Addition-subtraction chain

An addition-subtraction chain, a generalization of addition chains to include subtraction, is a sequence a₀, a₁, a₂, a₃, ... that satisfies

a_{0}=1,\,

{\text{for }}k>0,\ a_{k}=a_{i}\pm a_{j}{\text{ for some }}0\leq i,j<k.

An addition-subtraction chain for n, of length L, is an addition-subtraction chain such that $a_{L}=n$ . 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: $a_{0}=1$ , $a_{1}=2=1+1$ , $a_{2}=4=2+2$ , $a_{3}=3=4-1$ . This is not a minimal addition-subtraction chain for n=3, however, because we could instead have chosen $a_{2}=3=2+1$ . 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):

a_{0}=1,\ a_{1}=2=1+1,\ a_{2}=4=2+2,\ a_{3}=8=4+4,\ a_{4}=16=8+8,\ a_{5}=32=16+16,\ a_{6}=31=32-1.

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 $x^{n}$ 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).

References

Volger, Hugo (8 April 1985). "Some results on addition/subtraction chains". Information Processing Letters. 20 (3): 155–160. doi:10.1016/0020-0190(85)90085-7.
Morain, François; Olivos, Jorge (1990). "Speeding up the computations on an elliptic curve using addition-subtraction chains" (PDF). Informatique théorique et applications. 24 (6): 531–543. CiteSeerX 10.1.1.56.347.
Downey, Peter J.; Leong, Benton L.; Sethi, Ravi (1981). "Computing sequences with addition chains". SIAM J. Comput. 10 (3): 638–646. doi:10.1137/0210047.
Sloane, N. J. A. (ed.). "Sequence A128998 (length of minimum addition-subtraction chain)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.