Military Communications and Information Technology: A Trusted ...

462 Military Communications and Information Technology...
3 : x 0 + x 6 + x 12 + x 13 + x 16 + x 20 + x 21 + x 22 + x 3 x 18 + x 13 x 19 + x 13 x 20 + x 5 x 12 x 20 +
+ x 8 x 18 x 22 + x 12 x 15 x 21
4 : x 0 + x 6 + x 11 + x 14 + x 16 + x 17 + x 18 + x 19 + x 23 + x 4 x 19 + x 4 x 21 + x 5 x 22 + x 9 x 19 +
+ x 1 x 17 x 23 + x 5 x 7 x 18 + x 5 x 12 x 19
The NLFSRs of order 27:
5 : x 0 + x 4 + x 8 + x 9 + x 11 + x 12 + x 15 + x 16 + x 23 + x 12 x 22 + x 13 x 23 + x 13 x 25 +
+ x 22 x 23 + x 7 x 8 x 24 + x 12 x 14 x 26 + x 6 x 11 x 19 x 22
6 : x 0 + x 1 + x 8 + x 10 + x 11 + x 12 + x 17 + x 19 + x 21 + x 22 + x 23 + x 6 x 25 + x 9 x 15 +
+ x 18 x 23 + x 23 x 26 + x 2 x 20 x 21 + x 13 x 21 x 23 + x 5 x 18 x 19 x 23
7 : x 0 + x 1 + x 2 + x 5 + x 10 + x 13 + x 15 + x 24 + x 26 + x 7 x 22 + x 11 x 18 + x 13 x 19 +
+ x 16 x 25 + x 24 x 25 + x 15 x 25 x 26 + x 8 x 10 x 25 x 26
VI. Conclusions
We used the implementation of Nonlinear Feedback Shift Registers in Field
Programmable Gate Arrays to perform a search of NLFSRs of the order up to
n = 27, the maximum period equal to 2 n – 1 and a possibly simple algebraic form
of the feedback function. The structure of the Algebraic Normal Form of the feedback
function was fixed in our search. We put the algebraic degree of ANF equal to four
and randomly chose linear and higher order terms. The hardware implementation
of NLFSRs and verification modules enabled to speed our search about 100 times
up comparing to software implementation on current PCs. The future task would
be to find NLFSRs with bigger number of stages n. This requires an improvement
of searching methods and the use more hardware resources.
References
[1] N.G. deBruijn, A combinatorial problem. Indag. Math., 8(1946), pp. 461-467.
[2] E. Dubrova, A list of maximum period NLFSRs. Cryptology ePrint Archive, 2012/166.
www.iacr.org
[3] C. Flye Sainte-Marie, Solution to question nr. 48. L’Intermédiaire des Mathématiciens
1(1894). pp. 107-110.
[4] B.M. Gammel, R. Goetffert, O. Kniffler, Achterbahn 128/80. The eSTREAM
project, www.ecrypt.eu.org/stream/, www.matpack.de/achterbahn
[5] S.W. Golomb, Shift Register Sequences. San Francisco, Holden-Day, 1967, revised
edition, Laguna Hills, CA, Aegean Park Press, 1982.
[6] S.W. Golomb, G. Gong, Signal Design for Good Correlation. For Wireless
Communication, Cryptography, and Radar. Cambridge University Press, 2005.
[7] G.M. Kyureghyan, Minimal polynomials of the modified de Bruijn sequences.
Discrete Applied Math., 156(2008), pp. 1549-1553.