An addition-subtraction chain for n, of length L, is an addition-subtraction chain such that . That is, one can thereby compute n by L additions and/or subtractions. (Note that n need not be positive. In this case, one may also include a−1 = 0 in the sequence, so that n = −1 can be obtained by a chain of length 1.)
By definition, every addition chain is also an addition-subtraction chain, but not vice versa. Therefore, the length of the shortest addition-subtraction chain for n is bounded above by the length of the shortest addition chain for n. In general, however, the determination of a minimal addition-subtraction chain (like the problem of determining a minimum addition chain) is a difficult problem for which no efficient algorithms are currently known. The related problem of finding an optimal addition sequence is NP-complete (Downey et al., 1981), but it is not known for certain whether finding optimal addition or addition-subtraction chains is NP-hard.
For example, one addition-subtraction chain is: , , , . This is not a minimal addition-subtraction chain for n=3, however, because we could instead have chosen . The smallest n for which an addition-subtraction chain is shorter than the minimal addition chain is n=31, which can be computed in only 6 additions (rather than 7 for the minimal addition chain):
Like an addition chain, an addition-subtraction chain can be used for addition-chain exponentiation: given the addition-subtraction chain of length L for n, the power can be computed by multiplying or dividing by x L times, where the subtractions correspond to divisions. This is potentially efficient in problems where division is an inexpensive operation, most notably for exponentiation on elliptic curves where division corresponds to a mere sign change (as proposed by Morain and Olivos, 1990).
Some hardware multipliers multiply by n using an addition chain described by n in binary:
n = 31 = 0 0 0 1 1 1 1 1 (binary).
Other hardware multipliers multiply by n using an addition-subtraction chain described by n in Booth encoding:
n = 31 = 0 0 1 0 0 0 0 -1 (Booth encoding).
- Hugo Volger, "Some results on addition/subtraction chains," Information Processing Letters 20, pp. 155–160 (1985).
- François Morain and Jorge Olivos, "Speeding up the computations on an elliptic curve using addition-subtraction chains[permanent dead link]," RAIRO Informatique théoretique et application 24, pp. 531–543 (1990).
- Peter Downey, Benton Leong, and Ravi Sethi, "Computing sequences with addition chains," SIAM J. Comput. 10 (3), 638-646 (1981).
- Sequence A128998, length of minimum addition-subtraction chain, The On-Line Encyclopedia of Integer Sequences.