Simultaneous Diophantine Approximation Problem Research Articles

Another Look at the Security Analysis of the Modulus N = p2q by Utilizing an Approximation Approach for ϕ(N)

Newly developed techniques have been recently documented, which capitalize on the security provided by prime power modulus denoted as N = prqs where2 ≤ s < r. Previousresearchprimarilyconcentrated on the factorization of the modulus of type at minimum N = p3q2. In contrast, within the context of 2 ≤ s < r, we address scenarios in the modulus N = p2q (i.e. r = 2 and s = 1) still need to be covered, showing a significant result to the field of study. This work presents two factorization approaches for the multiple moduli Ni = p2 iqi, relying on a good approximation of the Euler’s totient function ϕ(Ni). The initial method for factorization deals with the multiple moduli Ni = p2 iqi derived from m public keys (Ni,ei) and is interconnected through the equation eid − kiϕ(Ni) = 1. In contrast, the second factorization method is associated with the eidi − kϕ(Ni) = 1. By reorganizing the equations as a simultaneous Diophantine approximation problem and implementing the LLL algorithm, it becomes possible to factorize the list of moduli Ni = p2 iqi concurrently, given that the unknowns d, di, k, and ki are suff iciently small. The key difference between our results and the referenced work is that we cover a real-world cryptosystem that uses the modulus N =p2q. In contrast, the previous work covers a hypothetical situation of modulus in the form of N = prqs.

PKCHD: Towards a Probabilistic Knapsack Public-Key Cryptosystem with High Density

By introducing an easy knapsack-type problem, a probabilistic knapsack-type public key cryptosystem (PKCHD) is proposed. It uses a Chinese remainder theorem to disguise the easy knapsack sequence. Thence, to recover the trapdoor information, the implicit attacker has to solve at least two hard number-theoretic problems, namely integer factorization and simultaneous Diophantine approximation problems. In PKCHD, the encryption function is nonlinear about the message vector. Under the re-linearization attack model, PKCHD obtains a high density and is secure against the low-density subset sum attacks, and the success probability for an attacker to recover the message vector with a single call to a lattice oracle is negligible. The infeasibilities of other attacks on the proposed PKCHD are also investigated. Meanwhile, it can use the hardest knapsack vector as the public key if its density evaluates the hardness of a knapsack instance. Furthermore, PKCHD only performs quadratic bit operations which confirms the efficiency of encrypting a message and deciphering a given cipher-text.

Computing Extremely Large Values of the Riemann Zeta Function

The paper summarizes the computation results pursuing peak values of the Riemann zeta function. The computing method is based on the RS-Peak algorithm by which we are able to solve simultaneous Diophantine approximation problems efficiently. The computation environment was served by the SZTAKI Desktop Grid operated by the Laboratory of Parallel and Distributed Systems at the Hungarian Academy of Sciences and the ATLAS supercomputing cluster of the Eotvos Lorand University, Budapest. We present the largest Riemann zeta value known till the end of 2016.

Lattice Points in Algebraic Cross-polytopes and Simplices

The number of lattice points \(| tP \cap \mathbb {Z}^d |\), as a function of the real variable \(t>1\) is studied, where \(P \subset \mathbb {R}^d\) belongs to a special class of algebraic cross-polytopes and simplices. It is shown that the number of lattice points can be approximated by an explicitly given polynomial of t depending only on P. The error term is related to a simultaneous Diophantine approximation problem for algebraic numbers, as in Schmidt’s theorem. The main ingredients of the proof are a Poisson summation formula for general algebraic polytopes, and a representation of the Fourier transform of the characteristic function of an arbitrary simplex in the form of a complex line integral.

SOME NOTES ON THE REGULAR GRAPH DEFINED BY SCHMIDT AND SUMMERER AND UNIFORM APPROXIMATION

Within the study of parametric geometry of numbers W. Schmidt and L. Summerer introduced so-called regular graphs. Roughly speaking the successive minima functions for the classical simultaneous Diophantine approximation problem have a very special pattern if the vector $\underline{\zeta}$ induces a regular graph. The regular graph is in particular of interest due to a conjecture by Schmidt and Summer concerning classic approximation constants. This paper aims to provide several new results on the behavior of the successive minima functions for the regular graph. Moreover, we improve the best known upper bounds for the classic approximation constants $\hat{w}_{n}(\zeta)$, valid uniformly for all transcendental $\zeta$, provided that the Schmidt-Summerer conjecture is true.

Efficient computing of n-dimensional simultaneous Diophantine approximation problems

Abstract In this paper we consider two algorithmic problems of simultaneous Diophantine approximations. The first algorithm produces a full solution set for approximating an irrational number with rationals with common denominators from a given interval. The second one aims at finding as many simultaneous solutions as possible in a given time unit. All the presented algorithms are implemented, tested and the PariGP version made publicly available.

Fast public-key encryption scheme based on Chinese remainder theorem

Traditional public-key cryptosystems suffer from a relatively low encryption/decryption speed, which hampers their applications in resource-constrained environments. A fast public-key cryptosystem is proposed to remedy this drawback. The new algorithm uses Chinese remainder theorem to hide the trapdoor information. The encryption of the system only carries out several modular multiplication operations, and the decryption only needs a modular multiplication and a low-dimensional matrixvector multiplication, which makes the speed of the encryption and the decryption of the scheme very high. The security of the system is based on two difficult number-theoretic problems. The attacker has to solve the integer factorization problem and the simultaneous Diophantine approximation problem simultaneously to recover the secret key from the public key. The proposed cryptosystem is also shown to be secure against lattice attack. The analysis shows that the encryption algorithm is a secure, fast and efficient public-key cryptosystem.

Cryptanalysis of a public key cryptosystem based on two cryptographic assumptions

Baocang and Yupu proposed a relatively fast public key cryptosystem. The authors claim that the security of their system is based on two number-theoretic hard problems, namely the simultaneous Diophantine approximation problem and the integer factorisation problem. In this article we present a polynomial time heuristic attack that enables us to recover the private key from the public key. In particular, we show that breaking the system can be reduced to finding a short vector in a lattice which can be achieved using the L3-lattice reduction algorithm.

Public key cryptosystem based on two cryptographic assumptions

A new fast public key cryptosystem is proposed, which is based on two dissimilar number-theoretic hard problems, namely the simultaneous Diophantine approximation problem and integer factorisation problem. The adversary has to solve the two hard problems simultaneously to recover the plaintext according to their knowledge about the public keys and the cipher-text. Therefore, the scheme is expected to gain a high level of security. The newly-designed public key cryptosystem is efficient with respect to encryption and decryption. The encryption of this system is about three times faster than that of RSA, and the decryption is six times faster than that of RSA. The cipher-text expansion of the system is about 8:3.

On finite-precision representations of geometric objects

The paper presents an experimental study on approximation of continuously distributed geometric objects by equations with integer coefficients. The problem considered here is, given an equation of a geometric object with real coefficients, to find a set of integer coefficients that gives a good approximation to the original equation. This problem is reduced to a problem called a simultaneous Diophantine approximation problem, for which no efficient method to give the best solution has yet been found. Here considered are three heuristic methods: a naive method, a method based on the continued fraction theory, and a method using the basis reduction technique. These methods together with a brute-force exhaustive search method are studied using randomly generated data, and also proposed is a hybrid method, which seems allowable from both the viewpoint of time complexity and the viewpoint of quality of approximation.

The Computational Complexity of Simultaneous Diophantine Approximation Problems

Simultaneous Diophantine approximation in d dimensions deals with the approximation of a vector ${\bf \alpha } = (\alpha _1 , \cdots ,\alpha _d )$ of d real numbers by vectors of rational numbers all having the same denominator. This paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector $ {\bf \alpha} $ of d rational numbers. We measure the goodness of an approximation using the sup norm. We show that a result of H. W. Lenstra, Jr. produces polynomial-time algorithms to find sup norm best approximations to a given vector ${\bf \alpha} $ when the dimension d is fixed. We show that a recent algorithm of A. K. Lenstra, H. W. Lenstra, Jr. and L. Lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector ${\bf \alpha} $ in d dimensions with a denominator $Q^ * $ satisfying $1 \leqq Q^ * \leqq 2^{d/2} N$ which is within a factor $\sqrt {5d} 2^{(d + 1)/2} $ of the best approximation with denominator Q with $1 \leqq Q \leqq N$. This algorithm runs in time polynomial in the input size, independent of the dimension d. We also prove complementary results showing that certain natural simultaneous Diophantine approximation problems are NP-hard. We show that the problem of deciding whether a given vector ${\bf \alpha} $ of rational numbers has a simultaneous approximation of specified accuracy with respect to the sup norm with denominator Q in a given interval $1 \leqq Q \leqq N$ is NP-complete. (Here the dimension d is allowed to vary.) We prove two other complexity results, which suggest that the problem of locating best (sup norm) simultaneous approximations is harder than this NP-complete problem.