Minimal weight digit set conversions

Phillips, Braden J.; Burgess, Neil

doi:10.1109/tc.2004.14

Cited by 26 publications

(20 citation statements)

References 29 publications

(43 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By "right-to-left" we mean that the the digits of α are determined in turn from least-to most-significant. Phillips and Burgess [13] generalize all previously known right-to-left constructions by presenting a construction which produces minimal weight 1 representations using the digit set D ,u , which is defined as follows:…”

Section: Algorithm 2 Scalar Multiplication Via Horner's Rulementioning

confidence: 99%

“…To compute nP using Algorithm 2 where n is encoded in binary, we can first compute a minimal weight radix-2 representation of n with digits in D ,u using the right-to-left method of Phillips and Burgess [13]. However, this approach presents a slight annoyance to implementors: Algorithm 2 processes the digits of α = a s−1 .…”

Section: Algorithm 2 Scalar Multiplication Via Horner's Rulementioning

confidence: 99%

“…This family of digit sets has been studied previously by Phillips and Burgess [13] and Heuberger and Muir [5]. Both works contain algorithms which construct minimal weight representations from right to left.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Unbalanced digit sets and the closest choice strategy for minimal weight integer representations

Heuberger

Muir

2009

Des. Codes Cryptogr.

View full text Add to dashboard Cite

An online algorithm is presented that produces an optimal radix-2 representation of an input integer n using digits from the set D ,u = {a ∈ Z : ≤ a ≤ u}, where ≤ 0 and u ≥ 1. The algorithm works by scanning the digits of the binary representation of n from leftto-right (i.e., from most-significant to least-significant). The output representation is optimal in the sense that, of all radix-2 representations of n with digits from D ,u , it has as few nonzero digits as possible (i.e., it has minimal weight). Such representations are useful in the efficient implementation of elliptic curve cryptography. The strategy the algorithm utilizes is to choose an integer of the form d2 i , where d ∈ D ,u , that is closest to n with respect to a particular distance function. It is possible to choose values of and u so that the set D ,u is unbalanced in the sense that it contains more negative digits than positive digits, or more positive digits than negative digits. Our distance function takes the possible unbalanced nature of D ,u into account.

show abstract

Section: Algorithm 2 Scalar Multiplication Via Horner's Rulementioning

confidence: 99%

Section: Algorithm 2 Scalar Multiplication Via Horner's Rulementioning

confidence: 99%

See 1 more Smart Citation

Unbalanced digit sets and the closest choice strategy for minimal weight integer representations

Heuberger

Muir

2009

Des. Codes Cryptogr.

View full text Add to dashboard Cite

show abstract

“…Integer representations using digit sets of the form {a ∈ Z : ≤ a ≤ u} where ≤ 0 and u ≥ 1 have been proposed previously in the literature. Phillips and Burgess [15] introduced a "generalized sliding window" transformation which is applied to an integer's standard radix-r representation where r ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

Minimal weight and colexicographically minimal integer representations

Heuberger¹,

Muir²

2007

Journal of Mathematical Cryptology

View full text Add to dashboard Cite

Redundant number systems (e.g., signed binary representations) have been utilized to efficiently implement algebraic operations required by public-key cryptosystems, especially those based on elliptic curves. Several families of integer representations have been proposed that have a minimal number of nonzero digits (so-called minimal weight representations). We observe that many of the constructions for minimal weight representations actually work by building representations which are minimal in another sense. For a given set of digits, these constructions build colexicographically minimal representations; that is, they build representations where each nonzero digit is positioned as far left (toward the most significant digit) as possible. We utilize this strategy in a new algorithm which constructs a very general family of minimal weight dimension-d joint representations for any d ≥ 1. The digits we use are from the set {a ∈ Z : ≤ a ≤ u} where ≤ 0 and u ≥ 1 are integers. By selecting particular values of and u, it is easily seen that our algorithm generalizes many of the minimal weight representations previously described in the literature. From our algorithm, we obtain a syntactical description of a particular family of dimension-d joint representations; any representation which obeys this syntax must be both colexicographically minimal and have minimal weight; moreover, every vector of integers has exactly one representation that satisfies this syntax. We utilize this syntax in a combinatorial analysis of the weight of the representations.

show abstract

“…The requirement that the nonzero digits must not be adjacent makes CSD representation unique and differentiates it from the non unique Minimal Signed Digit (MSD) representation [3] which also has minimal Hamming weight. One drawback of representing a variable in CSD is that the conversion from binary to CSD can only be generated recursively, whether it is determined from the Least Significant Bit (LSB) to the Most Significant Bit (MSB), i.e., right-to-left or from the MSB to the LSB, i.e., left-to-right.…”

mentioning

confidence: 99%

Fast and VLSI efficient binary-to-CSD encoder using bypass signal

2011

View full text Add to dashboard Cite

The generation of a canonical signed digit representation from a binary representation is revisited. Based on the property that each nonzero digit is surrounded by a zero digit, a hardware-efficient conversion method using bypass instead of carry propagation is proposed. The proposed method requires less area per digit and the required bypass signal can be generated or propagated with only a single NOR gate. It is shown that the proposed converter outperforms previous converters and a look-ahead circuitry to speed up the generation of bypass signals is also proposed.Original Publication:M Faust, Oscar Gustafsson and C-H Chang, Fast and VLSI efficient binary-to-CSD encoder using bypass signal, 2011, ELECTRONICS LETTERS, (47), 1, 18-19.http://dx.doi.org/10.1049/el.2010.3055Copyright: Iethttp://www.theiet.org

show abstract

Minimal weight digit set conversions

Cited by 26 publications

References 29 publications

Unbalanced digit sets and the closest choice strategy for minimal weight integer representations

Unbalanced digit sets and the closest choice strategy for minimal weight integer representations

Minimal weight and colexicographically minimal integer representations

Fast and VLSI efficient binary-to-CSD encoder using bypass signal

Contact Info

Product

Resources

About