 Karatsuba digit-operation cost

Count a fixed-base digit addition, subtraction, digit copy, or single-digit multiplication as constant cost. Base $B$ stays fixed as $n$ grows; an entire arbitrary-length integer multiplication is not one constant-cost step. With this model, splitting, sum formation, shifts and recombination require $O(n)$ digit operations. Ordinary digit-by-digit multiplication forms $n^2$ digit products for two $n$-position operands. The GMP base-case description uses this rectangular product arrangement.^[ref]

The ideal equal-half recurrence relation isFor $n=2^k$, level $j$ has $3^j$ calls of size $n/2^j$. If each internal call incurs exactly $\alpha$ times its size in additional digit work, that level contributes $\alpha n(3/2{)}^j$. There are $3^k=n^{{\log }_{2}3}$ leaves. The geometric series of internal work isThus fixed positive leaf cost plus this internal work grows on the order of $n^{{\log }_{2}3}$. In Big O notation, $T(n)=O(n^{{\log }_{2}3})$. Removing one of four products at every level matters much more than saving just one at the root.

The literal sum-based implementation instead admits the upper boundfor a nondecreasing worst-case cost envelope above a fixed base-case threshold. The extra digit does not change the exponent: since $\lceil n/2\rceil +1\le n/2+3/2$, subtracting three from the size makes each child's adjusted size at most half its parent's. Stop this comparison at any fixed threshold greater than three; its recursion depth is ${\log }_{2}n+O(1)$ and the level-work sum is bounded by the same geometric series up to constant factors. Exact call counts can depend on carries, zero shortcuts and the threshold, so they need not equal the ideal $3^j$ at each depth.