Abstract
In this paper, we describe a solution to the register synthesis problem for a class of sequence generators known as algebraic feedback shift registers (AFSRs). These registers are based on the algebra of π-adic numbers, where π is an element in a ring R, and produce sequences of elements in R/(π). We give several cases where the register synthesis problem can be solved by an efficient algorithm. Consequently, any keystreams over R/(π) used in stream ciphers must be unable to be generated by a small register in these classes. This paper extends the analyses of feedback with carry shift registers and algebraic feedback shift registers by Goresky, Klapper, and Xu.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Bonnecaze, P. Sole, C. Bachoc and B. Mourrain, Type II codes over Z 4, IEEE Trans. Info. Theory, Vol. IT-43 (1997) pp. 969–976.
Z. Borevich and I. Shafarevich, Number Theory, Academic Press, New York (1966).
J. Conway and N. J. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory, Ser. A, Vol. 62 (1993) pp. 30–45.
J. Fields and P. Gaborit, On the non Z4-linearity of certain good binary codes, IEEE Trans. Info. Theory, Vol. IT-45 (1999) pp. 1674–1677.
D. Gollman and W. Chambers, Clock-controlled shift registers: a review, IEEE Journal on Selected Areas in Communication, Vol. 7 (1989) pp. 525–533.
S. Golomb, Shift Register Sequences, Aegean Park Press, Laguna Hills, CA (1982).
A. Klapper and M. Goresky, 2-adic shift registers, Fast Software Encryption (ed. R. Anderson), Lecture Notes in Computer Science, Springer-Verlag, Berlin, 809 (1994) pp. 174–178.
N. A. Hammons, P. Kumar, A. Calderbank, N. Sloane and P. Sole, Z 4 linearity of kerdock, preparata, goethals, and related codes, IEEE Trans. Infor. Theory, Vol. 40 (1994) pp. 301–319.
N. Jacobson, Basic Algebra I, W.H. Freeman, San Francisco (1974).
N. Jacobson, Basic Algebra II, W.H. Freeman, San Francisco (1980).
A. Klapper and M. Goresky, Feedback shift registers, 2-adic span, and combiners with memory, Journal of Cryptology, Vol. 10 (1997) pp. 111–147.
A. Klapper and J. Xu, Algebraic feedback shift registers, Theoretical Computer Science, Vol. 226 (1999) pp. 61–93.
N. Koblitz, p-Adic Numbers, p-Adic Analysis, and Zeta Functions, Graduate Texts in Mathematics, Vol. 58, Springer-Verlag, New York (1984).
K. Mahler, On a geometrical representation of p-adic numbers, Ann. of Math., Vol. 41 (1940) pp. 8–56.
J. Massey, Shift-register synthesis and BCH decoding, IEEE Trans. Infor. Theory, Vol. IT-15 (1969) pp. 122–127.
J. Massey and R. Rueppel, Methods of, and Apparatus for, Transforming a Digital Data Sequence into an Encoded Form, Vol. 4797922 of U.S. Patent (1989).
J. Reeds and N. Sloane, Shift-register synthesis (modulo m), SIAM J. Comp., Vol. 14 (1985) pp. 505–513.
R. Rueppel, Analysis and Design of Stream Ciphers, Springer-Verlag, New York (1986).
A. Shanbhag, P. Kumar and T. Helleseth, Improved binary codes and sequence families from Z 4 linear codes, IEEE Trans. Info. Theory, Vol. IT-42 (1996) pp. 1582–1587.
B. M. M. de Weger, Approximation lattices of p-adic numbers, J. Num. Thy., Vol. 24 (1986) pp. 70–88.
J. Xu and A. Klapper, Feedback with carry shift registers over Z/(n), Proceedings of SETA '98, Springer-Verlag, New York (1998).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klapper, A., Xu, J. Register Synthesis for Algebraic Feedback Shift Registers Based on Non-Primes. Designs, Codes and Cryptography 31, 227–250 (2004). https://doi.org/10.1023/B:DESI.0000015886.71135.e1
Issue Date:
DOI: https://doi.org/10.1023/B:DESI.0000015886.71135.e1