A more efficient multiplication method, the Schönhage–Strassen algorithm based on the Fast Fourier transform, requires O(p log p log log p) time to square a p-bit number, reducing the complexity to O(p2 log p log log p) or Õ(p2).
Even with grade-school integer multiplication, this is only O((log n)4) time; using the multiplication algorithm with best-known asymptotic running time, the Schönhage–Strassen algorithm, we can lower this to O((log n)3(log log n)(log log log n)) time, or using soft-O notation Õ((log n)3).
In 1971 Strassen published another paper together with Arnold Schönhage on asymptotically-fast integer multiplication based on the fast Fourier transform; see the Schönhage–Strassen algorithm.
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