Editing Addition-chain exponentiation (section)

==Addition-subtraction–chain exponentiation==
If both multiplication and division are allowed, then an [[addition-subtraction chain]] may be used to obtain even fewer total multiplications+divisions (where subtraction corresponds to division). However, the slowness of division compared to multiplication makes this technique unattractive in general.  For exponentiation to [[negative number|negative]] integer powers, on the other hand, since one division is required anyway, an addition-subtraction chain is often beneficial.  One such example is ''a''<sup>&minus;31</sup>, where computing 1/''a''<sup>31</sup> by a shortest addition chain for ''a''<sup>31</sup> requires 7 multiplications and one division, whereas the shortest addition-subtraction chain requires 5 multiplications and one division:
:<math>a^{-31} = a / ((((a^2)^2)^2)^2)^2 \!</math> (addition-subtraction chain, 5 mults + 1 div).

For exponentiation on [[elliptic curve]]s, the inverse of a point (''x'',&nbsp;''y'') is available at no cost, since it is simply (''x'',&nbsp;&minus;''y''), and therefore addition-subtraction chains are optimal in this context even for positive integer exponents.<ref>François Morain and Jorge Olivos, "[http://www.numdam.org/item/ITA_1990__24_6_531_0/ Speeding up the computations on an elliptic curve using addition-subtraction chains]", ''RAIRO Informatique théoretique et application'' '''24''', pp. 531-543 (1990).</ref>