Polynomial Multiplication over Finite Fields in Time \( O(n \log n \)

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than \( n \) over a finite field \( \mathbb {F}_q \) with \( q \) elements can be multiplied in time \( O (n \log q \log (n \log q)) \) , uniformly in \( q \) . Under the same hypothesis, we show how to multiply two \( n \) -bit integers in time \( O (n \log n) \) ; this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [22]. Our results hold in the Turing machine model with a finite number of tapes.