S. Anand
ABSTRACT
Feedback-with-carry shift registers (FCSRs) are nonlinear analogues of linear feedback shift registers (LFSRs). Like the LFSRs, FCSRs are easy to implement and are important primitives in stream cipher design and pseudorandom number generation. In this paper, we investigate the properties of combiner generators that use two 2-adic feedback-with-carry shift registers as primitives. The combining function is simply the XOR function. This choice is motivated by an observation of Arnault and Berger on the high nonlinearity of the FCSR and that of Siegenthaler on the trade-off between resilience and correlation immunity of boolean functions. When the two FCSRs have odd prime power connection integers with 2 as a primitive root, we determine the exact period of the output sequence. We also prove that if the prime factors of the connection integers of the two FCSRs belong to different equivalence classes modulo 4, then the output sequence is symmetrically complementary. We use this fact to derive upper bounds on the linear complexity and the 2-adic complexity of the output sequence of the FCSR-combiner.
- F. Arnault and T.-P. Berger. Design of new pseudorandom generators based on a filtered fcsr automaton. In Proceedings of the SASC Workshop, pages 109--120, October 2004.Google Scholar
- F. Arnault and T.-P. Berger. F-FCSR: Design of a new class of stream ciphers. In H. Gilbert and H. Handschuh, editors, 12th. International Workshop, Fast Software Encryption 2005, Paris, France. Lecture Notes in Computer Science 3557, pages 83--97. Springer, February 2005. Google ScholarDigital Library
- F. Arnault, T.-P. Berger, and A. Necer. A new class of stream ciphers combining LFSR and FCSR architectures. In A. Menezes and P. Sarkar, editors, Progress in Cryptology -- INDOCRYPT 2002, Lecture Notes in Computer Science, volume 2551, pages 22--33. Springer, New York, 2002. Google ScholarDigital Library
- H. Beker and F. Piper. Cipher Systems. John Wiley, 1982.Google Scholar
- R. Couture and P. L'Ecuyer. Distribution properties of multiply-with-carry random number generators. Mathematics of Computation, 66:591--607, 1997. Google ScholarDigital Library
- B. M. M. de Weger. Approximation lattices of p-adic numbers. Journal of Number Theory, 24:70--88, 1986.Google ScholarCross Ref
- C. F. Gauß. Disquisitiones Arithmeticæ. (Reprinted English translation, Yale University Press, New Haven, 1966), 1801.Google Scholar
- S. W. Golomb. Shift Register Sequences. Holden-Day, San Francisco, 1967. Google ScholarDigital Library
- M. Goresky and A. Klapper. Large period nearly de Bruijn FCSR sequences. In Advances in Cryptology -- EUROCRYPT'95, Lecture Notes in Computer Science, volume 921, pages 263--273. Springer, New York, 1995. Google ScholarDigital Library
- F. Gouvěa. p-adic Numbers: An Introduction. Springer--Verlag, 2nd. edition, 2003.Google Scholar
- G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 5th edition, 1979.Google Scholar
- A. Klapper and M. Goresky. 2-adic shift registers. In Fast Software Encryption, Cambridge Security Workshop, Lecture Notes in Computer Science, volume 809. Springer--Verlag, December 1993. Google ScholarDigital Library
- A. Klapper and M. Goresky. Feedback shift registers, 2-adic span and combiners with memory. Journal of Cryptology, 10:111--147, 1997.Google ScholarDigital Library
- N. Koblitz. p-adic Numbers, p-adic Analysis, and Zeta Functions. Springer--Verlag, New York, GTM Vol. 58 edition, 1984.Google Scholar
- K. Mahler. Introduction to p-adic Numbers and their Functions. Cambridge University Press, 1973.Google Scholar
- G. Marsaglia. yet another rng. Posted to the Usenet newsgroup sci.stat.math, August 1, 1994.Google Scholar
- J. L. Massey. Shift-register synthesis and BCH decoding. IEEE Transactions on Information Theory, IT-15:122--127, January 1969.Google ScholarDigital Library
- M. Mittelbach and A. Finger. Investigation of FCSR-based pseudorandom sequence generators for stream ciphers. In Proceedings of the 3rd. International Conference on Networking, February 2004.Google Scholar
- R. A. Rueppel. Analysis and Design of Stream Ciphers. Springer--Verlag, 1986. Google ScholarDigital Library
- B. Schneier. Applied Cryptography. John Wiley & Sons, 2nd edition, 1996.Google Scholar
- T. Siegenthaler. Correlation immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30:776--780, 1984.Google ScholarDigital Library
- Z. Tasheva, B. Bedzhev, and B. Stoyanov. N-adic summation shrinking generator -- basic properties and empirical evidences. Submitted to the IACR e-print archive, 2004.Google Scholar
- J. Xu. Stream Cipher Analysis Based on FCSRs. Ph.D. dissertation, University of Kentucky, Lexington, Kentucky, 2000. Google ScholarDigital Library
Index Terms
- Periodicity, complementarity and complexity of 2-adic FCSR combiner generators
Recommendations
Clock-controlled FCSR sequence with large linear complexity
SETA'10: Proceedings of the 6th international conference on Sequences and their applicationsIn this paper, we investigate the stop-and-go clock-controlled generator based on FCSR. The output sequence is proven to have large linear complexity. Further, the experimental results show that most of the output sequences also have almost optimal 2-...
Continued Fraction Expansion as Isometry – The Law of the Iterated Logarithm for Linear, Jump, and 2-Adic Complexity
In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2-adic complexity arise naturally to measure the (non)randomness of a given string. Here, we define an isometry K on Fq infin which is the precise ...
Extended Games-Chan algorithm for the 2-adic complexity of FCSR-sequences
Binary sequences generated by feedback shift registers with carry operation (FCSR) share many of the important properties enjoyed by sequences generated by linear feedback shift registers. We present an FCSR analog of the (extended) Games Chan algorithm,...
Comments