Simultaneous Diophantine Approximation Research Articles

A note on weighted simultaneous Diophantine approximation on manifolds

A note on weighted simultaneous Diophantine approximation on manifolds

Measure theoretic laws for limsup sets defined by rectangles

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles. Applications are given for simultaneous Diophantine approximation, Khintchine-Groshev theorem for rectangles and higher dimensional shrinking target problems.

On the dimension drop conjecture for diagonal flows on the space of lattices

Let X=G/Γ, where G is a Lie group and Γ is a lattice in G, let U be an open subset of X, and let {gt} be a one-parameter subgroup of G. Consider the set of points in X whose gt-orbit misses U; it has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when G is a simple Lie group of real rank 1. In this paper we prove this conjecture for the case G=SLm+n(R), Γ=SLm+n(Z) and gt=diag(ent,…,ent,e−mt,…,e−mt), in fact providing an effective estimate for the codimension. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on SLm+n(R)/SLm+n(Z). We also discuss an application to the problem of improving Dirichlet's theorem in simultaneous Diophantine approximation.

Diophantische Approximationen

This workshop was focused on a large variety of problems which have seen important progress during the last few years, such as extensions and refinements of Schmidt Subspace Theorem, together with new applications, works on the Zilber-Pink conjecture and on unlikely intersections, geometry of numbers, simultaneous Diophantine approximation, theory of heights, continued fractions, and arithmetic dynamics.

Simultaneous p-adic Diophantine approximation

AbstractThe aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approximation in the p-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on p-adic manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.

Public key exponent attacks on multi-prime power modulus using continued fraction expansion method

This paper proposes three public key exponent attacks of breaking the security of the prime power modulus 𝑁=𝑝2𝑞2 where 𝑝 and 𝑞 are distinct prime numbers of the same bit size. The first approach shows that the RSA prime power modulus 𝑁=𝑝2𝑞2 for q<𝑝<2q using key equation 𝑒𝑑−𝑘𝜙(𝑁)=1 where 𝜙(𝑁)= 𝑝2𝑞2(𝑝−1)(𝑞−1) can be broken by recovering the secret keys 𝑘 /𝑑 from the convergents of the continued fraction expansion of e/𝑁−2𝑁3/4 +𝑁1/2 . The paper also reports the second and third approaches of factoring 𝑛 multi-prime power moduli 𝑁𝑖=𝑝𝑖2 𝑞𝑖2 simultaneously through exploiting generalized system of equations 𝑒𝑖𝑑−𝑘𝑖𝜑(𝑁𝑖)=1 and 𝑒𝑖𝑑𝑖−𝑘𝜑(𝑁𝑖)=1 respectively. This can be achieved in polynomial time through utilizing Lenstra Lenstra Lovasz (LLL) algorithm and simultaneous Diophantine approximations method for 𝑖=1,2,…,𝑛.

Random walks, spectral gaps, and Khintchine’s theorem on fractals

This work addresses problems on simultaneous Diophantine approximation on fractals, motivated by a long standing problem of Mahler regarding Cantor’s middle 1/3 set. We obtain the first instances where a complete analogue of Khintchine’s Theorem holds for fractal measures. Our results apply to fractals which are self-similar by a system of rational similarities of \({\mathbb {R}}^d\) (for any \(d\ge 1\)) and have sufficiently small Hausdorff co-dimension. A concrete example of such measures in the context of Mahler’s problem is the Hausdorff measure on the “middle \(1/5\) Cantor set”; i.e. the set of numbers whose base \(5\) expansions miss a single digit. The key new ingredient is an effective equidistribution theorem for certain fractal measures on the homogeneous space \({\mathcal {L}}_{d+1}\) of unimodular lattices; a result of independent interest. The latter is established via a new technique involving the construction of S-arithmetic operators possessing a spectral gap and encoding the arithmetic structure of the maps generating the fractal. As a consequence of our methods, we show that spherical averages of certain random walks naturally associated to the fractal measures effectively equidistribute on \({\mathcal {L}}_{d+1}\).

Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For β>1 let Tβ be the β-transformation on [0,1]. We determine the Lebesgue measure and Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβnx−f(x,y)|<φ(n) for infinitely many n∈N}, where f:[0,1]2→[0,1] is a Lipschitz function and φ is a positive function on N. Let β2≥β1>1, f1,f2:[0,1]→[0,1] be two Lipschitz functions, τ1,τ2 be two positive continuous functions on [0,1]. We also determine the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−f1(x)|<β1−nτ1(x)|Tβ2ny−f2(y)|<β2−nτ2(y) for infinitely many n∈N}. Under certain additional assumptions, the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−g1(x,y)|<β1−nτ1(x)|Tβ2ny−g2(x,y)|<β2−nτ2(y) for infinitely many n∈N} is also determined, where g1,g2:[0,1]2→[0,1] are two Lipschitz functions.

Identifiability of Asymmetric Circular and Cylindrical Distributions

Identifiability of statistical models is a fundamental and essential condition that is required to prove the consistency of maximum likelihood estimators. The identifiability of the skew families of distributions on the circle and cylinder for estimating model parameters has not been fully investigated in the literature. In this paper, a new method combining the trigonometric moments and the simultaneous Diophantine approximation is proposed to prove the identifiability of asymmetric circular and cylindrical distributions. Using this method, we prove the identifiability of general sine-skewed circular distributions, including the sine-skewed von Mises and sine-skewed wrapped Cauchy distributions, and that of a M\"{o}bius transformed cardioid distribution, which can be regarded as asymmetric distributions on the unit circle. In addition, we prove the identifiability of two cylindrical distributions wherein both marginal distributions of a circular random variable are the sine-skewed wrapped Cauchy distribution, and conditional distributions of a random variable on the non-negative real line given the circular random variable are a Weibull distribution and a generalized Pareto-type distribution, respectively.

Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in ${\mathbb {R}}^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine–Groshev theorem that does not carry a monotonicity assumption when $nm&amp;gt;2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani [11] that monotonicity is not required when $nm&amp;gt;1$. That result resolved a conjecture of Beresneich et al. [5], and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin–Schaeffer conjecture. When $nm=1$, it is known by work of Duffin and Schaeffer [16] that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine–Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.

Factoring the Modulus of Type N = p2q by Finding Small Solutions of the Equation er − (Ns + t) = αp2 + βq2

The modulus of type N=p2q is often used in many variants of factoring-based cryptosystems due to its ability to fasten the decryption process. Faster decryption is suitable for securing small devices in the Internet of Things (IoT) environment or securing fast-forwarding encryption services used in mobile applications. Taking this into account, the security analysis of such modulus is indeed paramount. This paper presents two cryptanalyses that use new enabling conditions to factor the modulus N=p2q of the factoring-based cryptosystem. The first cryptanalysis considers a single user with a public key pair (e,N) related via an arbitrary relation to equation er−(Ns+t)=αp2+βq2, where r,s,t are unknown parameters. The second cryptanalysis considers two distinct cases in the situation of k-users (i.e., multiple users) for k≥2, given the instances of (Ni,ei) where i=1,…,k. By using the lattice basis reduction algorithm for solving simultaneous Diophantine approximation, the k-instances of (Ni,ei) can be successfully factored in polynomial time.

Increment of insecure RSA private exponent bound through perfect square RSA diophantine parameters cryptanalysis

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e. In this work, first we show that if e satisfies the Diophantine equation of the form ex2−ϕ(N)y2=z for appropriate values of x,y and z under certain specified conditions, then one is able to factor N. That is, the unknown yx can be found amongst the convergents of eN via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given (Ni,ei) for i=1,2,⋯,k and a fixed x that fulfills the Diophantine equation eix2−yi2ϕ(Ni)=zi. This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

Exploiting the security of N = prqs through approximation of ϕ(N)

This paper reports new techniques that exploit the security of the prime power moduli [Formula: see text] using continued fraction method. Our study shows that the key equation [Formula: see text] can be exploited using [Formula: see text] as good approximation of [Formula: see text]. This enables us to get [Formula: see text] from the convergents of the continued fractions expansion of [Formula: see text] where the bound of the private exponent is [Formula: see text] which leads to the polynomial time factorization of the moduli [Formula: see text]. We further report the polynomial time attacks that can break the security of the generalized prime power moduli [Formula: see text] using generalized system of equation of the form [Formula: see text] and [Formula: see text] by applying simultaneous Diophantine approximations and LLL algorithm techniques where [Formula: see text] and [Formula: see text].

Extension of simultaneous Diophantine approximation algorithm for partial approximate common divisor variants

A simultaneous Diophantine approximation (SDA) algorithm takes instances of the partial approximate common divisor (PACD) problem as input and outputs a solution. While several encryption schemes have been published and their securities depend on the presumed hardness of variant of the PACD problem, fewer studies have attempted to extend the SDA algorithm to be applicable to these variants. In this study, the SDA algorithm is extended to solve the general PACD problem. In order to proceed, first the variants of the PACD problem are classified and how to extend the SDA algorithm for each is suggested. Technically, the authors show that a short vector of some lattice used in the SDA algorithm gives an algebraic relation between secret parameters. Then, all the secret parameters can be recovered by finding this short vector. It is also confirmed experimentally that this algorithm works well.

Lyapunov exponent of random dynamical systems on the circle

AbstractWe consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

Cryptanalysis of Modular Exponentiation Outsourcing Protocols

Abstract Public-key cryptographic primitives are time consuming for resource-constrained devices. A classical problem is to securely offload group exponentiations from a (comparatively) weak device—the client—to an untrusted more powerful device—the server. A delegation protocol must usually meet two security objectives: privacy—the exponent or the base should not be revealed to a passive adversary—and verifiability—a malicious server should not be able to make the client accept an invalid value as the result of the delegated computation. Most proposed protocols relies on a secret splitting of the exponent and the base, and a considerable amount of literature has been devoted to their analysis. Recently, Su et al. (Su, Q., Zhang, R. and Xue, R. (2020) Secure outsourcing algorithms for composite modular exponentiation based on single untrusted cloud. Comput. J., 63, 1271.) and Rangasamy and Kuppusamy (Rangasamy, J. and Kuppusamy, L. (2018) Revisiting Single-Server Algorithms for Outsourcing Modular Exponentiation. In Chakraborty, D. and Iwata, T. (eds), Progress in Cryptology - INDOCRYPT 2018: 19th International Conference in Cryptology in India, New Delhi, India, December 912, Vol. 11356, Lecture Notes in Computer Science. Springer, Heidelberg, Germany, pp. 320. proposed outsourcing protocols for modular exponentiations. They claim that their protocols achieve security (privacy and verifiability). We show that these claims are flawed and that their schemes are broken beyond repair. They remain insecure even if one increases significantly the proposed parameters (and consequently the protocols computational and communication complexities). Our attacks rely on standard lattice-based cryptanalytic techniques, namely the Coppersmith methods to find small integer zeroes of modular multivariate polynomials and simultaneous Diophantine approximation methods for the so-called approximate greatest common divisor problem.

Quasiperiodic Patterns of the Complex Dimensions of Nonlattice Self-Similar Strings, via the LLL Algorithm

The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász.

Strategic Way of Factoring Modulus ``\(N = p^r q\)

Cryptography is fundamental to the provision of a wider notion of information security. Electronic information can easily be transmitted and stored in relatively insecure environments. This research was present to factor the prime power modulus \(N = p^r q\) for \(r \geq 2\) using the RSA key equation, if \(\frac{y}{x}\) is a convergents of the continued fractions expansions of \(\frac{e}{N - \left(2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}} - 2^{\frac{r-1}{r+1}} N^{\frac{r-1}{r+1}}\right)}\). We furthered our analysis on \(n\) prime power moduli \(N_i = p_i^r q_i\) by transforming the generalized key equations into Simultaneous Diophantine approximations and using the LLL algorithm on \(n\) prime power public keys \((N_i,e_i)\) we were able to factorize the \(n\) prime power moduli \(N_i = p_i^r q_i\), for \(i = 1,....,n\) simultaneously in polynomial time.

Equidistribution of random walks on compact groups II. The Wasserstein metric

We consider a random walk Sk with i.i.d. steps on a compact group equipped with a bi-invariant metric. We prove quantitative ergodic theorems for the sum ∑ k=1Nf(Sk) with Hölder continuous test functions f, including the central limit theorem, the law of the iterated logarithm and an almost sure approximation by a Wiener process, provided that the distribution of Sk converges to the Haar measure in the p-Wasserstein metric fast enough. As an example, we construct discrete random walks on an irrational lattice on the torus Rd/Zd, and find their precise rate of convergence to uniformity in the p-Wasserstein metric. The proof uses a new Berry–Esseen type inequality for the p-Wasserstein metric on the torus, and the simultaneous Diophantine approximation properties of the lattice. These results complement the first part of this paper on random walks with an absolutely continuous component and quantitative ergodic theorems for Borel measurable test functions.

Reductions between short vector problems and simultaneous approximation

In 1982, Lagarias showed that solving the approximate Shortest Vector Problem also solves the problem of finding good simultaneous Diophantine approximations. Here we provide a deterministic, dimension-preserving reduction in the reverse direction. It has polynomial time and space complexity, and it is gap-preserving under the appropriate norms. We also give an alternative to the Lagarias algorithm by first reducing his version of simultaneous approximation to one with no explicit range in which a solution is sought.