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%I #17 Sep 25 2023 18:22:10
%S 0,1,2,2,3,3,4,3,4,4,5,4,5,5,5,4,5,5,6,5,6,6,6,5,6,6,6,6,7,6,6,5,6,6,
%T 7,6,7,7,7,6,7,7,7,7,7,7,7,6,7,7,7,7,8,7,8,7,8,8,8,7,8,7,7,6,7,7,8,7,
%U 8,8,8,7,8,8,8,8,8,8,8,7,8,8,8,8,8,8,9
%N Length of shortest addition-subtraction chain for n.
%C Equivalently, the minimal total number of multiplications and divisions required to compute an n-th power. This is useful for exponentiation on, for example, elliptic curves where division is cheap (as proposed by Morain and Olivos, 1990). Addition-subtraction chains are also defined for negative n. Various bounds and a rules to construct a(n) up to n=42 can be found in Volger (1985).
%C a(n) < A003313(n) for n in A229624. - _T. D. Noe_, May 02 2007
%H Jens Groth and Victor Shoup, <a href="https://eprint.iacr.org/2023/1175">Fast batched asynchronous distributed key generation</a>, Cryptology ePrint Archive, 2023. See p. 31.
%H F. Morain and J. Olivos, <a href="ftp://ftp.inria.fr/INRIA/publication/Theses/TU-0144/ch4.ps">Speeding up the computations on an elliptic curve using addition-subtraction chains</a>, RAIRO Informatique theoretique et application, vol. 24 (1990), pp. 531-543.
%H Hugo Volger, <a href="https://doi.org/10.1016/0020-0190(85)90085-7">Some results on addition/subtraction chains</a>, Information Processing Letters, Vol. 20 (1985), pp. 155-160.
%H <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%e a(31) = 6 because 31 = 2^5 - 1 and 2^5 can be produced by 5 additions (5 doublings) starting with 1.
%Y Cf. A003313.
%K nonn,nice
%O 1,3
%A Steven G. Johnson (stevenj(AT)math.mit.edu), May 01 2007
%E More terms from _T. D. Noe_, May 02 2007
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