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Taking Roots over High Extensions of Finite Fields
"... We present a new algorithm for computing mth roots over the finite field Fq, where q = p n, with p a prime, and m any positive integer. In the particular case m = 2, the cost of the new algorithm is an expected O(M(n) log(p) + C(n) log(n)) operations in Fp, where M(n) and C(n) are bounds for the co ..."
Abstract

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We present a new algorithm for computing mth roots over the finite field Fq, where q = p n, with p a prime, and m any positive integer. In the particular case m = 2, the cost of the new algorithm is an expected O(M(n) log(p) + C(n) log(n)) operations in Fp, where M(n) and C(n) are bounds for the cost of polynomial multiplication and modular polynomial composition. Known results give M(n) = O(n log(n) log log(n)) and C(n) = O(n 1.67), so our algorithm is subquadratic in n.