Solving Norm Form Equations Via Lattice Basis Reduction
نویسنده
چکیده
The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm is then utilized to show that the integer solutions to NK/Q(x 4 √ N4 − 1 + y 4 √ N4 + 1 + z) = ±1 (where K = Q( 4 √ N4 − 1, 4 √ N4 + 1)) are given by (x, y, z) = (0, 0,±1), (±1, 0,±N) and (0,±1,±N), for 5 ≤ N ≤ 100.
منابع مشابه
Short review of lattice basis reduction types and his applications. (Russian)
This article presets a review of lattice lattice basis reduction types. Paper contains the main five types of lattice basis reduction: size reduced (weak Hermit), c-reduced, Lovasz condition, Hermit-Korkin-Zolotarev, Minkowski reduced. The article provides references to applications in: information theory (decoding of coding group in MIMO), calculus (minimize of the positive quadratic form), co...
متن کاملApplication of new basis functions for solving nonlinear stochastic differential equations
This paper presents an approach for solving a nonlinear stochastic differential equations (NSDEs) using a new basis functions (NBFs). These functions and their operational matrices are used for representing matrix form of the NBFs. With using this method in combination with the collocation method, the NSDEs are reduced a stochastic nonlinear system of equations and unknowns. Then, the error ana...
متن کاملLattice Basis Reduction in Infinity Norm
In the high-tech world of today, the demand for security is constantly rising. That is why identifying hard computational problems for cryptographical use has become a very important task. It is crucial to find computational problems, which complexity could provide a basis for the security of the cryptosystems. However, there are only very few hard computational problems that are useful for cry...
متن کاملCryptanalysis of a Public Key Cryptosystem Based on Diophantine Equations via Weighted LLL Reduction
In this paper, we give an attack against a public key cryptosystem based on Diophantine equations of degree increasing type (DEC) proposed by the third author ([Oku15]). We show that the security of DEC depends on the difficulty of finding special (relatively) short vectors in some lattices obtained from a public key and a ciphertext. The most important target vector in our attack is not necess...
متن کاملLattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems
We report on improved practical algorithms for lattice basis reduc tion We propose a practical oating point version of the L algorithm of Lenstra Lenstra Lov asz We present a variant of the L algorithm with deep insertions and a practical algorithm for block Korkin Zolotarev reduction a concept introduced by Schnorr Empirical tests show that the strongest of these algorithms solves al most all ...
متن کامل