2022/01/26 by Kedlaya, Kiran, Umans, Christopher
#Chinese Remaindering #Modular composition #multipoint evaluation #polynomial factorization
paper · doi:10.4230/dagsemproc.08381.5
We obtain randomized algorithms for factoring degree n univariate polynomials over Fq requiring O(n1.5 + o(1) log1+o(1) q+ n1 + o(1)log2+o(1) q) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ge n, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree n univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use O(n(omega + 1)/2) field operations, where omega is the exponent of matrix multiplication (Brent & Kung (1978)), with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication (Huang & Pan (1997)). We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent alpha implies an algorithm for the other with exponent alpha + o(1), and vice versa. We then give two new algorithms that solve the problem optimally (up to lower order terms): an algebraic algorithm for fields of characteristic at most no(1), and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of em multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese Remainder Theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques which are commonly employed in practice, so they may be competitive for real problem sizes. This contrasts with all previous subquadratic algorithsm for these problems, which rely on fast matrix multiplication. This is joint work with Kiran Kedlaya.