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PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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Cited by 26 (2 self)
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Sharpening PRIMES is in P for a large family of numbers
 Math. Comp
, 2005
"... We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algo ..."
Abstract

Cited by 9 (0 self)
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We present algorithms that are deterministic primality tests for a large family of integers, namely, integers n ≡ 1 (mod 4) for which an integer a is given such that the Jacobi symbol ( a) = −1, and n integers n ≡ −1 (mod 4) for which an integer a is given such that ( a 1−a) = ( ) = −1. The algorithms n n we present run in 2 − min(k,[2 log log n]) Õ(log n) 6 time, where k = ν2(n − 1) is the exact power of 2 dividing n − 1 when n ≡ 1 (mod 4) and k = ν2(n + 1) if n ≡ −1 (mod 4). The complexity of our algorithms improves up to Õ(log n)4 when k ≥ [2 log log n]. We also give tests for more general family of numbers and study their complexity.