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A note on the signed sliding window integer recoding and a lefttoright analogue
 in “Selected Areas in Cryptography – SAC 2004”, Lecture Notes in Computer Science 3357 (2005), 130– 143
, 2004
"... Abstract. Additionsubtractionchains obtained from signed digit recodings of integers are a common tool for computing multiples of random elements of a group where the computation of inverses is a fast operation. Cohen and Solinas independently described one such recoding, the wNAF. For scalars of ..."
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Cited by 19 (5 self)
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Abstract. Additionsubtractionchains obtained from signed digit recodings of integers are a common tool for computing multiples of random elements of a group where the computation of inverses is a fast operation. Cohen and Solinas independently described one such recoding, the wNAF. For scalars of the size commonly used in cryptographic applications, it leads to the current scalar multiplication algorithm of choice. However, we could find no formal proof of its optimality in the literature. This recoding is computed righttoleft. We solve two open questions regarding the wNAF. We first prove that the wNAF is a redundant radix2 recoding of smallest weight among all those with integral coefficients smaller in absolute value than 2 w−1. Secondly, we introduce a lefttoright recoding with the same digit set as the wNAF, generalizing previous results. We also prove that the two recodings have the same (optimal) weight. Finally, we sketch how to prove similar results for other recodings.
Fractional windows revisited: improved signeddigit representations for efficient exponentiation
 in “Information Security and Cryptology – ICISC 2004”, Lecture Notes in Computer Science 3506
, 2004
"... Abstract. This paper extends results concerning efficient exponentiation in groups where inversion is easy (e.g. in elliptic curve cryptography). It examines the righttoleft and lefttoright signed fractional window (RLSFW and LRSFW) techniques and shows that both RLSFW and LRSFW representati ..."
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Abstract. This paper extends results concerning efficient exponentiation in groups where inversion is easy (e.g. in elliptic curve cryptography). It examines the righttoleft and lefttoright signed fractional window (RLSFW and LRSFW) techniques and shows that both RLSFW and LRSFW representations have minimal weight among all signeddigit representations with digit set {±1, ±3,..., ±m, 0}. (Fractional windows generalize earlier slidingwindow techniques, providing more flexibility for exponentiation algorithms in order to make best use of the memory that is available for storing intermediate results.) Then it considers the length of representations: LRSFW representations are an improvement over RLSFW representations in that they tend to be shorter; further length improvements are possible by postprocessing the representations.
The alternating greedy expansion and applications to lefttoright algorithms
 in Cryptography Theoret. Comput. Sci IEICE TRANS. FUNDAMENTALS, VOL.E90–A, NO.5 MAY 2007 341 (2005
, 2004
"... Abstract. In [4], we introduced the alternating greedy expansion of integers, which turned out to be useful in several lefttoright algorihms in cryptography. In this paper, we collect known results about this alternating greedy expansion and complement it with other useful properties and algorithm ..."
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Cited by 9 (3 self)
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Abstract. In [4], we introduced the alternating greedy expansion of integers, which turned out to be useful in several lefttoright algorihms in cryptography. In this paper, we collect known results about this alternating greedy expansion and complement it with other useful properties and algorithms. In the second part, we apply it to give an algorithm for computing a joint expansion of d integers of minimal joint Hamming weight from left to right, i.e., from the column with the most significant bits towards the column with the least significant bits. Furthermore, we can also compute an expansion equivalent to the socalled wNAF from left to right using the alternating greedy expansion. 1.
Minimal weight and colexicographically minimal integer representations
, 2006
"... Redundant number systems (e.g., signed binary representations) have been utilized to efficiently implement algebraic operations required by publickey cryptosystems, especially those based on elliptic curves. Several families of integer representations have been proposed that have a minimal number o ..."
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Cited by 8 (7 self)
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Redundant number systems (e.g., signed binary representations) have been utilized to efficiently implement algebraic operations required by publickey cryptosystems, especially those based on elliptic curves. Several families of integer representations have been proposed that have a minimal number of nonzero digits (socalled 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 dimensiond joint representations for any d ≥ 1. The digits we use are from the set {a ∈ Z: ℓ ≤ a ≤ u} where ℓ ≤ 0 and u ≥ 1 are integers. By selecting particular values of ℓ 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 dimensiond 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.
COUNTING OPTIMAL JOINT DIGIT EXPANSIONS
 INTEGERS 5(3), 2005, A09
, 2005
"... This paper deals with pairs of integers, written in base two expansions using digits 0, ±1. Representations with minimal Hamming weight (number of nonzero pairs of digits) are of special importance because of applications in Cryptography. The interest here is to count the number of such optimal rep ..."
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Cited by 2 (2 self)
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This paper deals with pairs of integers, written in base two expansions using digits 0, ±1. Representations with minimal Hamming weight (number of nonzero pairs of digits) are of special importance because of applications in Cryptography. The interest here is to count the number of such optimal representations.
Unbalanced digit sets and the closest choice strategy for minimal weight integer representations
, 2008
"... ..."
A New Upper Bound for the Minimal Density of Joint Representations in Elliptic Curve Cryptosystems
, 2007
"... SUMMARY The most time consuming operation to verify a ..."
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SUMMARY The most time consuming operation to verify a
Complements and Signed . . . a MultiExponentiationAlgorithm of Wu, Lou, Lai and Chang
, 2008
"... Wu, Lou, Lai and Chang proposed a multiexponentiation algorithm using binary complements and the nonadjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the co ..."
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Wu, Lou, Lai and Chang proposed a multiexponentiation algorithm using binary complements and the nonadjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the complement operation does not have significant influence on the performance of the algorithm and can therefore be omitted.