## Cyclotomy primality proofs and their certificates. Mathematica Goettingensis (2006)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Mihăilescu06cyclotomyprimality,

author = {Preda Mihăilescu},

title = {Cyclotomy primality proofs and their certificates. Mathematica Goettingensis},

year = {2006}

}

### OpenURL

### Abstract

Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein

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Citation Context ...verification is done in O ( p 2k · log(q) ) binary operations. Proof. A first condition which must be fulfilled by a Jacobi sum is the local p - adic norming condition j ≡ ±1 mod (1 − ξ) 2 , see e.g. =-=[16]-=-, and this fixes the choice of a root of unity factor 3 . Thus one starts by verifying that (51) α × α = q, and α ≡ ±1 mod (1 − ξ) 2 , in O(pk · log(q)) operations – note that the coefficients of a Ja... |

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Citation Context ... general limitation remains the necessity to know some large factored divisors s|(nk −1). Tests of this kind are denoted in general by the name of Lucas - Lehmer tests. The idea of Adleman et. al. in =-=[1]-=- was to bypass the above mentioned restriction, by choosing k so large, that an integer s > √ n and which splits completely in small – albeit, not polynomial – prime factors is granted to exist by ana... |

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Citation Context ... practical alternative for proving primality on computers is the random polynomial test using the group of points of an elliptic curve over finite fields, originally invented by Goldwasser and Kilian =-=[13]-=-. The test was made practical by a contribution of A. O. L. Atkin [5] and has been implemented at the same time by F. Morain, who maintained and improved [28] a program ECPP [27] since more than a dec... |

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Citation Context ... − 1), then ord2m(n) = 2u with 2. If n ≡ 3 mod 4 and v = v2(n + 1), then ord2 m(n2 ) = 2 u+1 with u = max(0, m − v), u = max(0, m − v). Proof. The proof is left as an exercise to the reader, see also =-=[31]-=-, Chapter II, §3. □ The remarkable phenomenon above consists in the fact that the order ordp m(n) starts from an initial value t = ordp(n) which is constant for m ≤ vp(n t − 1) and then increases by f... |

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Citation Context ... is odd and s = p k+1 . (iii) p = 2, k ≥ 2 and s = 2 k+2 . (iv) p = 2 and k = 1. Proof. The ring O(K) is a free Z - module of rank t and discriminant which divides sN for some integer N > 1, see e.g. =-=[30]-=-. With this, the assertions become simple verifications based upon the definition of K and the one of a galois extension. The assumption that t = pk can be dropped, by using the linear independence of... |

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Citation Context ...esolvents encountered when solving the equation Xs = 1 with radicals, over Q. Or, equivalently, when building the s−th cyclotomic field Ls/Q by a succession of prime power galois extensions, see e.g. =-=[17]-=-. Let n, m > 1 be integers with m squarefree and let λ(m) be the exponent of (Z/m · Z) ∗ , where λ is the Carmichael function. In this section, u = λ(m) and f will be some divisor of u and we assume t... |

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Citation Context ... kind t/u ∑ T = N[η], with η = σ ui (ζ) ∈ R, where u|t. Such subextensions are galois (even abelian). They have been considered recently by Lenstra and Pomerance in their version of the AKS algorithm =-=[21]-=-; the term of pseudo - fields was coined in that context. Remark 3. Let (R, ζ, σ) be some s−th cyclotomic extension of N, with R = N[ζ] and t = [R : N]. Suppose that there is an integer u > 1, and β ∈... |

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Citation Context ...·Z) a certification method for CPP; such a method was not known or predicted to exist previously. The same frame yields also a simple understanding of (a generalized form of) the Berrizbeitia variant =-=[9]-=- of the celebrated polynomial time deterministic test of Agrawal, Kayal and Saxena [3]; this is presented in [6] and, independently, by Bernstein in [8]. Finally, the notion of cyclotomic extension of... |

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Citation Context ... yields also a simple understanding of (a generalized form of) the Berrizbeitia variant [9] of the celebrated polynomial time deterministic test of Agrawal, Kayal and Saxena [3]; this is presented in =-=[6]-=- and, independently, by Bernstein in [8]. Finally, the notion of cyclotomic extension of rings can be extended to elliptic extensions of rings - closely connected to the Schoof-Elkies-Atkins algorithm... |

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Citation Context ...−1)/r = 1, find a solution of the equation y r = x in Fq. The problem has an efficient polynomial time solution, if a r a −th root of unity ρ ∈ Fq is known and the algorithm was described by Huang in =-=[14, 15]-=-. We shall treat here the generalization of the problem to cyclotomic extensions of rings. The basic idea is the same and it is well illustrated by the case r = 2 and q = p ≡ 5 mod 8. In this case we ... |

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Citation Context ...l values of k; it was also not possible to combine informations4 PREDA MIH ĂILESCU from tests for different values of k [33], [25]. The Theorem 1 solves both questions elegantly. As we have shown in =-=[23]-=-, the Theorem 1 not only generalizes the notion of Lucas - Lehmer tests and builds a bridge to combining them with the test of Adleman, Pomerance and Rumely, it also indicates a way for a new comprehe... |

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Citation Context ... Las Vegas version, which is mostly the version used in implementations, and a computationally more complicated deterministic version. The ideas of CPP were improved by Lenstra et. al. in [20], [19], =-=[12]-=-, [10], [23], [24],; their constructive base can be described as building a frame, in which a factor Ψ(X)|Φs(X) mod n can be constructed for some large s and such that, if n is prime, the factor is ir... |

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Citation Context ...onnection to classical test was hard to recognize. This connection was brought to light by H. W. Lenstra Jr. in his presentation of the result of Adleman, Pomerance and Rumely at the Bourbaki Seminar =-=[18]-=-. Let us consider again the Lucas - Lehmer test described above, where q|(n−1) is a prime with q > √ n. One can assert that this test constructs a primitive q−th root of unity modulo n, in the sense t... |

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Citation Context ...robabilistic Las Vegas version, which is mostly the version used in implementations, and a computationally more complicated deterministic version. The ideas of CPP were improved by Lenstra et. al. in =-=[20]-=-, [19], [12], [10], [23], [24],; their constructive base can be described as building a frame, in which a factor Ψ(X)|Φs(X) mod n can be constructed for some large s and such that, if n is prime, the ... |

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Citation Context ...Goldwasser and Kilian [13]. The test was made practical by a contribution of A. O. L. Atkin [5] and has been implemented at the same time by F. Morain, who maintained and improved [28] a program ECPP =-=[27]-=- since more than a decade. The purpose of this paper is to give a compact presentation of the theoretical background of the CPP algorithms and an overview of the basic variants. We also present a new ... |

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Citation Context ...cular, the required factored part was often larger than √ n even for small values of k; it was also not possible to combine informations4 PREDA MIH ĂILESCU from tests for different values of k [33], =-=[25]-=-. The Theorem 1 solves both questions elegantly. As we have shown in [23], the Theorem 1 not only generalizes the notion of Lucas - Lehmer tests and builds a bridge to combining them with the test of ... |

1 | Dual Elliptic Primes and applications to cyclotomic primality proving, Mathematic Goettingensis 07/2006 - Mihăilescu |

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Citation Context ...raic integer. The method is straightforward and was implemented in the Master Thesis [22]. For larger conductors, it is preferable to use methods of lattice reduction. These have been investigated in =-=[11]-=-, [23], [32] and are based on the following observation. Let Q ⊂ Z [ ] ξpk be a prime ideal above q; note that the choice of p implies that q splits completely and Q has inertial degree one. Let G = G... |

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Citation Context ...s and Jacobi sums in C too; finally, from the conjugates of a Jacobi sum, one recovers its coefficients as an algebraic integer. The method is straightforward and was implemented in the Master Thesis =-=[22]-=-. For larger conductors, it is preferable to use methods of lattice reduction. These have been investigated in [11], [23], [32] and are based on the following observation. Let Q ⊂ Z [ ] ξpk be a prime... |

1 | Dual elliptic primes and applications to cyclotomy primality proving. Mathematica Gottingensis
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Citation Context ...elatives of twin primes in imaginary quadratic extensions of Q, this leads to a new and very efficient combination of CPP and ECPP (elliptic curve primality proving) algorithms, which is presented in =-=[26]-=-. The present paper is herewith both an overview of the recent developments in CPP and a foundation for the description of new results. 1.1. Some notations. Throughout this paper we let n > 1 be an in... |

1 | Jacobi sums over finite fields
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Citation Context ...N of degree t; if such an extension is provided, one seeks a p−th power non residue, like one would do if R was a field. Let us recall some facts and usual notations about cyclotomic fields (see also =-=[32]-=-). The s−th cyclotomic field is Ls = Q(ζs) = Q[X]/(Φs(X)), an abelian extension of degree ϕ(s) with ring of integers O(Ls) = Z[ζs] and galois group Gs = Gal(Ls/Q) = {σa : ζs ↦→ ζ a s ; where (a, s) = ... |

1 |
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Citation Context ... particular, the required factored part was often larger than √ n even for small values of k; it was also not possible to combine informations4 PREDA MIH ĂILESCU from tests for different values of k =-=[33]-=-, [25]. The Theorem 1 solves both questions elegantly. As we have shown in [23], the Theorem 1 not only generalizes the notion of Lucas - Lehmer tests and builds a bridge to combining them with the te... |