A Fast Algorithm for Computing Binomial Coefficients Modulo Powers of Two

Abstract

I present a new algorithm for computing binomial coefficients modulo 2N. The proposed method has an O(N 3 · Multiplication(N) + N 4) preprocessing time, after which a binomial coefficient C(P, Q) with 0 ≤ Q ≤ P ≤ 2N − 1 can be computed modulo 2N in O(N 2 · log(N) · Multiplication(N)) time. Multiplication(N) denotes the time complexity of multiplying two N-bit numbers, which can range from O(N 2) to O(N · log(N) · log(log(N))) or better. Thus, the overall time complexity for evaluating M binomial coefficients C(P, Q) modulo 2N with 0 ≤ Q ≤ P ≤ 2N − 1 is O((N 3 + M · N 2 · log(N)) · Multiplication(N) + N 4). After preprocessing, we can actually compute binomial coefficients modulo any 2R with R ≤ N. For larger values of P and Q, variations of Lucas' theorem must be used first in order to reduce the computation to the evaluation of multiple (O(log⁡(P))) binomial coefficients C(P′, Q′) (or restricted types of factorials P′!) modulo 2N with 0 ≤ Q′ ≤ P′ ≤ 2N − 1.