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Efficient algorithms for computing the Jacobi symbol

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Algorithmic Number Theory (ANTS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1122))

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Abstract

We present two new algorithms for computing the Jacobi Symbol: the right-shift and left-shift κ-ary algorithms. For inputs of at most n bits in length, both algorithms take O(n2/log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be about two to three times faster for inputs of 100 to 1000 decimal digits in length.

We also present parallel versions of both algorithms for the CRCW PRAM. One version takes O ɛ (n/log log n) time using O(n 1+ɛ) processors, giving the first sublinear parallel algorithms for this problem, and the other version takes polylog time using a subexponential number of processors.

Computing equipment provided through a grant from the Holcomb Research Institute. A preliminary version of this paper was presented on November 3, 1995 at the AMS meeting at Kent State University, Kent, Ohio.

Supported by the Butler Summer Institute.

Supported by a Butler University Faculty Research Fellowship.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Butler University, 46208, Indianapolis, Indiana, USA

    Shawna M. Meyer & Jonathan P. Sorenson

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  1. Shawna M. Meyer
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  2. Jonathan P. Sorenson
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Henri Cohen

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© 1996 Springer-Verlag Berlin Heidelberg

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Meyer, S.M., Sorenson, J.P. (1996). Efficient algorithms for computing the Jacobi symbol. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61581-4_58

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  • DOI: https://doi.org/10.1007/3-540-61581-4_58

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61581-1

  • Online ISBN: 978-3-540-70632-8

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