Stretching de Bruijn sequences
We give a one-step construction of de Bruijn sequences of general alphabet size and with order n + k , given a de Bruijn sequence of order n and any integer k > 1 . This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer k...
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| Published in | Designs, codes, and cryptography Vol. 85; no. 2; pp. 381 - 394 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.11.2017
Springer Nature B.V |
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| Abstract | We give a one-step construction of de Bruijn sequences of general alphabet size and with order
n
+
k
, given a de Bruijn sequence of order
n
and any integer
k
>
1
. This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer
k
. The method starts with a lower order de Bruijn cycle, finds its inverse cycles in the higher order digraph, which are then cross-joined into one full cycle. Therefore, this generalizes the Lempel’s binary construction and the Alhakim–Akinwande construction for non-binary alphabets and a wide class of homomorphisms. |
|---|---|
| AbstractList | We give a one-step construction of de Bruijn sequences of general alphabet size and with order n + k , given a de Bruijn sequence of order n and any integer k > 1 . This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer k. The method starts with a lower order de Bruijn cycle, finds its inverse cycles in the higher order digraph, which are then cross-joined into one full cycle. Therefore, this generalizes the Lempel’s binary construction and the Alhakim–Akinwande construction for non-binary alphabets and a wide class of homomorphisms. We give a one-step construction of de Bruijn sequences of general alphabet size and with order n + k , given a de Bruijn sequence of order n and any integer k > 1 . This is achieved by using an appropriate class of graph homomorphisms between de Bruijn digraphs whose orders differ by an integer k . The method starts with a lower order de Bruijn cycle, finds its inverse cycles in the higher order digraph, which are then cross-joined into one full cycle. Therefore, this generalizes the Lempel’s binary construction and the Alhakim–Akinwande construction for non-binary alphabets and a wide class of homomorphisms. |
| Author | Alhakim, Abbas Nouiehed, Maher |
| Author_xml | – sequence: 1 givenname: Abbas surname: Alhakim fullname: Alhakim, Abbas email: aa145@aub.edu.lb organization: Department of Mathematics, American University of Beirut – sequence: 2 givenname: Maher surname: Nouiehed fullname: Nouiehed, Maher organization: Department of Industrial and Systems Engineering, University of Southern California |
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| Cites_doi | 10.1016/S0019-9958(79)90708-3 10.1137/1.9781611970081.ch10 10.1137/1024041 10.1007/s10623-010-9423-7 10.1109/18.761285 10.1109/T-C.1970.222859 |
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| Keywords | 05C45 68R01 de Bruijn sequence 05C60 Recursive construction Linear feedback shift register de Bruijn graph homomorphism Lempel’s D-morphism 05C38 |
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| References | Hell M., Johansson T., Meier W.: Grain: a stream cipher for constrained environments. ECRYPT (European Network of Excellence for Cryptology), eSTREAM Project (2005). FredricksenHA survey of full length nonlinear shift register cycle algorithmsSIAM Rev.19822419522165246610.1137/10240410482.68033 MykkeltveitJSiuM-KTongPOn the cyclic structure of some nonlinear shift register sequencesInf. Control197943220221510.1016/S0019-9958(79)90708-30431.68059 AlhakimAAkinwandeMA recursive construction of nonbinary de Bruijn sequencesDes. Codes Cryptogr.201060155169279569310.1007/s10623-010-9423-71227.68080 HornRAJohnsonCRMatrix Analysis20132New YorkCambridge University Press1267.15001 GolombSShift Register Sequences1967San FranciscoHolden-Day0267.94022 MandalKGongGKnudsenLRWuHCryptographically strong de Bruijn sequences with large periodsSAC 2012. LNCS2012HeidelbergSpringer104118 Akinwande M.B.O.: Homomorphisms of nonbinary de Bruijn graphs with applications. Ph.D. Dissertation, Clarkson University, New York (2010). ChangTParkBKimYHAn efficient implementation of the D-homomorphism for generation of de Bruijn sequencesIEEE Trans. Inf. Theory19994512801283168626910.1109/18.7612850957.94020 LempelAOn a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registersIEEE Trans. Comput. C1970191204120939609810.1109/T-C.1970.2228590225.94028 GeffePRHow to protect data with ciphers that are really hard to breakElectronics197346199101 NiederreiterHRandom Number Generation and Quasi Monte Carlo Methods1992PhiladelphiaSIAM20520610.1137/1.97816119700810761.65002 H Fredricksen (314_CR4) 1982; 24 S Golomb (314_CR6) 1967 RA Horn (314_CR8) 2013 T Chang (314_CR3) 1999; 45 PR Geffe (314_CR5) 1973; 46 314_CR7 K Mandal (314_CR10) 2012 314_CR1 A Alhakim (314_CR2) 2010; 60 A Lempel (314_CR9) 1970; 19 J Mykkeltveit (314_CR11) 1979; 43 H Niederreiter (314_CR12) 1992 |
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| Snippet | We give a one-step construction of de Bruijn sequences of general alphabet size and with order
n
+
k
, given a de Bruijn sequence of order
n
and any integer
k... We give a one-step construction of de Bruijn sequences of general alphabet size and with order n + k , given a de Bruijn sequence of order n and any integer k... |
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| SubjectTerms | Circuits Coding and Information Theory Computer Science Cryptology Data Structures and Information Theory Discrete Mathematics in Computer Science Graph theory Homomorphisms Information and Communication Sequences Stretching |
| Title | Stretching de Bruijn sequences |
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