The Metrical Theory of Jacobi-Perron Algorithm

Schweiger, Fritz

doi:10.1007/bfb0059845

Cited by 52 publications

(35 citation statements)

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“…For more details, see for instance [13]. The dynamical system defined by ~ on the unit square has been much studied [40], in particular its invariant measures and its unique invariant ergodic probability measure equivalent to Lebesgue measure [14], [15] (generalization of the classical Gauss measure) .…”

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confidence: 99%

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux¹,

Berthé²,

Ito³

2002

Annales De L’institut Fourier

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confidence: 99%

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Arnoux¹,

Berthé²,

Ito³

2002

Annales De L’institut Fourier

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“…There are a lot of generalizations of continued fractions for the simultaneous case, starting with the work of Jacobi [11] which lead to the Jacobi-Perron-Algorithm (JPA) [20,28,26,2,8]. However, the JPA is not able to compute solutions to such approximation quality as we will require in our proposed commitment scheme (cf.…”

Section: Theoremmentioning

confidence: 99%

Using the Inhomogeneous Simultaneous Approximation Problem for Cryptographic Design

Armknecht

Elsner

Schmidt

2011

Lecture Notes in Computer Science

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Abstract. Since the introduction of the concept of provable security, there has been the steady search for suitable problems that can be used as a foundation for cryptographic schemes. Indeed, identifying such problems is a challenging task. First, it should be open and investigated for a long time to make its hardness assumption plausible. Second, it should be easy to construct hard problem instances. Third, it should allow to build cryptographic applications on top of them. Not surprisingly, only a few problems are known today that satisfy all conditions, e. g., factorization, discrete logarithm, and lattice problems. In this work, we introduce another candidate: the Inhomogeneous Simultaneous Approximation Problem (ISAP), an old problem from the field of analytic number theory that dates back to the 19th century. Although the Simultaneous Approximation Problem (SAP) is already known in cryptography, it has mainly been considered in its homogeneous instantiation for attacking schemes. We take a look at the hardness and applicability of ISAP, i. e., the inhomogeneous variant, for designing schemes. More precisely, we define a decisional problem related to ISAP, called DISAP, and show that it is NP-complete. With respect to its hardness, we review existing approaches for computing a solution and give suggestions for the efficient generation of hard instances. Regarding the applicability, we describe as a proof of concept a bit commitment scheme where the hiding property is directly reducible to DISAP. An implementation confirms its usability in principle (e. g., size of one commitment is slightly more than 6 KB and execution time is in the milliseconds).

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“…Werke published in 1891, [19]. In 1907, Perron [27] developed a generalization of this algorithm for higher dimensions (for a complete survey about the Jacobi-Perron algorithm see [10] and [30]). Further results on the Jacobi-Perron algorithm can be found in [28] and [36].…”

Section: Introductionmentioning

confidence: 99%

On the periodic writing of cubic irrationals and a generalization of Rédei functions

Murru

2015

Int. J. Number Theory

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In this paper, we provide a periodic representation (by means of periodic rational or integer sequences) for any cubic irrationality. In particular, for a root α of a cubic polynomal with rational coefficients, we study the Cerruti polynomials µUsing these polynomials, we show how any cubic irrational can be written periodically as a ternary continued fraction. A periodic multidimensioanl continued fraction (with pre-period of length 2 and period of length 3) is proved convergent to a given cubic irrationality, by using the algebraic properties of cubic irrationalities and linear recurrent sequences.

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The Metrical Theory of Jacobi-Perron Algorithm

Cited by 52 publications

References 0 publications

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Discrete planes, ${\Bbb Z}^2$-actions, Jacobi-Perron algorithm and substitutions

Using the Inhomogeneous Simultaneous Approximation Problem for Cryptographic Design

On the periodic writing of cubic irrationals and a generalization of Rédei functions

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