ISIT 1998. Cambridge. M A , USA, August 16 - August 2 1
On the Complexity of Reeonfigurable Feedback Shift Register Sequences Jung Yuehl College of Computer Science Northeastern University Boston, MA. 02115, USA Email
[email protected]
Agnes Hui Chan’ College of Computer Science Northeastern University Boston, MA. 02115, USA Email ahchanQccs.neu.edu
Abstract - Reconfigurable Feedback Shift Registers can generate sequences at t h e optical rate while exploits t h e complexity of a slower speed generator. In this paper, we show that the complexity of such a RFSR sequence is at least as strong as t h a t of t h e electronic generator, and discuss s o m e of the properties required of t h e feedback functions in order to ensure realization of complex sequences.
I. INTRODUCTION
Muriel M6dard2 Lincoln Laboratory M.I.T. Lexington, MA 02173, USA Email
[email protected]
If the fj’s are linear, then such a RFSR is called a Reconfigurable Linear Feedback Shift Rgister (RLFSR). Most of our results are valid for both linear and nonlinear feedback functions. We believe that similar analysis can be applied to general RFSR sequences with more than two feedback functions.
111. CORRELATION FACTORS AND COMPLEXITY Intuitively, if the two feedback functions are identical, then the RLFSR sequence generated is as complex as the feedback function itself. Thus, to analyse the complexity of such a sequence, we need to measure the similarity of two functions defined by the correlation factor.
File transfer through high speed optical TDM networks is fast becoming a reality, which leads to the need of cost-effective ultrafast pseudorandom key stream generations [l].Chan and Definition 2 Let fjj(8) be the j - t h output bit generated b y Mkdard first introduced the design of a Reconfigurable Feedback Shift Register (RFSR) [2] that relies upon electronic logic f ; with intial loading 8 E (0, l } L , i = 0 , l . For j 2 1, the correlation factor ej is defined by for reconfiguring an optical FSR generating a key stream at the optical speed. Such a generator has been shown to generate sequences with long periods and good statistical properties. In this paper, we show the complexity of the sequence Proposition 1 Let f o , f i be any linear feedback functions of generated by a RFSR can be reduced to the complexity of the length L, then el = 112 slower electronic controller, thus establishes a lower bound on the complexity of the RFSR sequence. We also discuss T h e o r e m 1 Let s be a given RFSR sequence generated by properties of the feedback functions required for realizing high feedback functions fo and f l with 6-correlation factor e,j. Ifs, complexity sequences. fo, fl and 6 are known, then on average, all but re6 bits of the control sequence t can be determined eficiently, where r 11. RECONFIGURABLE FEEDBACK SHIFTREGISTERS is the period o f t . To design an ultrafast sequence generator, we consider an optical FSR of length L ( 2 l o 4 ) with limited gate counts that If e6 = 0 then the control sequence can be determined uniquely outputs a symbol at every clock cycle. Due to the high cost and completely from the RFSR sequence. We define th’e comand technology constraints in optics, the number of taps in plexity of a sequence to be the smallest number of bits required the FSR has to be small, but such a sequence can be eas- to determine the whole sequence. ily broken by exhaustive search. We use a “slow” electronic strong sequence generator to reconfigure the optical feedback function. Let G be such an electronic sequence generator that generates key stream t , known as the control sequence, and 6 be the ratio of the optical data rate to the electronic data rate. At every 6 clock cycles, the taps of our FSR are reconfigured according to the output symbol t , of G. For simplicity, we consider only two feedback functions. Definition 1 A 2-functzon RFSR consast of a FSR of length L , two feedbackfunctaons f o , f l , and a controller that outputs a bznary sequence t . The RFSR generates the sequence s given by S z 6 + j + L = ft.(sz6+2, S16+3+lr . . . $ S z 6 + 3 + L - l ) (1) where 0 5 i , 0 5 j < 6. ‘This work was supported by NSF grant, NCR-9753048 2This work is supported by The US. Air Force. Opinions, interpretations, conclusions, are those of the authors, and are not necessarily endorsed by the U.S. Air Force.
0-7803-5000-6/98/$10.00 0 1998 IEEE.
T h e o r e m 2 Let s be the RFSR sequence with control sequence t. Assume that the speed of the ratio is given b y 6 and the feedback functions f o and f 1 have 6-correlation factor e6 = 0. If the complexity o f t is ct, then the complexity of the sequence s is at least 6(ct - 1). We have established that fast, complex RFSR sequences can be generated by using known complex slower sequences. Further work is needed in classifying feedback functions that realize strong RFSR sequences efficiently.
REFERENCES V.W.S. Chan All Optical Networks, Scientific Americam, Vol.
273, no.3, September 1995 A.H. Chan, M. Medard, Reconfigurable Feedback Shift Registers, International Symposium on Information Theory, Ulm,
Germany, June 1997. J.Dj. Golic, Towards Fast Correlation Attacks on Irre:gularly Clocked Shift Registers, Eurocrypt95, Springer-Verlag, Berlin, Germnay, 1995.
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