates, intercropping, routine planting of tropical forages, especially livestock that feed on plants, could be key to climate change mitigation, also, expansion of the planting area as well as the incorporation of improved seedlings that are highly resistant to extreme temperatures will accelerate food security across the country within the periods studied.

Public-Key Cryptography Based on Tropical Circular Matrices

Some public-key cryptosystems based on the tropical semiring have been proposed in recent years because of their increased efficiency, since the multiplication is actually an ordinary addition of numbers and there is no ordinary multiplication of numbers in the tropical semiring. However, most of these tropical cryptosystems have security defects because they adopt a public matrix to construct commutative semirings. This paper proposes new public-key cryptosystems based on tropical circular matrices. The security of the cryptosystems relies on the NP-hard problem of solving tropical nonlinear systems of integers. Since the used commutative semiring of circular matrices cannot be expressed by a known matrix, the cryptosystems can resist KU attacks. There is no tropical matrix addition operation in the cryptosystem, and it can resist RM attacks. The new cryptosystems can be considered as a potential post-quantum cryptosystem.

Ternary arithmetic, factorization, and the class number one problem

Ordinary multiplication of natural numbers can be generalized to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of ‘3-primality’ -primality with respect to ternary multiplication- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields Q(√-n), n > 0, with odd discriminant and whose ring of integers admits unique factorization. We also describe how to determine representations of numbers as ternary products and related algorithms for usual primality testing and integer factorization.

Public Key Cryptosystem Based on Singular Matrix

The algorithms such as RSA, ElGamal and ECC work on integers. Commutative operations on integer multiplication leave these algorithms vulnerable to attack by eavesdroppers. For this reason, experts develop the concept of non-commutative algebra in the public key cryptosystem by adding non-commutative properties to groups, semirings, semiring division, matrices and matrix decomposition. However, the key generating process in some public key cryptosystems is quite complicated to carry out. Therefore, in previous research, Liu used nonsingular matrices to form a simpler public key cryptosystem. However, eavesdroppers use the inverse of nonsingular matrices to construct the private key. As a result, this public key cryptosystem is still vulnerable to attacks. Therefore, we use a singular matrix to modify and build the proposed public key cryptosystem in this study. This study indicates that the singular matrix can be used to modify the public key cryptosystem. The results also show that the key generating algorithm only uses ordinary matrix multiplication (without using matrix power operations), so it is not too complicated. Furthermore, the proposed public key cryptosystem works on a matrix over integers so that the possible brute force attack trials are endless. The proposed public key cryptosystem also cannot be attacked by matrix inversion because it uses a singular matrix.
 HIGHLIGHTS
 
 Public key cryptosystems that use commutative operations are vulnerable to eavesdropping attacks
 This study uses a singular matrix to modify and build a public key cryptosystem
 The proposed public key cryptosystem works on ordinary matrix multiplication operations and cannot be attacked by matrix inversion

Iwasawa–Greenberg main conjecture for nonordinary modular forms and Eisenstein congruences on GU(3,1)

AbstractIn this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on$\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s$\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.

Semigroup operations which are distributive over a given semigroup operation on positive real numbers

Let R+ be the space of positive real numbers with the ordinary topology and let ⋆ be an arbitrary cancellative continuous semigroup operation on R+ or some special noncancellative continuous semigroup operation on R+. We characterize the set D −1 ⋆ (R+) of all cancellative continuous semigroup operations on R+ which are distributive over ⋆ in terms of homeomorphism. As a consequence, it is shown that if ⋆ is homeomorphically isomorphic to the ordinary addition + on R+, any element of D −1 ⋆ (R+) is homeomorphically isomorphic to the ordinary multiplication on R+, and that if ⋆ is cancellative and not homeomorphically isomorphic to +, then D −1 ⋆ (R+) is empty. Moreover, if ⋆ is homeomorphically isomorphic to some special noncancellative continuous semigroup operation on R+, D −1 ⋆ (R+) is also shown to be empty.

Convolutions on the complex torus

“Quasi-elliptic” functions can be given a ring structure in two different ways, using either ordinary multiplication, or convolution. The map between the corresponding standard bases is calculated. A related structure has appeared recently in the computation of Feynman integrals. The two approaches are related by a sequence of polynomials closely tied to the Eulerian polynomials.

Finite basis problem for involution monoids of unitriangular boolean matrices

Let $$(BU_n, \,^*\,)$$ be the involution monoid of all Boolean upper triangular $$n\times n$$ matrices with 1s on the main diagonal under the ordinary matrix multiplication and the skew transposition. The involution monoid $$(BU_2, \,^*\,)$$ is easily seen to be finitely based. In this paper, we shown that $$(BU_n, \,^*\,)$$ is non-finitely based for each $$n \ge 3$$, which answers an open question posed by Auinger et al. Therefore involution monoid $$(BU_n, \,^*\,)$$ is finitely based if and only if $$n = 2$$.

The monoid of $$2 \times 2$$ triangular boolean matrices under skew transposition is non-finitely based

Let $$\mathscr {T\!B}_n$$ be the involution semigroup of all upper triangular boolean $$n\times n$$ matrices under the ordinary matrix multiplication and the skew transposition. It is shown by Auinger et al. that the involution semigroup $$\mathscr {T\!B}_n$$ is non-finitely based if $$n > 2$$, but the case when $$n=2$$ still remains open. In this paper, we give a sufficient condition under which an involution semigroup is non-finitely based. As an application, we show that the involution semigroup $$\mathscr {T\!B}_2$$ is non-finitely based. Hence $$\mathscr {T\!B}_n$$ is non-finitely based for all $$n \ge 2$$.

Semigroup operations distributed by natural noncancellative semigroup operations on the positive real numbers

Let $${\mathbf {R}}_+$$ be the space of positive real numbers with the ordinary topology. Let $$\star $$ be the natural noncancellative continuous semigroup operation $$\times $$ or $$\diamond $$ on $${\mathbf {R}}_+$$ as defined in first section. We characterize the set $${\mathcal {D}}_\star ({\mathbf {R}}_+)$$ of all cancellative continuous semigroup operations on $${\mathbf {R}}_+$$ which are distributed by $$\star $$ in terms of homeomorphism. As a consequence, we show that an arbitrary semigroup operation in $${\mathcal {D}}_\star ({\mathbf {R}}_+)$$ is homeomorphically isomorphic to the ordinary multiplication on $${\mathbf {R}}_+$$ .

An axiomatic property based triangular fuzzy extension of Saaty’s consistency

The fuzzy analytic hierarchy process (FAHP) is a widely applied fuzzy multi-criteria decision making method. In triangular FAHP, consistency of triangular fuzzy multiplicative preference relations (TFMPRs) plays a very important role for checking the quality of human fuzzy judgments. This paper introduces some indices to measure fuzziness of triangular fuzzy judgments and row fuzziness proportionalities in a TFMPR. An expected interval based possibility degree formula is developed to compare two triangular fuzzy judgments and used to judge whether the average-based intensity of a triangular fuzzy judgment is changed. The paper proposes four axiomatic properties aimed at characterizing multiplicative consistency of TFMPRs, and illustrates that four known consistency definitions fail to meet all the four axiomatic properties. By analyzing the axiomatic property of restriction on ambiguity of single judgments, a constrained triangular fuzzy multiplication based transitivity equation is devised and used to construct a triangular fuzzy matrix from a TFMPR. Based on the constructed fuzzy matrix, the paper establishes an ordinary triangular fuzzy multiplication based transitivity equation to define consistency of TFMPRs. Important properties of consistent TFMPRs are offered to demonstrate that all the four axiomatic properties are satisfied by the proposed consistency definition. Three numerical examples are given to illustrate the use of the proposed consistency model and the obtained properties for judging consistency of TFMPRs and determining unknown values of incomplete TFMPRs.

Axiomatic property based consistency analysis and decision making with interval multiplicative reciprocal preference relations

This paper focuses on obtaining an interval extension of Saaty's consistency and eliciting normalized interval weights in analytic form from interval multiplicative reciprocal preference relations (IMRPRs) as well as checking acceptability of IMRPRs. Four axiomatic properties are presented to characterize multiplicative consistency of IMRPRs. The paper shows that six existing consistency definitions fail to satisfy all the four axiomatic properties. A constrained interval multiplication based transitivity equation is devised and an interval matrix is constructed to capture original preferences and row uncertainty proportionalities of an IMRPR. Based on the constructed interval matrix, an ordinary interval multiplication based transitivity equation is developed to define consistency of IMRPRs. A logarithmic least square model is established to find normalized interval weights from IMRPRs and its analytic solution is obtained by the Lagrangian multiplier method. The paper introduces an index to measure uniformity of row uncertainties in an IMRPR and proposes a geometric consistency index to measure inconsistency of IMRPRs. A novel method is put forward to determine acceptability of IMRPRs by checking acceptable consistency and acceptable uncertainty. The proposed models are illustrated by eight numerical examples and a hierarchical multi-criteria decision making problem dealing with recommending undergraduate students to graduate admission.

Large proton cumulants from the superposition of ordinary multiplicity distributions

We construct a multiplicity distribution characterized by large factorial cumulants (integrated correlation functions) from a simple combination of two ordinary multiplicity distributions characterized by small factorial cumulants. We find that such a model, which could be interpreted as representing two event classes, reproduces the preliminary data for the proton cumulants measured by the STAR collaboration at $7.7$ GeV very well. This model then predicts very large values for the fifth and sixth order factorial cumulants, which can be tested in experiment.

A Delay Efficient Vedic Multiplier

Vedic mathematics is the ancient Indian method of mathematics based on 16 Sutras applicable to various branches of mathematics like trigonometry, calculus, geometry, conics etc. Multiplication is effectively used in modern communication and Digital Signal Processing applications. Ordinary multiplication requires propagation of carry from LSB to MSB while adding binary partial products, which limits the overall speed of multiplication. Vedic mathematics helps in generation of partial products and sums in one step, and ensures reduction in overall propagation delay. Urdhva Tiryakbhyam Sutra and Nikhilam Sutra are the two multiplication techniques used in Vedic mathematics. In this paper, an 8 * 8 Nikhilam Sutra multiplier for three different sets of bases is realized. The concepts of Urdhva Tiryakbhyam Sutra multiplication are used for the implementation of the proposed multiplier. The implementation results are compared with that of a Modified Booth’s multiplier in terms of delay, area and power. The design is synthesized in Synopsys Design Compiler using CMOS 90 nm technology, and results show that the proposed multiplier using Nikhilam Sutra with 25 bases is faster than the Modified Booth’s multiplier by 51.28%.

Semigroup operations distributed by the ordinary multiplication or addition on the real numbers

Semigroup operations distributed by the ordinary multiplication or addition on the real numbers

Operator dimensions from moduli

We consider the operator spectrum of a three-dimensional mathcal{N}=2 superconformal field theory with a moduli space of one complex dimension, such as the fixed point theory with three chiral superfields X, Y, Z and a superpotential W = XYZ. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large R-charge J. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly J + 1, a fact that cannot be seen directly from the XYZ Lagrangian. The third-lowest scalar primary lies in along multiplet with dimension J + 2−c−3J−3 + O(J−4), where c−3 is an unknown positive coefficient. The coefficient c−3 is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of c−3 does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large J, this proves the existence of scalar semi-short states at all values of J. Thus the combination of mathcal{N}=2 superconformal symmetry with the large-J expansion is more powerful than the sum of its parts.

A GPU-Adapted Structure for Unstructured Grids

A key advantage of working with structured grids e.g., images is the ability to directly tap into the powerful machinery of linear algebra. This is not much so for unstructured grids where intermediate bookkeeping data structures stand in the way. On modern high performance computing hardware, the conventional wisdom behind these intermediate structures is further challenged by costly memory access, and more importantly by prohibitive memory resources on environments such as graphics hardware. In this paper, we bypass this problem by introducing a sparse matrix representation for unstructured grids which not only reduces the memory storage requirements but also cuts down on the bulk of data movement from global storage to the compute units. In order to take full advantage of the proposed representation, we augment ordinary matrix multiplication by means of action maps, local maps which encode the desired interaction between grid vertices. In this way, geometric computations and topological modifications translate into concise linear algebra operations. In our algorithmic formulation, we capitalize on the nature of sparse matrix-vector multiplication which allows avoiding explicit transpose computation and storage. Furthermore, we develop an efficient vectorization to the demanding assembly process of standard graph and finite element matrices.

Malliavin calculus for generalized and test stochastic processes

We extend the Malliavin calculus from the classical finite variance setting to generalized processes with infinite variance and their test processes. The domain and the range of the basic Malliavin operators is characterized in terms of test processes and generalized processes. Various properties are proved such as the duality of the integral and the derivative in strong and in weak sense, the product rule with respect to ordinary and Wick multiplication and the chain rule in classical and in Wick sense.

Fundamental invariants of many-body Hilbert space

Many-body Hilbert space is a functional vector space with the natural structure of an algebra, in which vector multiplication is ordinary multiplication of wave functions. This algebra is finite-dimensional, with exactly [Formula: see text] generators for N identical particles, bosons or fermions, in d dimensions. The generators are called shapes. Each shape is a possible many-body vacuum. Shapes are natural generalizations of the ground-state Slater determinant to more than one dimension. Physical states, including the ground state, are superpositions of shapes with symmetric-function coefficients, for both bosons and fermions. These symmetric functions may be interpreted as bosonic excitations of the shapes. The algebraic structure of Hilbert space described here provides qualitative insights into long-standing issues of many-body physics, including the fermion sign problem and the microscopic origin of bands in the spectra of finite systems.