Fast Modular Composition in any Characteristic

We give an algorithm for modular composition of degree n univariate polynomials over a finite field F q requiring n 1 + o(1) log 1 + o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F q requiring (n 1. 5 + o(1) + n 1 + o(1) log q) log 1 + o(1) q bit operations, improving upon the methods of von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998). Our results also imply algorithms for irreducibility testing and computing minimal polynomials whose running times are best-possible, up to lower order terms. As in Umans (2008), we reduce modular composition to certain instances of multipoint evaluation of multivariate polynomials. We then give an algorithm that solves this problem optimally (up to lower order terms), in arbitrary characteristic. The main idea is to lift to characteristic 0, apply a small number of rounds of multimodular reduction, and finish with a small number of multidimensional FFTs. The final evaluations are then reconstructed using the Chinese Remainder Theorem. As a bonus, we obtain a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithm uses techniques which are commonly employed in practice, so it may be competitive for real problem sizes. This contrasts with previous asymptotically fast methods relying on fast matrix multiplication.

Authors

• Kiran S. Kedlaya
• Chris Umans

Keywords

• Polynomials
• Galois fields
• Computer science
• Multidimensional systems
• Flexible printed circuits
• Mathematics
• Application software
• Testing
• Data structures
• Engineering profession
• Data Structure
• Running Time
• Matrix Multiplication
• Characteristic Zero
• Finite Field
• Lower Order Terms
• Bitwise Operations
• Univariate Polynomial
• Transposable
• Sufficiently Large
• Field Size
• Evaluation Of Algorithms
• Evaluation Points
• Word Size
• Modulo Operation
• Polynomials In Variables
• modular composition
• multipoint evaluation
• polynomial factorization
• multimodular reduction