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Limits RegistryDavid Harvey and Joris van der Hoeven published an integer-multiplication algorithm running in O(n log n) time in 2019 (formally published in the Annals of Mathematics in 2021), achieving the complexity that Schönhage and Strassen had conjectured in 1971 to be the best possible for multiplying two n-digit numbers. It settled a nearly 50-year-old open question about achievability of this bound, although the deeper question of whether O(n log n) is a proven lower bound (i.e., that no faster algorithm can exist) remains formally open.
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David Harvey and Joris van der Hoeven published an integer-multiplication algorithm running in O(n log n) time in 2019 (formally published in the Annals of Mathematics in 2021), achieving the complexity that Schönhage and Strassen had conjectured in 1971 to be the best possible for multiplying two n-digit numbers. It settled a nearly 50-year-old open question about achievability of this bound, although the deeper question of whether O(n log n) is a proven lower bound (i.e., that no faster algorithm can exist) remains formally open.
The Harvey-van der Hoeven algorithm multiplies two n-digit integers in O(n log n) time, matching a complexity long conjectured (by Schönhage and Strassen in 1971) to be optimal for integer multiplication, though a matching unconditional lower bound has not been proven.
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Permanent ID limitsregistry.com/limits/LR-INTEGER-MULTIPLICATION-COMPLEXITY
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Limits Registry. LR-INTEGER-MULTIPLICATION-COMPLEXITY. Fastest known integer multiplication algorithm complexity. 2026.