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Construction Of Hilbert Class Fields Of Imaginary Quadratic Fields And Dihedral Equations Modulo p
, 1989
"... . The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, note ..."
Abstract

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. The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, noted W(X), has a solvable Galois group. When this group is dihedral, we show how to express the roots of this polynomial in terms of radicals. We then use these expressions to solve the equation W(X) j 0 mod p, where p is a prime. 1 Hilbert polynomials 1.1 Weber's functions We first introduce some functions. Let z be any complex number and put q = exp(2ißz). Dedekind's j function is defined by [21, x24 p. 85] j(z) = j(q) = q 1=24 Y m1 (1 \Gamma q m ): (1) We can expand j as [21, x34 p. 112] j(q) = q 1=24 0 @ 1 + X n1 (\Gamma1) n (q n(3n\Gamma1)=2 + q n(3n+1)=2 ) 1 A : (2) The Weber's functions are [21, x34 p. 114] f(z) = e \Gammaiß=24 j( z+1 2 ) j(z) ; (3) f 1 (z) = j...
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 ..."
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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.