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Anatoly Karatsuba coauthored Multiplication of many-digital numbers by automatic computers by A. Karatsuba and Yu. Ofman. Karatsuba multiplication replaces four half-size products by three, using an algebraic rearrangement to recover the middle term. The resulting recurrence relation $T(n)=3T(n/2)+O(n)$ gives the exponent ${\log }_{2}3$ instead of two in the usual digit-cost model. This is an early example of changing an algorithm's growth rate through an identity.^[ref]

Karatsuba multiplication is a divide-and-conquer algorithm for multiplication of integers. Splitting each operand into two blocks seems to require four smaller products; the Karatsuba three-product identity recovers the two cross terms together from just three products. Repeating that saving through recursion changes the time complexity from quadratic to $O(n^{{\log }_{2}3})$, where $n$ counts digits in a fixed positional notation. The exponent is about $1.585$: the benefit comes from the algorithm, even though each splitting step needs more addition and subtraction.

The Karatsuba worked multiplication below computes $1234\cdot 5678=7,006,652$ using $12\cdot 56$, $34\cdot 78$, and $46\cdot 134$. The last operand already shows why “half-size” needs care: Karatsuba carry handling and Karatsuba recursion termination account for the extra digit. Karatsuba digit-operation cost states the operation model; Karatsuba practical crossover explains why the asymptotic saving alone does not select a useful threshold.

The original 1962 paper presents a binary numeral system automaton construction: its Theorem 2 and proof reduce multiplication to taking squares of numbers, then reduce a larger square of a number to three smaller squares of numbers. Its footnote explicitly addresses the extra digit in a sum of halves. The sequential time complexity analysis below uses a different operation model from that paper's automaton size and depth bounds, including its Boolean circuit realization without feedback.^[ref]

Yu. Ofman coauthored the 1962 paper Multiplication of many-digital numbers by automatic computers by A. Karatsuba and Yu. Ofman. The work reduces a multiplication of large integers to three smaller multiplications rather than four, yielding the divide-and-conquer recurrence behind Karatsuba multiplication. It is an early example where an algebraic rearrangement changes an algorithm's asymptotic cost.^[ref]