Modular composition is the problem to compute the composition of two univariate polynomials modulo a third one. For polynomials with coefficients in a finite field, Kedlaya and Umans proved in 2008 that the theoretical bit complexity for performing this task could be made arbitrarily close to linear. Unfortunately, beyond its major theoretical impact, this result has not led to practically faster implementations yet. In this article, we explore particular cases of moduli over finite fields for which modular composition turns out to be cheaper than in the general case. In the most favourable cases, our algorithms achieve quasi-linear costs.

Authors: Joris van der Hoeven, Grégoire Lecerf

Keywords: modular composition, finite field, factorization, complexity

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