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Iteration of quadratic polynomials over finite fields

Abstract:
For a finite field of odd cardinality $q$, we show that the sequence of iterates of $aX2+c,$ starting at $0$, always recurs after $O(q/loglogq)$ steps. For $X2+1$ the same is true for any starting value. We suggest that the traditional "Birthday Paradox" model is inappropriate for iterates of $X3+c,$ when $q$ is 2 mod 3.
Publication status:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.1112/S0025579317000328

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Institution:
University of Oxford
Oxford college:
Worcester College
Role:
Author
Publisher:
Cambridge University Press Publisher's website
Journal:
Mathematika Journal website
Volume:
63
Issue:
3
Pages:
1041-1059
Publication date:
2017-11-29
Acceptance date:
2017-03-29
DOI:
EISSN:
2041-7942
ISSN:
0025-5793
Source identifiers:
687410
Pubs id:
pubs:687410
UUID:
uuid:4019b5ed-b4e7-46f0-aa75-51fc391f5f69
Local pid:
pubs:687410
Deposit date:
2017-03-29

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