4 January 1999 Physics Letters A 251 (1999) 31138 Using chaos to produce synchronized stochastic dynamics in non-homogeneous map arrays with a random scalar coupling Ali A. Minai 1 Department of Electrical & Computer Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221-0030, USA Received 20 July 1998; revised manuscript received 25 September 1998; accepted for publication 14 October 1998 Communicated by A.R. Bishop Abstract Unidirectionally coupled chaotic systems hold great interest from the information processing and communications perspec- tive. In this Letter, we report on a novel method for synchronizing two identical but internally non-homogeneous populations of chaotic maps using a scalar random coupling between them. The resulting synchronized dynamics is stochastic, and can be used in secure multi-user communication applications. c  1999 Published by Elsevier Science B.V. PACS: 05.45.+b; 43.72.+q; 07.05.Pj 1. Introduction Synchronization between unidirectionally coupled chaotic systems [1,2] is of considerable interest from the theoretical and applied standpoint. In particular, it has been suggested that such systems may be useful for secure communications [318]. In most such pro- posals, the chaotic coupling signal is used as the car- rier of information 1 either directly or through modu- lation 1 and synchronization between chaotic systems in the transmitter and receiver allows the recovery of the message. While there are several impediments to such applications, two key issues of interest are: (1) Making the coupling signal maximally unstruc- tured [9], so that the hidden message cannot be recov- ered through reconstruction and noise-removal meth- ods [10112]; and (2) multiplexing several messages onto the same scalar coupling signal [13]. In this Let- 1 E-mail: ali.minai@uc.edu. ter, we report on a novel synchronization technique that is relevant to both issues. In particular, we show that identical but internally inhomogeneous arrays of chaotic maps can be synchronized if coupled through a random scalar signal from the drive array to the re- sponse array. As a result, corresponding maps in each array generate synchronized high-dimensional noise rather than low-dimensional chaos, though chaos is essential for the synchronization to emerge. Thus, the method we present contrasts with all other schemes for chaotic synchronization, where the coupling is deter- ministic 1 though perhaps intermittent [14] 1 and the synchronized signals are deterministic 1 albeit chaotic. The availability of truly random synchronized signals has obvious utility in real-time encrypted communi- cation, where the signals can be used as encryption keys [15,16,8]. The method we report is also applica- ble to secure multi-user spread-spectrum digital com- munication [17,18], with the random signals gener- ated by individual maps used as aperiodic spreading 0375-9601/99/$ 1 see front matter c  1999 Published by Elsevier Science B.V. All rights reserved. PII S0375-9601 ( 98 ) 00799-3 PLA 8521