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On Taking Square Roots without Quadratic Nonresidues over Finite Fields
, 2009
"... We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in Õ(log2 q) bit operations over f ..."
Abstract

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We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in Õ(log2 q) bit operations over finite fields with q elements. As an application, we construct a deterministic primality proving algorithm, which runs in Õ(log3 N) for some integers N. 1
Improving the Berlekamp algorithm for binomials x n − a
"... In this paper, we describe an improvement of the Berlekamp algorithm for binomials x n −a over prime fields Fp. We implement the proposed method for various cases and compare the results with the original Berlekamp method. The proposed method can be extended easily to the case where the base field i ..."
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In this paper, we describe an improvement of the Berlekamp algorithm for binomials x n −a over prime fields Fp. We implement the proposed method for various cases and compare the results with the original Berlekamp method. The proposed method can be extended easily to the case where the base field is not a prime field.
Fast Primality Proving on Cullen Numbers
, 2009
"... We present a unconditional deterministic primality proving algorithm for Cullen numbers. The expected running time and the worst case running time of the algorithm are Õ(log2 N) bit operations and Õ(log 3 N) bit operations, respectively. 1 ..."
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We present a unconditional deterministic primality proving algorithm for Cullen numbers. The expected running time and the worst case running time of the algorithm are Õ(log2 N) bit operations and Õ(log 3 N) bit operations, respectively. 1
Using the smoothness of p − 1 for computing roots modulo p
, 2008
"... We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo p of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if p − 1 has a (ln p) O(1)smooth divisor exceeding (p − 1) 1 2 +δ for some arbitrary small δ. ..."
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We prove, without recourse to the Extended Riemann Hypothesis, that the projection modulo p of any prefixed polynomial with integer coefficients can be completely factored in deterministic polynomial time if p − 1 has a (ln p) O(1)smooth divisor exceeding (p − 1) 1 2 +δ for some arbitrary small δ. We also address the issue of computing roots modulo p in deterministic time. 1