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).