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• W2000840326 abstract "Let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w greater-than-or-equal-to 2> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>w geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an integer and let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript w> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>D_w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the set of integers that includes zero and the odd integers with absolute value less than <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 Superscript w minus 1> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>w</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>2^{w-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Every integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be represented as a finite sum of the form <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n equals sigma-summation a Subscript i Baseline 2 Superscript i> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>n = sum a_i 2^i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript i Baseline element-of upper D Subscript w> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>a_i in D_w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, such that of any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consecutive <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a Subscript i> <mml:semantics> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>a_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s at most one is nonzero. Such representations are called <italic>width-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nonadjacent forms</italic> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-NAFs). When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w equals 2> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>w=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> these representations use the digits <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartSet 0 comma plus-or-minus 1 EndSet> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>±<!-- ± --></mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{0,pm 1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and coincide with the well-known <italic>nonadjacent forms</italic>. Width-<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-NAF. We show that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-NAFs have a minimal number of nonzero digits and we also give a new characterization of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=w> <mml:semantics> <mml:mi>w</mml:mi> <mml:annotation encoding=application/x-tex>w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-NAFs and show that any base 2 representation of an integer, with digits in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper D Subscript w> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>w</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>D_w</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that has a minimal number of nonzero digits is at most one digit longer than its binary representation." @default.
• W2000840326 created "2016-06-24" @default.
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• W2000840326 date "2005-07-12" @default.
• W2000840326 modified "2023-09-23" @default.
• W2000840326 title "Minimality and other properties of the width-𝑤 nonadjacent form" @default.
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• W2000840326 doi "https://doi.org/10.1090/s0025-5718-05-01769-2" @default.
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