In-place fast polynomial modular remainder
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We consider the fast in-place computation of the Euclidean polynomial modular remainder R(X) ≢ A(X) mod B(X) with A and B of respective degrees n and m ≤ n. If the multiplication of two polynomials of degree k can be performed with M(k) operations and O(k) extra space, then standard algorithms for the remainder require O(n/m M(m)) arithmetic operations and, apart from that of A and B, at least O(n – m) extra memory. This extra space is notably usually used to store the whole quotient Q(X) such that A = BQ + R with deg R < deg B.We avoid the storage of the whole of this quotient, and propose an algorithm still using O(n/m M(m)) arithmetic operations but only O(m) extra space.When the divisor B is sparse with a constant number of non-zero terms, the arithmetic complexity bound reduces to O(n).When it is allowed to use the input space of A or B for intermediate computations, but putting A and B back to their initial states after the completion of the remainder computation, we further propose an in-place algorithm (that is with its extra required space reduced to O(1) only) using at mostO(n/m M(m) log(m) arithmetic operations.To achieve this, we develop techniques for Toeplitz matrix operations which output is also part of the input. In-place accumulated versions are obtained for the latter and for polynomial remaindering via reductions to accumulated polynomial multiplication, for which a recent fast in-place algorithm hasbeen developed.
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