That is, one can thereby compute n by L additions and/or subtractions.
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.
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).