Computing Sequences with Addition Chains

“…Downey, Leong, and Sethi [6] proved that the decision problem whose instances are the tuples (k, n 1 ,n 2 ,...,n k , ) such that there is an addition chain of length containing n 1 ,n 2 ,...,n k is NP-complete. It is conjectured that the subset of instances with fixed k, such as k = 1, is also NP-complete.…”

Section: Introductionmentioning

confidence: 99%

A Note on the Signed Sliding Window Integer Recoding and a Left-to-Right Analogue

Avanzi

2004

Selected Areas in Cryptography

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Abstract. Addition-subtraction-chains 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 w-NAF. 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 right-to-left.We solve two open questions regarding the w-NAF. We first prove that the w-NAF is a redundant radix-2 recoding of smallest weight among all those with integral coefficients smaller in absolute value than 2 w−1 . Secondly, we introduce a left-to-right recoding with the same digit set as the w-NAF, 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.

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“…As the problem of finding the shortest addition chain is NPcomplete [17], practical algorithms for exponentiation without a division operation have been studied such as the binary method, the m-ary method, window methods [31,34,41,52], exponent-folding methods [33,49], and precomputation methods [7,32]. For a group where the inversion operation is efficient like elliptic curve cryptosystems [30,35], an addition-subtraction chain can yield a good performance [38].…”

Section: Introductionmentioning

confidence: 99%