Canonical number systems in imaginary quadratic fields

Катаи, И.; Kovács, Beáta

doi:10.1007/bf01904880

Cited by 84 publications

(52 citation statements)

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“…Many papers are devoted to this problem. Generalizing former results of I. Kátai & B. Kovács [6], [7], B. Kovács proved:…”

supporting

confidence: 69%

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New criteria for canonical number systems

Akiyama

Rao

2004

Acta Arith.

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“…Many papers are devoted to this problem. Generalizing former results of I. Kátai & B. Kovács [6], [7], B. Kovács proved:…”

supporting

confidence: 69%

“…, |p 0 |−1}) is said to form a canonical number system if P (x) is a CNS polynomial. Here we only refer to the original studies of I. Kátai & J. Szabó [8], I. Kátai & B. Kovács [6], [7], W. Gilbert [3] and B. Kovács & A. Pethő [9].…”

mentioning

confidence: 99%

New criteria for canonical number systems

Akiyama

Rao

2004

Acta Arith.

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“…Extensions to rings of integers of algebraic number fields were worked out by Knuth [133], Penney [143], Kátai and Szabó [128], Kátai and Kovács [129,130], Kovács [134], Gilbert [103], Kovács and Pethő [135], Kátai and Környei [131], and many others. The main directions of research concern the description of rings having a canonical number system, and the description of the canonical number systems if they exist.…”

Section: Some Of the Most Important Joint Research Topics And Resultsmentioning

confidence: 99%

30 years of collaboration

Fuchs

Hajdu

2016

Period Math Hung

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Abstract. We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem), which will be presented in turn.

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“…Thus, {0, 1} is a 1-NADS. This is equivalent to saying that τ is the base of a canonical number system in Z[τ ] in the sense of [13], and is a particular case of results from [12].…”

Section: Remarkmentioning

confidence: 99%

On Redundant τ-Adic Expansions and Non-adjacent Digit Sets

Avanzi

Heuberger

Prodinger

2007

Selected Areas in Cryptography

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Abstract. This paper studies τ -adic expansions of scalars, which are important in the design of scalar multiplication algorithms on Koblitz Curves, and are less understood than their binary counterparts.At Crypto '97 Solinas introduced the width-w τ-adic non-adjacent form for use with Koblitz curves. It is an expansion of integers z = È i=0 ziτ i , where τ is a quadratic integer depending on the curve, such that zi = 0 implies zw+i−1 = . . . = zi+1 = 0, like the sliding window binary recodings of integers. We show that the digit sets described by Solinas, formed by elements of minimal norm in their residue classes, are uniquely determined. However, unlike for binary representations, syntactic constraints do not necessarily imply minimality of weight.Digit sets that permit recoding of all inputs are characterized, thus extending the line of research begun by Muir and Stinson at SAC 2003 to Koblitz Curves.Two new useful digit sets are introduced: one set makes precomputations easier, the second set is suitable for low-memory applications, generalising an approach started by Avanzi, Ciet, and Sica at PKC 2004 and continued by several authors since. Results by Solinas, and by Blake, Murty, and Xu are generalized.Termination, optimality, and cryptographic applications are considered. We show how to perform a "windowed" scalar multiplication on Koblitz curves without doing precomputations first, thus reducing memory storage dependent on the base point to just one point.

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Canonical number systems in imaginary quadratic fields

Cited by 84 publications

References 0 publications

New criteria for canonical number systems

New criteria for canonical number systems

30 years of collaboration

On Redundant τ-Adic Expansions and Non-adjacent Digit Sets

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