Modular Equations Research Articles

New Cryptanalysis of Prime Power RSA with Two Private Exponents

Prime Power RSA is a variant of the RSA scheme due to Takagi with modulus N=prq for r⩾2, where p,q are of the same bit-size. In this paper, we concentrate on one type of Prime Power RSA which assumes e·d≡1modpr−1(p−1)(q−1). Two new attacks on this type of Prime Power RSA are presented when given two pairs of public and private exponents, namely, (e1,d1) and (e2,d2) with the same modulus N. Suppose that d1<Nβ1,d2<Nβ2. In 2015, Zheng and Hu showed that when β1β2<(r−1)3/(r+1)3, N may be factored in probabilistic polynomial time. The first attack of this paper shows that one can obtain the factorization of N in probabilistic polynomial time, provided that β1β2<r/(r+1)3. Later, in the second attack, we improve both the first attack and the attack of Zheng and Hu, and show that the condition β1β2<r(r−1)2/(r+1)3 already suffices to break the Prime Power RSA. By introducing multiple parameters, our lattice constructions take full advantage of known information, and obtain the best known attack. Specifically, we make full use of the information that pr is a divisor of N, while the attack of Zheng and Hu only assumes that pr−1 is a divisor of N. As a consequence, this method implies a better lattice construction, and thus improves the previous attack. The experiments which reach a better upper bound than before also verify it. Our approaches are based on Coppersmith’s method for finding small roots of bivariate modular polynomial equations.

On some Schläfli type modular equations

On page 90 of his first notebook Ramanujan has recorded many Schl¨afli type modular equations. This article aims to give new and simple proof of Sch¨alflitype modular equation of degree 11 which was recorded by Ramanujan and a degree 23 modular equation. We use parametrization method to prove these modular equations.

On Ramanujan’s modular equations of degree three

The object of this article is to give new and simple proof of all but one Ramanujan’s modular equations of degree 3.

On intermediate exceptional series

The Freudenthal–Tits magic square m(A1,A2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {m}(\\mathbb {A}_1,\\mathbb {A}_2)$$\\end{document} for A=R,C,H,O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {A}=\\mathbb {R},\\mathbb {C},\\mathbb {H},\\mathbb {O}$$\\end{document} of semi-simple Lie algebras can be extended by including the sextonions S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}$$\\end{document}. A series of non-reductive Lie algebras naturally appear in the new row associated with the sextonions, which we will call the intermediate exceptional series, with the largest one as the intermediate Lie algebra E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} constructed by Landsberg–Manivel. We study various aspects of the intermediate vertex operator (super)algebras associated with the intermediate exceptional series, including rationality, coset constructions, irreducible modules, (super)characters and modular linear differential equations. For all gI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g}_I$$\\end{document} belonging to the intermediate exceptional series, the intermediate VOA L1(gI)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1(\\mathfrak {g}_I)$$\\end{document} has characters of irreducible modules coinciding with those of the simple rational C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-cofinite W-algebra W-h∨/6(g,fθ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{-h^\\vee /6}(\\mathfrak {g},f_\ heta )$$\\end{document} studied by Kawasetsu, with g\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {g} $$\\end{document} belonging to the Cvitanović–Deligne exceptional series. We propose some new intermediate VOA Lk(gI)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_k(\\mathfrak {g}_I)$$\\end{document} with integer level k and investigate their properties. For example, for the intermediate Lie algebra D6+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{6+1/2}$$\\end{document} between D6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_6$$\\end{document} and E7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_7$$\\end{document} in the subexceptional series and also in Vogel’s projective plane, we find that the intermediate VOA L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2(D_{6+1/2})$$\\end{document} has a simple current extension to a SVOA with four irreducible Neveu–Schwarz modules, and the supercharacters can be realized by a fermionic Hecke operator on the N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=1$$\\end{document} Virasoro minimal model L(c16,2,0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L(c_{16,2},0)$$\\end{document}. We also provide some (super) coset constructions such as L2(E7)/L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_2(E_7)/L_2(D_{6+1/2})$$\\end{document} and L1(D6+1/2)⊗2/L2(D6+1/2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1(D_{6+1/2})^{\\otimes 2}\\!/L_2(D_{6+1/2})$$\\end{document}. In the end, we find that the theta blocks associated with the intermediate exceptional series produce some new holomorphic Jacobi forms of critical weight and lattice index.

Modularity of the Schur index, modular differential equations, and high-temperature asymptotics

In this paper we analytically explore the modularity of the flavored Schur index of 4d N=2 superconformal field theories. We focus on the A1 theories of class-S and N=4 theories with SU(N) gauge group. We work out the modular orbit of the flavored index and defect index, compute the dimension of the space spanned by the orbit, and provide complete basis for computing modular transformation matrices. The dimension obtained from the flavored analysis predicts the minimal order of the unflavored modular differential equation satisfied by the unflavored Schur index. With the help of modularity, we also study analytically the high temperature asymptotics of the Schur index. In the high temperature limit τ→+i0, we identified the (defect) Schur index of the genus-zero A1 theories of class S with the S3 partition function of the SU(2)×U(1)n star-shaped quiver (with Wilson line insertion). In the identification, we observe an interesting relation between the linear-independence of defect indices and the convergence of the Wilson line partition functions. Published by the American Physical Society 2024

New Modular Equations of Composite Degrees and Partition Identities

New Modular Equations of Composite Degrees and Partition Identities

Performance comparison of quantum-safe multivariate polynomial public key encapsulation algorithm

A novel quantum-safe key encapsulation algorithm, called Multivariate Polynomial Public Key (MPPK), was recently proposed by Kuang, Perepechaenko, and Barbeau. Security of the MPPK key encapsulation mechanism does not rely on the prime factorization or discrete logarithm problems. It builds upon the NP-completeness of the modular Diophantine equation problem, for which there are no known efficient classical or quantum algorithms. Hence, it is resistant to known quantum computing attacks. The private key of MPPK comprises a pair of multivariate polynomials. In a companion paper, we analyzed the performance of MPPK when these polynomials are quadratic. The analysis highlighted the MPPK high decapsulation time. We found that, while maintaining the security strength, the polynomials can be linear. Considerable performance gains are obtained for the decapsulation process. In this article, we benchmark the linear case and compare the results with the previous quadratic case.

On Ramanujan's modular equations and Hecke groups

Inspired by the work of Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is no developed theory about how to find the degrees of these polynomial modular equations explicitly. In this paper, we determine the degrees of the polynomial modular equations explicitly and study the relation between Hecke groups and modular equations in Ramanujan's theories of signatures 2, 3, and 4.

Modularity in Argyres-Douglas theories with a = c

We consider a family of Argyres-Douglas theories, which are 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 strongly coupled superconformal field theories (SCFTs) but share many features with 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theories. In particular, the two central charges of these theories are the same, namely a = c. We derive a simple and illuminating formula for the Schur index of these theories, which factorizes into the product of a Casimir term and a term referred to as the Schur partition function. While the former is controlled by the anomaly, the latter is identified with the vacuum character of the corresponding chiral algebra and is expected to satisfy the modular linear differential equation. Our simple expression for the Schur partition function, which can be regarded as the generalization of MacMahon’s generalized sum-of-divisor function, allows one to numerically compute the series expansions efficiently, and furthermore find the corresponding modular linear differential equation. In a special case where the chiral algebra is known, we are able to derive the corresponding modular linear differential equation using Zhu’s recursion relation. We further study the solutions to the modular linear differential equations and discuss their modular transformations. As an application, we study the high temperature limit or the Cardy-like limit of the Schur index using its simple expression and modular properties, thus shedding light on the 1/4-BPS microstates of genuine N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs with a = c and their dual quantum gravity via the AdS/CFT correspondence.

Modular differential equations and algebraic systems

Modular differential equations and algebraic systems

Modular approximate discrete modeling for blisk system with triple closed-loop equivalence method

At present, the lumped-parametric model is often used in the preliminary design of bladed disks. However, the lack of systematic modeling methods leads to high errors. In order to develop the systematic modeling method and improve the equivalence accuracy, this paper considers the comprehensive equivalence from the aspects of natural characteristics, transient forced responses, and steady-state forced responses under aerodynamic excitations. This paper includes two novelties. First, based on the modular equation of blisk system, an approximate discrete model is derived for computational efficiency. Second, a triple closed-loop equivalence method with modal weight optimization is developed innovatively for optimal equivalence parameters. The method can significantly improve the accuracy of element parameters in the lumped-parametric model and achieve the comprehensive equivalence to actual blisk. Compared with the existing equivalence method, the average errors of natural frequencies in the whole mode interval and in the high-density mode interval are reduced by 41.75 % and 36.76 %, respectively. The maximum and average displacement errors of the transient response are reduced by 51.76 % and 58.98 %, respectively. Furthermore, vibration characteristics of the mass and stiffness mistuned blisks are analyzed under traveling wake excitation during acceleration and deceleration processes by approximate discrete model and determined parameters.

Solving modular cubic equations with Coppersmith's method

Several cryptosystems based on Elliptic Curve Cryptography such as KMOV and Demytko process the message as a point M=(x0,y0) of an elliptic curve with an equation of the form y2≡x3+ax+b(modn) over a finite field when n is a prime number, or over a finite ring when n=pq is an RSA modulus. Other systems use singular cubic curves such as y2≡x3+ax2(modn) and y2+axy≡x3(modn). In this paper, we present a method to find the small solutions of the former modular cubic equations. Our method is based on Coppersmith's technique and enables one to find the solutions (x0,y0) when |x0|3|y0|2 is smaller than the modulus.

The Hopf equation with certain modular nonlinearities

The dynamics of waves of sign-variable shape has been studied within the framework of the Hopf equation (ut +Fux = 0) with a non-analytic propagation velocity containing the modulus of the function at the zero crossing (F ∼ |u|α). It is shown that Riemann waves exist only for a certain smoothness of the function F[u(x)] at the initial moment of time. Otherwise, the wave immediately overturns (gradient catastrophe). Popular in nonlinear physics the modular Hopf equation, the dispersionless Schamel equation, and the dispersionless logarithmic Korteweg-de Vries equation are considered as examples.

Schur indices for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills with more general gauge groups

We study the unflavored Schur indices in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory for the Bn, Cn, Dn, G2 gauge groups. We explore two methods, namely the character expansion method and the Fermi gas method, to efficiently compute the q-series expansion of the Schur indices to some high orders. Using the available data and the modular properties, we are able to fix the exact formulas for the general gauge groups up to some high ranks and discover some interesting new features. We also identify some empirical modular anomaly equations, but unlike the case of An groups, they are quite complicated and not sufficiently useful to fix exact formulas for gauge groups of arbitrary rank.

Some new Ramanujan-type modular equations of degree 15

Some new Ramanujan-type modular equations of degree 15

On intermediate Lie algebra E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document}

E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} is an intermediate Lie algebra filling a hole between E7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_7$$\\end{document} and E8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_8$$\\end{document} in the Deligne–Cvitanović exceptional series. It was found independently by Mathur, Muhki, Sen in the classification of 2d RCFTs via modular linear differential equations (MLDE) and by Deligne, Cohen, de Man in representation theory. In this paper we propose some new vertex operator algebras (VOA) associated with E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} and give some useful information at small levels. We conjecture that the affine VOA (E7+1/2)k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(E_{7+1/2})_k$$\\end{document} is rational if and only if the level k is at most 5, and provide some evidence from the viewpoint of MLDE. We propose a conjectural Weyl dimension formula for infinitely many irreducible representations of E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document}, which generates almost all irreducible representations of E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} with level k≤4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\le 4$$\\end{document}. More concretely, we propose the affine VOA E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} at level 2 and the rank-two instanton VOA associated with E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document}. We compute the VOA characters and provide some coset constructions. These generalize the previous works of Kawasetsu for affine VOA E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} at level 1 and of Arakawa–Kawasetsu at level -5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-5$$\\end{document}. We then predict the conformal weights of affine VOA E7+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_{7+1/2}$$\\end{document} at level 3, 4, 5.

An Improved Coppersmith Algorithm Based on Block Preprocessing

Since Coppersmith proposed the use of the LLL algorithm to solve univariate modular polynomial equations at EUROCRYPT’96, it has sparked a fervent research interest in lattice analysis among cryptographers. Despite its polynomial-time nature, the LLL algorithm exhibits a high-order polynomial upper bound in terms of theoretical complexity, particularly with longer computation times when applied to high-dimensional lattices. In addressing this issue, we propose an improved algorithm based on block preprocessing, building on the original Coppersmith algorithm and thus providing proof of correctness for this algorithm. This approach effectively reduces the solution time of the algorithm, offering a maximum improvement of 8.1% compared to the original Coppersmith algorithm. Additionally, we demonstrate the compatibility of our algorithm with the rounding algorithm proposed at PKC 2014. The combined utilization of these approaches further enhances the efficiency of our algorithm. The experimental results show that the combined algorithm achieves a maximum improvement of 22.4% in solution time compared to the original Coppersmith algorithm. It also outperforms the standalone rounding algorithm with a maximum improvement of 12.1%. When compared to the improved Coppersmith algorithm based on row common factor extraction, our proposed algorithm demonstrates comparable or even superior performance in certain dimensions. The block preprocessing algorithm in our approach enables independent execution without data exchange, making it suitable for leveraging multi-processing advantages in scenarios involving higher degrees of modular polynomial equations. This offers a new perspective for achieving the parallel computation of the Coppersmith algorithm, facilitating parallel execution and providing valuable insights.

Some new identities for colored partitions with parts in multiples of $ 4 $

<p>To start, we provided the deformations of certain specific modular equations derived from Ramanujan's notebook, Part Ⅲ, by using the complete square formula. Subsequently, we established fifteen identities for partitions with distinct colors that arose from these deformed modular equations in the theory of modular equations. Throughout our investigation, we found that the fifteen identities have parts that were in multiples of $ 4 $.</p>

Advancing IoT Device Security in Smart Cities: Through Innovative Key Generation and Distribution With D_F, Gf, and Multi-Order Recursive Sequences

In today's mass communication landscape, security is a paramount concern, notably with the rapid expansion of the Internet of Things (IoT). Various methods aim to bolster IoT communication security, particularly by regulating access between IoT devices and networks. Encrypting data with a shared secret key is crucial, considering the limited capabilities of these devices, demanding a lightweight yet robust control mechanism. While traditional methods like Diffie-Hellman facilitated secure communication, vulnerabilities arose from modular and exponential equations. Our paper proposed a mathematical refinement of the Diffie Hellman (D_H) protocol. By leveraging GF finite fields and multi-order recursive sequences, this enhanced method aims to fortify confidentiality and complexity in exchanged keys, enabling secure data transmission while remaining efficient for resource restricted IoT devices. Validation using the Affine encryption method demonstrates considerable improvements in complexity, security, and speed. Incorporating Galois field (GF) and third-order sequencing enhances secrecy and complexity, ensuring swift computational processes.

ON AMODULAR EQUATION OF DEGREE 23

On page 249 of his second notebook, Ramanujan incorrectly recorded and crossed out a modular equation of degree 23. B. C. Berndt has corrected and proved the same using the theory of modular forms. The main objective of this article is to prove the modular equation of degree 23 corrected by Berndt by employing the method of parametrization. Using a similar technique, we also prove a modular equation of degree 11 due to Ramanujan.