The aim of this paper is to generalize some results on addition chains to addition-multiplication chains. The pa-
per concerns with generating shortest addition-multiplication chains. It rst presents two methods for generating short
addition-multiplication chains. Second, it presents an algorithm for generating a shortest addition-multiplication chain.
Then it proposes three main improvements for generating a shortest addition-multiplication chain. The practical results
show that the proposed improvements reduce, on the average, the running time and storage of the algorithm by about
95% and 35% respectively for data range 12 􀀀 20 bits. Similar practical results are obtained for generating all shortest
addition-multiplication chains. Finally, the paper discusses how to apply the algorithm to obtain some results that have
been uncovered previously.