## Computation of Irregular Primes up to Eight Million (1996)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Shokrollahi96computationof,

author = {M.A. Shokrollahi},

title = {Computation of Irregular Primes up to Eight Million},

institution = {},

year = {1996}

}

### OpenURL

### Abstract

We report on a joint project with Joe Buhler, Richard Crandall, Reijo Ernvall, and Tauno Metsankyla dealing with the computation of irregular primes and cyclotomic invariants for primes between four and eight million. This extends previous computations of Buhler et al. [4]. Our computation of the irregular primes is based on a new approach which has originated in the study of Stickelberger codes [13]. It reduces the problem to that of finding zeros of a polynomial over F p of degree ! (p \Gamma 1)=2 among the quadratic residues. Use of fast polynomial gcd-algorithms gives an O(p log 2 p log log p)-algorithm for this task. By employing the Schonhage-Strassen algorithm for fast integer multiplication combined with a version of fast multiple evaluation of polynomials we design an algorithm with running time O(p log p log log p). This algorithm is particularly efficient when run on primes p for which p \Gamma 1 has small prime factors. We also give some improvements on the previous imple...

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Citation Context ... s \Gammab ) b p\Gamma1\Gammat mod L; where s is a square root of a pth root of 1 modulo L. If q m 6j 1 mod L for all t for a given p, then Vandiver's conjecture is true for p. (See Washington's book =-=[18]-=- for details.) The next application is concerned with the problem of computing the so-called cyclotomic invariants. By Iwasawa's general result and the theorem of Ferrero and Washington we have ord p ... |

128 |
The complexity of partial derivatives
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Citation Context ... interested in a lower bound for the non scalar complexity L(F ) of F over K. Let @ i denote the derivation morphism @=@x i of K(x 0 ; : : : ; x n\Gamma1 ). By the Baur-Strassen derivative inequality =-=[1]-=- we have 3L(F )sL(F; @ 0 F; : : : ; @ n\Gamma1 F ): Note that @ i F = Ff 0 (x i )=f(x i ). Hence, the right-hand side of the above inequality is at least L(f 0 (x 0 )=f(x 0 ); : : : ; f 0 (x n\Gamma1 ... |

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Shokrollahi Algebraic Complexity Theory, Volume 315, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag
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Citation Context ...orithm for computing the above product. (For a thorough discussion of the model of straight-line programs and the lower bound techniques used in the sequel we refer the reader to the forthcoming book =-=[5]-=-.) 8 Let f be a polynomial of degree n with indeterminate coefficients over a field k and denote the field generated over k by the coefficients of f by K. Let x 0 ; : : : ; x n\Gamma1 be indeterminate... |

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Citation Context ...clean and efficient implementations made TP the natural choice to implement our algorithm in. For more information on this software and how to obtain it via ftp, the reader is referred to the TP-book =-=[10]-=-. We finish this section by providing some timings for the three versions of our algorithm run on different primes. The computations were done on a SPARC-10 with 40 MHZ. The TP-code was compiled with ... |

32 |
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Citation Context ...(x 0 )=f(x 0 ); : : : ; f 0 (x n\Gamma1 )=f(x n\Gamma1 )), which itself is greater than or equal to L(f 0 (x 0 ); : : : ; f 0 (x n\Gamma1 )) \Gamma 2n. A simple application of Strassen's Degree Bound =-=[14]-=- gives a lower bound of n log(n \Gamma 1) \Gamma 2n for the latter. 6 Acknowledegements The use of fast integer multiplication to find the zeros of a polynomial over a finite prime field has been a su... |

28 | Fast modular transforms
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Citation Context ...n some specific primes are provided by the table at the end of this section.) A major improvement can be gained by employing an idea related to the multiple evaluation algorithm of Borodin and Moenck =-=[2]-=-. First, note that by multiplying the ith 3 coefficient of j with w i , we are left with the problem of finding the zeros of the resulting polynomial among the quadratic residues of F \Theta p . By ab... |

21 |
Prasuhn private communication
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Citation Context ...rm of v, and transform the product back. This reduces the number of Fourier transforms per polynomial multiplication from three to two. Another improvement can be gained using an idea of D. Reischert =-=[8]-=-: If p is not a Fermat prime, then the largest prime factor q t of p \Gamma 1 is odd, hence we have to perform a negacyclic convolution of h(x) and v(x). Translated into integer multiplication, this m... |

13 |
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Citation Context ... The irregular pairs for p can be used to compute several other interesting number theoretic data, as described in Section 3. The irregular pairs were computed for primes less than 125000 by Wagstaff =-=[17]-=-. This paper also contains a nice account on the history of these computations prior to its appearance. Wagstaff's tables were extended by Tanner and Wagstaff [15] to primes below 150000. Both these a... |

12 | Irregular primes to one million
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Citation Context ...aff's tables were extended by Tanner and Wagstaff [15] to primes below 150000. Both these approaches were quadratic, i.e., their running time was proportional to p 2 . Buhler, Crandall, and Sompolski =-=[3] were-=- the first to invent an O(p 1+" )-algorithm for this task and used their method to extend the computations of irregular pairs to one million. Their elegant approach is based on the inversion of t... |

9 |
New Congruences for the Bernoulli Numbers
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Citation Context ... primes less than 125000 by Wagstaff [17]. This paper also contains a nice account on the history of these computations prior to its appearance. Wagstaff's tables were extended by Tanner and Wagstaff =-=[15] to p-=-rimes below 150000. Both these approaches were quadratic, i.e., their running time was proportional to p 2 . Buhler, Crandall, and Sompolski [3] were the first to invent an O(p 1+" )-algorithm fo... |

3 |
onhage: Asymptotically fast algorithms for the numerical multiplication and division of polynomials with complex coefficients
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Citation Context ...sign, according to whether d is even or odd). We have thus reduced the problem to that of multiplying two polynomials of degree less than d over F p . We now use Schonhage's technique as presented in =-=[9]-=- to reduce this problem to that of multiplying integers. Let m := dlog(dp 2 )e. (Here and in the sequel log denotes log 2 .) Further, let / d\Gamma1 X i=0 h i 2 mi ! \Delta / d\Gamma1 X i=0 v i 2 mi !... |

3 |
onhage: Private communication
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Citation Context ... the factorization step, this algorithm would be of order O(p log 2 p log log p). We are interested in a faster algorithm, i.e., one of order O(p log p log log p). Following a suggestion of Schonhage =-=[11]-=-, we will first transform this problem to an instance of integer multiplication. For reasons to become clear later, we will consider a more general setup: Problem 2. Given a factor d of p \Gamma 1 suc... |

2 |
Mets ankyl a: Irregular primes and cyclotomic invariants to four million
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(Show Context)
Citation Context ...Ernvall, and Tauno Metsankyla dealing with the computation of irregular primes and cyclotomic invariants for primes between four and eight million. This extends previous computations of Buhler et al. =-=[4]-=-. Our computation of the irregular primes is based on a new approach which has originated in the study of Stickelberger codes [13]. It reduces the problem to that of finding zeros of a polynomial over... |

2 |
Mets ankyl a: Cyclotomic invariants for primes between 125000 and 150000
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Citation Context ...K 0 generated over Q by a primitive pth root of 5 unity. In this section we will report on two such applications, which, up to now have always accompanied the computations of the irregular pairs, see =-=[4, 6, 7, 15, 17]-=-. For ns1 let K n denote the cyclotomic field of p n+1 st roots of unity, and let h n and A n be the class number and p-class group of K n , respectively. It is well known that h n = h + n h \Gamma n ... |

2 |
Mets ankyl a: Cyclotomic invariants for primes to one million
- Ernvall, T
- 1992
(Show Context)
Citation Context ...K 0 generated over Q by a primitive pth root of 5 unity. In this section we will report on two such applications, which, up to now have always accompanied the computations of the irregular pairs, see =-=[4, 6, 7, 15, 17]-=-. For ns1 let K n denote the cyclotomic field of p n+1 st roots of unity, and let h n and A n be the class number and p-class group of K n , respectively. It is well known that h n = h + n h \Gamma n ... |

2 |
Shokrollahi: Stickelberger Codes
- A
- 1996
(Show Context)
Citation Context ...and eight million. This extends previous computations of Buhler et al. [4]. Our computation of the irregular primes is based on a new approach which has originated in the study of Stickelberger codes =-=[13]-=-. It reduces the problem to that of finding zeros of a polynomial over F p of degree ! (p \Gamma 1)=2 among the quadratic residues. Use of fast polynomial gcd-algorithms gives an O(p log 2 p log log p... |

2 |
Private communication
- Vetter
- 1995
(Show Context)
Citation Context ... 1)bytes: A secondary program generated all primes between four and eight million, estimated the amount of memory used, and passed the data to a resource management program (RMP) written by E. Vetter =-=[16]-=-. The RMP passed the prime to the first machine available which had the amount of RAM necessary to compute the irregular pairs corresponding to the prime in question. Furthermore, the RMP also allowed... |

1 |
onhage: Schnelle Multiplikation groer Zahlen
- Sch
- 1971
(Show Context)
Citation Context ... = P 2d\Gamma2 i=0 (c i mod p)x i . The running time of this algorithm is clearly dominated by the time needed to multiply two integers of bit-length dm, which, using the Schonhage-Strassen algorithm =-=[12]-=-, is O(dm log dm log log dm). The first version of our algorithm was the implementation of this idea, with d = (p\Gamma1)=2, C the coset consisting of the quadratic nonresidues, and f = j. (Timings fo... |