Chain composition
We use mainly two operation on addition chains:
Adding a number
Let c=(1,2,...,p,...,n) be a chain,
then we build the chain c+p=((1,2,...,n,n+p).
For instance (1,2,3,6,7) + 3=
(1,2,3,6,7,10)
Chain multiplication
Let c=(1,2,...,n) and
c'=(1,2,...,p) be two addition chains,
Then the product c c' is the chain
(1,2,...,n,2n,3n, ..., pn).
For instance, (1,2,3,6,7,10) (1,2,4,8) = (1,2,3,6,7,10,20,40,80).
Remark
*-chains (see ...) are closed under the two preceding operations.
Chains and Euclidean division
The two preceding operations are naturally combined in the following way:
To build a [*]chain for qb+r, build a [*]chain
c=(1,2,...,r,...,b), a [*]chain
c'=(1,2,...,q), then compute the
[*]chain c c' + r = (1,2,...,r,...,q,2q,...,qb,
qb+r).