On Generalized Addition Chains

Given integers d 1, and g 2, a g-addition chain for d is a sequence of integers a0 = 1, a1, a2, . . . , ar 1, ar = d where ai = aj1 +aj2 + · · ·+ajk , with 2  k  g, and 0  j1  j2  · · ·  jk  i 1. The length of a g-addition chain is r, the number of terms following 1 in the sequence. We denote by lg(d) the length of a shortest addition chain for d. Many results have been established in the case g = 2. Our aim is to establish the same sort of results for arbitrary fixed g. In particular, we adapt methods for constructing g-addition chains when g = 2 to the case g > 2 and we study the asymptotic behavior of lg.