Some Results on Addition/Subtraction Chains

Abstract Let l(n) (respectively l (n)) be the length of the shortest addition chain respectively addition/subtraction chain for n. We shall present several results on l (n). In particular, we determine l (n) for all n satisfying s ( n ) ⩽ 3 and show ⌊ log n ⌋ + 2 ⩽ l ( n ) for all n satisfying s ( n ) ⩾ 3 , where s ( n ) is the extended sum of digits of n. These results are based on analogous results for l(n) and on the following two inequalities: |n| ⩽ 2d−1Ff+3 f + b ⩾ log s (n) for a chain of length k = d + f + b with d doublings, f short steps, and b back steps for n. Moreover, we show that the difference l( n )− l ( n ) (respectively l ( n )−⌈ log n ⌉ ) can be made arbbitrarily large. In addition, we prove that l ( m ) ⩽ l (− m ) ⩽ l ( m ) + 1 for m > 0 and characterize the case l (− m ) = l ( m ) . Finally, we determine l k ( n 1 ,…, n k ) , the k-dimensional generalization of l , with the help of l ( n 1 ,…, n k ) , the k-element generalization of l .