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If g and h are polynomials of degrees r and s over a field, their functional composition f = g(h) has degree n = rs. The functional decomposition problem is: given f of degree n = rs, determine whether such g and h exist, and, in the affirmative case, compute them. We first deal with univariate polynomials, and present sequential algorithms that use O(nlog 2 n loglogn) arithmetic operations, and a parallel algorithm with optimal depth O(logn). Then we consider the case where f and h aredoi:10.1016/s0747-7171(08)80014-4 fatcat:td25whsh6rgsvoocwu3dsloxem