In mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbers v and a sequence of index pairs w such that each term in v is the sum of two previous terms, the indices of those terms being specified by w:

v =(v0,...,vs), with v0 = 1 and vs = n
for each 0< is holds: vi = vj + vk, with wi=(j,k) and 0 ≤ j,ki − 1

Often only v is given since it is easy to extract w from v, but sometimes w is not uniquely reconstructible. An introduction is given by Knuth.

Exponentiation By Squaring - Alternatives and Generalizations
... Exponentiation by squaring can be viewed as a suboptimal addition-chain exponentiation algorithm it computes the exponent via an addition chain consisting of ... this occurs is for n=15 (squaring, 6 multiplies) (optimal addition chain, 5 multiplies if x3 is re-used) In general, finding the optimal addition chain for a given exponent is a ... in compilers where the chains for small powers have been pre-tabulated) ...

### Famous quotes containing the words chain and/or addition:

It could not have come down to us so far,
Through the interstices of things ajar
On the long bead chain of repeated birth,
To be a bird while we are men on earth,
Robert Frost (1874–1963)

As easy mayst thou fall
A drop of water in the breaking gulf,
And take unmingled thence that drop again,