The Jacobi-Perron Algorithm Its Theory and Application

Bernstein, Leon

doi:10.1007/bfb0069405

Cited by 129 publications

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“…Bernstein [1] gave some classes of periodic continued fractions obtained by the Jacobi-Perron algorithm. We can also give some examples of periodic expansions obtained by the AJPA, for example:…”

Section: The Cubic Casementioning

confidence: 99%

A new multidimensional continued fraction algorithm

Tamura¹,

Yasutomi

2009

Math. Comp.

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Abstract. It has been believed that the continued fraction expansion of (α, β) (1, α, β is a Q-basis of a real cubic field) obtained by the modified JacobiPerron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of () ( x denoting the fractional part of x). We present a new algorithm which is something like the modified Jacobi-Perron algorithm, and give some experimental results with this new algorithm. From our experiments, we can expect that the expansion of (α, β) with our algorithm always becomes periodic for any real cubic field. We also consider real quartic fields. IntroductionThe study of continued fractions has a long history dating back to J. Wallis (1616-1703) and Ch. Huygens (1629-1695) [8]. In particular, many kinds of higherdimensional continued fractions have been studied starting with K.G. Jacobi (1804-1851) [7]. A central problem has been to find a higher-dimensional generalization of Legendre's theorem concerning the periodic continued fractions. In fact, the following conjecture has been believed. 1, α 1 Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of ( Conjecture. Let) by the modified Jacobi-Perron algorithm.In this paper, we give some candidates of algorithms of continued fraction expansion of dimensions 2 and 3, which can be easily generalized to any dimension.By numerical experiments, we checked that, for instance, ( √ m / ∈ Q) obtained by our algorithm become periodic. We showed the periodicity

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Section: The Cubic Casementioning

confidence: 99%

A new multidimensional continued fraction algorithm

Tamura¹,

Yasutomi

2009

Math. Comp.

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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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“…Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13].…”

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confidence: 99%

Multi-continued fraction algorithm on multi-formal Laurent series

Dai¹,

Wang²,

Ye³

2006

Acta Arith.

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1. Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13]. These algorithms are adapted to study the same problem for multi-formal Laurent series [2,4,11,12]. But none of them guarantees the best rational approximation in general. In this paper, we deal with the multi-rational approximation problem over the formal Laurent series field F ((z −1 )): given an element r ∈ F ((z −1 )) m , find p ∈ F [z] m and q ∈ F [z] such that p/q approximates r as close as possible while deg(q) is bounded.We propose a new continued fraction algorithm for multi-formal Laurent series. It is proved that this algorithm guarantees best rational approximations for multi-formal Laurent series.The paper is organized as follows: Section 2 deals with the indexed valuation of F ((z −1 )) m . Section 3 contains the detailed definition of the problem of optimal rational approximation of multi-formal Laurent series. Section 4 proposes an algorithm called multidimensional continued fraction algorithm (m-CFA, for short), which produces a multi-continued fraction expansion C(r) for any given multi-formal Laurent series r. Section 5 shows that C(r) satisfies three basic conditions. Section 6 states the main results of this pa-

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The Jacobi-Perron Algorithm Its Theory and Application

Cited by 129 publications

References 0 publications

A new multidimensional continued fraction algorithm

A new multidimensional continued fraction algorithm

Using the Inhomogeneous Simultaneous Approximation Problem for Cryptographic Design

Multi-continued fraction algorithm on multi-formal Laurent series

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