Alternative Digit Sets for Nonadjacent Representations

Muir, James A.; Stinson, Douglas R.

doi:10.1137/s0895480103437651

Cited by 10 publications

(23 citation statements)

References 7 publications

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“…Muir and Stinson [12,13] study digit sets D = {0, 1, x} for integers x. The following results have been proved in their paper:…”

Section: Nonadjacent Digit Setsmentioning

confidence: 99%

“…Muir and Stinson [12,13]) that n ∈ N admits a D-NAF if and only if r(n) admits a D-NAF and that if (. .…”

Section: Nonadjacent Digit Setsmentioning

confidence: 99%

“…Muir and Stinson [12,13] give a list of all NADS {0, 1, x} with |x| ≤ 10 000. The list starts with 3, −1, −5, −13, −17, −25, −29, −37, −53, −61, −65, .…”

Section: Algorithm 1 Check For Nadsmentioning

confidence: 99%

“…Recently, Muir and Stinson [12,13] asked for other sets of digits, in particular of the form {0, 1, x}, such that every (positive) integer has a unique representation using these digits, base 2, and obeying again the condition that two adjacent digits cannot both be different from zero.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of Alternative Digit Sets for Nonadjacent Representations

Heuberger

Prodinger

2006

Mh Math

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Abstract. It is known that every positive integer n can be represented as a finite sum of the form i a i 2 i , where a i ∈ {0, 1, −1} and no two consecutive a i 's are non-zero ("nonadjacent form", NAF). Recently, Muir and Stinson [12, 13] investigated other digit sets of the form {0, 1, x}, such that each integer has a nonadjacent representation (such a number x is called admissible). The present paper continues this line of research.The topics covered include transducers that translate the standard binary representation into such a NAF and a careful topological study of the (exceptional) set (which is of fractal nature) of those numbers where no finite look-ahead is sufficient to construct the NAF from left-to-right, counting the number of digits 1 (resp. x) in a (random) representation, and the non-optimality of the representations if x is different from 3 or −1.

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“…Muir and Stinson [12,13] study digit sets D = {0, 1, x} for integers x. The following results have been proved in their paper:…”

Section: Nonadjacent Digit Setsmentioning

confidence: 99%

“…Muir and Stinson [12,13]) that n ∈ N admits a D-NAF if and only if r(n) admits a D-NAF and that if (. .…”

Section: Nonadjacent Digit Setsmentioning

confidence: 99%

“…Muir and Stinson [12,13] give a list of all NADS {0, 1, x} with |x| ≤ 10 000. The list starts with 3, −1, −5, −13, −17, −25, −29, −37, −53, −61, −65, .…”

Section: Algorithm 1 Check For Nadsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of Alternative Digit Sets for Nonadjacent Representations

Heuberger

Prodinger

2006

Mh Math

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“…Such sets are called non-adjacent digit sets (NADS). Muir and Stinson [15,16] gave new results at SAC 2003, proposing some properties that x must verify in order to lead to a NADS and they gave some infinite families of x such that {0, 1, x} is or is not a NADS. In the latter case, we say that {0, 1, x} is a NON-NADS.…”

Section: Introductionmentioning

confidence: 99%

Advances in Alternative Non-adjacent Form Representations

Avoine

Monnerat

Peyrin

2004

Progress in Cryptology - INDOCRYPT 2004

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Abstract. From several decades, non-adjacent form (NAF) representations for integers have been extensively studied as an alternative to the usual binary number system where digits are in {0, 1}. In cryptography, the non-adjacent digit set (NADS) {−1, 0, 1} is used for optimization of arithmetic operations in elliptic curves. At SAC 2003, Muir and Stinson published new results on alternative digit sets: they proposed infinite families of integers x such that {0, 1, x} is a NADS as well as infinite families of integers x such that {0, 1, x} is not a NADS, so called a NON-NADS. Muir and Stinson also provided an algorithm that determines whether x leads to a NADS by checking if every integer n ∈ [0, −x 3] has a {0, 1, x}-NAF. In this paper, we extend these results by providing generators of NON-NADS infinite families. Furthermore, we reduce the search bound from. We introduce the notion of worst NON-NADS and give the complete characterization of such sets. Beyond the theoretical results, our contribution also aims at exploring some algorithmic aspects. We supply a much more efficient algorithm than those proposed by Muir and Stinson, which takes only 343 seconds to compute all x's from 0 to −10 7 such that {0, 1, x} is a NADS.

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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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Alternative Digit Sets for Nonadjacent Representations

Cited by 10 publications

References 7 publications

Analysis of Alternative Digit Sets for Nonadjacent Representations

Analysis of Alternative Digit Sets for Nonadjacent Representations

Advances in Alternative Non-adjacent Form Representations

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

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