Editing Lucas sequence (section)

==Applications==
* Lucas sequences are used in probabilistic [[Lucas pseudoprime]] tests, which are part of the commonly used [[Baillie–PSW primality test]].
* Lucas sequences are used in some primality proof methods, including the [[Lucas–Lehmer–Riesel test]], and the N+1 and hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975.<ref name="BLS75">{{ cite journal|author=John Brillhart|author2=Derrick Henry Lehmer|author3-link=John Selfridge|author3=John Selfridge|title=New Primality Criteria and Factorizations of 2<sup>m</sup> ± 1|journal=Mathematics of Computation |volume=29|number=130|date=April 1975|pages=620–647|jstor=2005583|doi=10.1090/S0025-5718-1975-0384673-1|author-link=John Brillhart|author2-link=Derrick Henry Lehmer|doi-access=free}}</ref>
* LUC is a [[public-key cryptosystem]] based on Lucas sequences<ref>{{cite journal |author1=P. J. Smith |author2=M. J. J. Lennon |title=LUC: A new public key system |journal=Proceedings of the Ninth IFIP Int. Symp. On Computer Security |year=1993 |pages=103–117 |citeseerx=10.1.1.32.1835 }}</ref> that implements the analogs of [[ElGamal]] (LUCELG), [[Diffie–Hellman]] (LUCDIF), and [[RSA (algorithm)|RSA]] (LUCRSA). The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead of using [[modular exponentiation]] as in RSA or Diffie–Hellman. However, a paper by Bleichenbacher et al.<ref>{{cite book |author1=D. Bleichenbacher |author2=W. Bosma |author3=A. K. Lenstra |title=Advances in Cryptology — CRYPT0' 95 |chapter=Some Remarks on Lucas-Based Cryptosystems |series=Lecture Notes in Computer Science |volume=963 |year=1995 |pages=386–396 |doi=10.1007/3-540-44750-4_31 |isbn=978-3-540-60221-7 |chapter-url=http://www.math.ru.nl/~bosma/pubs/CRYPTO95.pdf|doi-access=free }}</ref> shows that many of the supposed security advantages of LUC over cryptosystems based on modular exponentiation are either not present, or not as substantial as claimed.