Modular Equations Research Articles

Modular linear differential equations of fourth order and minimal [formula omitted]-algebras

A characterization of the minimal W-algebras associated with the Deligne exceptional series at level −h∨/6 is obtained by using one-parameter family of modular linear differential equations of order 4. In particular, the characters of the Ramond-twisted modules of minimal W-algebras related to the Deligne exceptional series satisfy one of these differential equations. In order to obtain the characterization, the differential equations in the one parameter family which have solutions of “CFT type” are classified, whose solutions are explicitly described.

Certain relationships between q- product identities, modular equations and combinatorial partition identities

Certain relationships between q- product identities, modular equations and combinatorial partition identities

Three Ramanujan continued fractions with modularity

We study three Ramanujan continued fractions c(τ),W(τ) and T(τ). In fact, c(τ) and W(τ) are modular functions of level 16, and T(τ) is a modular function of level 32. We first prove that the values of c(τ) and W(τ) can generate the ray class field modulo 4 over an imaginary quadratic field K. We also prove that 2/(1−c(τ)),1/W(τ),T(τ)+1/T(τ) are algebraic integers for any imaginary quadratic quantity τ. Furthermore, we find the modular equations of c(τ),T(τ) and W(τ) for any level, and we show that c(τ) and W(τ) satisfy the Kronecker's congruence. We can express the value c(rτ) (respectively, T(rτ),W(rτ)) in terms of radicals for any positive rational number r when the value c(τ) (respectively, T(τ),W(τ)) can be written as radicals.

Solving a class of modular polynomial equations and its relation to modular inversion hidden number problem and inversive congruential generator

In this paper we revisit the modular inversion hidden number problem (MIHNP) and the inversive congruential generator (ICG) and consider how to attack them more efficiently. We consider systems of modular polynomial equations of the form $$a_{ij}+b_{ij}x_i+c_{ij}x_j+x_ix_j=0~(\mathrm {mod}~p)$$ and show the relation between solving such equations and attacking MIHNP and ICG. We present three heuristic strategies using Coppersmith’s lattice-based root-finding technique for solving the above modular equations. In the first strategy, we use the polynomial number of samples and get the same asymptotic bound on attacking ICG proposed in PKC 2012, which is the best result so far. However, exponential number of samples is required in the work of PKC 2012. In the second strategy, a part of polynomials chosen for the involved lattice are linear combinations of some polynomials and this enables us to achieve a larger upper bound for the desired root. Corresponding to the analysis of MIHNP we give an explicit lattice construction of the second attack method proposed by Boneh, Halevi and Howgrave-Graham in Asiacrypt 2001. We provide better bound than that in the work of PKC 2012 for attacking ICG. Moreover, we propose the third strategy in order to give a further improvement in the involved lattice construction in the sense of requiring fewer samples.

Modular properties of 6d (DELL) systems

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge . The low-energy Seiberg-Witten prepotential ℱ(a), however, is not explicitly invariant, because the flat moduli also change a − → aD = ∂ℱ/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E2. This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N) theory with two independent modular parameters τ and widehat{tau} , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift of widehat{tau} and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

A continued fraction of order twelve as a modular function

We study a continued fraction U ( τ ) U(\tau ) of order twelve using the modular function theory. We obtain the modular equations of U ( τ ) U(\tau ) by computing the affine models of modular curves X ( Γ ) X(\Gamma ) with Γ = Γ 1 ( 12 ) ∩ Γ 0 ( 12 n ) \Gamma = \Gamma _1 (12) \cap \Gamma _0(12n) for any positive integer n n ; this is a complete extension of the previous result of Mahadeva Naika et al. and Dharmendra et al. to every positive integer n n . We point out that we provide an explicit construction method for finding the modular equations of U ( τ ) U(\tau ) . We also prove that these modular equations satisfy the Kronecker congruence relations. Furthermore, we show that we can construct the ray class field modulo 12 12 over imaginary quadratic fields by using U ( τ ) U(\tau ) and the value U ( τ ) U(\tau ) at an imaginary quadratic argument is a unit. In addition, if U ( τ ) U(\tau ) is expressed in terms of radicals, then we can express U ( r τ ) U(r \tau ) in terms of radicals for a positive rational number r r .

Recomposing Rational Functions

Let $A$ be a rational function. For any decomposition of $A$ into a composition of rational functions $A=U\circ V$ the rational function $\widetilde A=V\circ U$ is called an elementary transformation of $A$, and rational functions $A$ and $B$ are called equivalent if there exists a chain of elementary transformations between $A$ and $B$. This equivalence relation naturally appears in the complex dynamics as a part of the problem of describing of semiconjugate rational functions. In this paper we show that for a rational function $A$ its equivalence class $[A]$ contains infinitely many conjugacy classes if and only if $A$ is a flexible Latt\`es map. For flexible Latt\`es maps $L=L_j$ induced by the multiplication by 2 on elliptic curves with given $j$-invariant we provide a very precise description of $[ L]$. Namely, we show that any rational function equivalent to $ L_j$ necessarily has the form $ L_{j'}$ for some $j'\in \mathbb C$, and that the set of $j'\in \mathbb C$ such that $ L_{j'}\sim L_{j}$ coincides with the orbit of $j$ under the correspondence associated with the classical modular equation $\Phi_2(x,y)=0$.

On the optimality of lattices for the coppersmith technique

We investigate a method for finding small integer solutions of a univariate modular equation, that was introduced by Coppersmith (Proceedings of Eurocrypt 1996, LNCS, vol 1070, pp 155–165, 1996) and extended by May (New RSA vulnerabilities using lattice reduction methods, Ph.D. thesis, University of Paderborn, 2003). We will refer this method as the Coppersmith technique. This paper provides a way to analyze a general limitations of the lattice construction for the Coppersmith technique. Our analysis upper bounds the possible range of U that is asymptotically equal to the bound given by the original result of Coppersmith and May. This means that they have already given the best lattice construction. In addition, we investigate the optimality for the bivariate equation to solve the small inverse problem, which was inspired by Kunihiro’s (LNCS 7483:55–69, 2012) argument. In particular, we show the optimality for the Boneh–Durfee’s equation (Proceedings of Eurocrypt 1999, LNCS, vol 1592, pp 389–401, 1999) used for RSA cryptoanalysis, To show our results, we establish framework for the technique by following the relation of Howgrave-Graham (Proceedings of cryptography and coding, LNCS, vol 1355, pp 131–142, 1997), and then concretely define the conditions in which the technique succeed and fails. We then provide a way to analyze the range of U that satisfies these conditions. Technically, we show that the original result of Coppersmith achieves the optimal bound for U when constructing a lattice in the standard way. We then provide evidence which indicates that constructing a non-standard lattice is generally difficult.

Modular and duality properties of surface operators in mathcal{N}={2}^{star } gauge theories

We calculate the instanton partition function of the four-dimensional mathcal{N}={2}^{star } SU(N) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure mathcal{N}=2 or to mathcal{N}={2}^{star } gauge theories.

A level 16 analogue of Ramanujan series for 1/π

The modular functionh(τ)=q∏n=1∞(1−q16n)2(1−q2n)(1−qn)2(1−q8n) is called a level 16 analogue of Ramanujan's series for1/π. We prove that h(τ) generates the field of modular functions on Γ0(16) and find its modular equation of level n for any positive integer n. Furthermore, we construct the ray class field K(h(τ)) modulo 4 over an imaginary quadratic field K for τ∈K∩H such that Z[4τ] is the integral closure of Z in K, where H is the complex upper half plane. For any τ∈K∩H, it turns out that the value 1/h(τ) is integral, and we can also explicitly evaluate the values of h(τ) if the discriminant of K is divisible by 4.

New P–Q mixed modular equations and their applications

In his second notebook, Ramanujan recorded total of seven P–Q modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained for higher orders. As a consequence, several values of quotients of theta-function are evaluated. The cubic singular modulus is evaluated at \(q=\exp (-2\pi \sqrt{n/3})\) for \(n\in \{5k, 1/5k, 5/k, k/5\}\), where \(k\in \{4,7,16\}\).

The third order modular linear differential equations

We propose a third order generalization of the Kaneko–Zagier modular differential equation, which has two parameters. We describe modular and quasimodular solutions of integral weight in the case where one of the exponents at infinity is a multiple root of the indicial equation. We also classify solutions of “character type”, which are the ones that are expected to relate to characters of simple modules of vertex operator algebras and one-point functions of two-dimensional conformal field theories. Several connections to generalized hypergeometric series are also discussed.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus hypergeometric function example. We then focus on identities relating the same hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to , hypergeometric functions, and show that one just reduces to the previous cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

Hidden Number Problem and Its Applications in Crypto Scheme

This paper is based on the research result of Nguyen’s attacking against DSA .We extent the Hidden Number Method (HNM) from the univariate modular polynomials with one order to the univariate modular polynomials with higher order. There are many crypto schemes using the arithmetical and truncate operations. If the constant terms are partly known, these schemes can be written as the modular equations. We analysis a scheme using the arithmetical and truncate operations so as to improve the lattice model. On the supposed condition, we can recover the basic key with the probability 99%.

Inequalities and infinite product formula for Ramanujan generalized modular equation function

We present several inequalities for the Ramanujan generalized modular equation function $$\mu _{a}(r)=\pi F(a,1-a;1;1-r^2)/$$ $$[2\sin (\pi a)F(a,1-a;1;r^2)]$$ with $$a\in (0,1/2]$$ and $$r\in (0,1)$$ , and provide an infinite product formula for $$\mu _{1/4}(r)$$ , where $$F(a,b;c;x)={}_{2}F_{1}(a,b;c;x)$$ is the Gaussian hypergeometric function.

From Feynman rules to conserved quantum numbers, I

In the context of Quantum Field Theory (QFT) there is often the need to find sets of graph-like diagrams (the so-called Feynman diagrams) for a given physical model. If negative, the answer to the related problem ‘Are there any diagrams with this set of external fields?’ may settle certain physical questions at once. Here the latter problem is formulated in terms of a system of linear diophantine equations derived from the Lagrangian density, from which necessary conditions for the existence of the required diagrams may be obtained. Those conditions are equalities that look like either linear diophantine equations or linear modular (i.e. congruence) equations, and may be found by means of fairly simple algorithms that involve integer computations. The diophantine equations so obtained represent (particle) number conservation rules, and are related to the conserved (additive) quantum numbers that may be assigned to the fields of the model.

General Bounds for Small Inverse Problems and Its Applications to Multi-Prime RSA

In 1999, Boneh and Durfee introduced small inverse problems which solve bivariate modular equations \(x(N+y)\equiv 1 \pmod {e}\). Sizes of solutions for \(x,y\) are bounded by \(X=N^{\delta }\) and \(Y=N^{\beta }\), respectively. They solved the problems for \( \beta ={1/2}\) in the context of small secret exponents attacks on RSA. They proposed a polynomial time algorithm which works when \( \delta <(7-2 \sqrt{7})/6 \approx {0.284}\), and further improved a bound to \( \delta <1-1/\sqrt{2}\approx {0.292}\). So far, small inverse problems for arbitrary \({\beta }\) have also been considered. Generalizations of Boneh and Durfee’s lattices to achieve the stronger bound provide the bound \( \delta <1-\sqrt{\beta }\). However, the algorithm works only when \( \beta \ge 1/4\). When \(0<\beta <1/4\), there have been several works which claimed the best bounds. In this paper, we revisit the problems for arbitrary \( \beta \). At first, we summarize the previous results for \(0<\beta <1/4\). We reveal that there are some results which are not valid and show that Weger’s algorithm provide the best bounds. Next, we propose an improved algorithm to solve the problem for \(0<\beta <1/4\). Our algorithm works when \( \delta <1-2(\sqrt{\beta (3+4 \beta )}-\beta )/3\). Our algorithm construction is based on the combinations of Boneh and Durfee’s two forms of lattices. This construction is more natural compared with previous works. In addition, we introduce an application of our result, small secret exponent attacks on Multi-Prime RSA with small primes differences.

On the factorization of the discriminant of a classical modular equation

On the factorization of the discriminant of a classical modular equation

A continued fraction of Ramanujan and some Ramanujan-Weber class invariants

On Page 36 of his ``lost notebook, Ramanujan recorded four $q$-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two $q$-series representations of Ramanujan's continued fraction found in his ``lost notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants $g_n$ and $G_n$ and establish formulas connecting $g_n$ and $G_n$. We obtain relations between our continued fraction with the Ramanujan-G\{o}llnitz-Gordon and Ramanujan's cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction $A(q)$ which is related to Ramanujan's continued fraction $F(a,b,\lambda;q),$ the Ramanujan-G\{o}llnitz-Gordon continued fraction $H(q)$ and the Dedekind eta function $\eta(s)$.

Generalized Grötzsch ring function and generalized elliptic integrals

In this paper, we study the quotient of hypergeometric functions μa(r) in the theory of Ramanujan’s generalized modular equation for a ∈ (0, 1/2]. Several new inequalities are given for this and related functions. Our main results complement and generalize some known results in the literature.