1 chain with 2 bits set Notation: @(.) means 2^(.) p,q are free nonnegative integer variables A,B are also free nonnegative integer variables * in the index-list as a right sub_index indicates a doubling step. : is a delimiter in the sub-index to refine this sub-index by a further 'value'. Such a 'value' is in general a polynomial of degree 0 or 1 in one variable [i.e. an affine form with integer coefficients and dimension less or equal 1]. This 'value' indicates either in a doubling step how many doubling steps are here, or when using a doubling step in another index how many doubling steps are used there. If a doubling step is used with exactly as many doubling steps it was generated then this delimiter and 'value' is optional [E.g. It may be left out in the left sub-index for every star-chain]. In all two-power-sum representations the two-power-exponents are always strictly monotonic decreasing and >= 0. Shortest Addition Chains with 1 Small Step 2* @(p)+@(q) A=p, B=q 0( ) 1 1( 0, *:A) @(A) 2( 1, 1:B) @(A)+@(B)