Proceedings of the lASTED International Conference COMMUNICATIONS, INTERNET, & INFORMATION TECHNOLOGY November 18-20, 2002, St. Thomas, US Virgin Islands ASYMMETRIC KEY ENCRYPTION USING DISTRIBUTED CHAOTIC NONLINEAR DYNAMICS Roy Tennyl,2, Lev S. Tsirnringl,Henry D.I.Abarbanell,3, Larry Larson2 1 Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402 2 Dept. of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0354 3 Department of Physics Department and Marine Physical Laboratory (Scripps Institution of Oceanography), University of California, San Diego, La Jolla, CA 92093-0402 ABSTRACT was introduced in 1978 and is currently the most widely In this paper we introduce a new method for public key used public key cryptosystem. Other methods for pub- encryption by using continuous nonlinear dynamics. We lic key encryption that are based on discrete number the- distribute a high-dimensional dissipative nonlinear dynam- ory are: Elliptic curves [2], EIGamal discrete logarithm ical system between transmitter and receiver, so we call problem based crypt~system [3], McElliece Coding theory the method: Distributed Dynamics Encryption (DDE). One based system [4], and Chor-Rivest knapsack based system part is a transmitter with public dynamics and the other part [5]. is a receiver with secret dynamics. The transmitter and re- During the last decade a new class of secret (symmetric) ceiver are coupled using bi-directional signals that are pub- key encryption schemes using chaotic non linear dynam- lic. A message is encoded by modulating the dynamics of ics was developed. Most of the proposed schemes are the transmitter which results in a shift in the position of the based on the following methods: chaos synchronization system's attractor. An unauthorized receiver who does not ([6],[7],[8], [9]), controlling chaos ([10],[11]) chaotic shift know the secret dynamics of the receiver does not know keying [12]. Unlike classical encryption schemes, chaotic the position of the attractor and can not decode the mes- dynamics encryption schemes have a continuous state and sage. We show that the security of DDE can be enhanced either continuous or discrete time represented by differen- by modulating the secret dynamics of the receiver and ini- tial equations or discrete maps ([13] [14] [15]). Crypt- tializing the transmitter state with a random value at the analysis of such methods is discussed in ([16] [17] [18]). beginning of each transmitted bit. We implemented and Encryption schemes are usually embedded in a commu- cryptanalyzed DDE using the dynamics of a coupled map nication system that transfer data through a channel and lattice. therefore not only the security level should be considered but also communication efficiency aspects such as power KEY WORDS efficiency, bandwidth efficiency and spectral shape: Effi- Nonlinear dynamics, Chaos, Public Key Encryption, Cou- cient modulation schemes such as Chaotic Pulse Position pled Map Lattice Modulation (CPPM) [19] and chaotic Frequency Modu- lation (CFM) [20] can be used to send the signals that couple between the transmitter and receiver in the pres- 1 Introduction ence of channel distortions. Transmission of chaotic sig- nals through a bandlimited channel was studied in [21]. Encryption schemes for public key encryption and secret High speed secure communication by synchronization of key encryption have been thoroughly studied during the high-dimensional chaotic optical ring dynamics described past decades. In a secret (synlIDetric) key encryption in [22]. scheme, a secret key which is known only to the transmit- In this paper we introduce a new method for public key en- ter and the receiver is used to encrypt and decrypt a mes- cryption using a Distributed Dynamics Encryption scheme sage. In a public (asymmetric) key encryption a public key (DDE). In Section 2 we introduce DDE and overview the is used at the transmitter to encrypt a message. It is as- encryption and decryption algorithms. In Section 3 we dis- sumed that the public key is known to both an authorized cuss various cryptanalysis attacks on the system and sug- receiver and an unauthorized receiver. The message is de- gest methods to protect against those attacks. Simulation coded by a secret key which is different from the public key of DDE is described in Section 4. Summary and sugges- and is assumed to be known only by the authorized receiver. tions for future research are presented in Section 5. Decoding the message using the public key which was used to encrypt the message is made computationally infeasible. Classical cryptography is founded mainly on discrete num- ber theory. The RSA public encryption scheme [1], named after it's inventors (R.L.Rivest, A.Shanrir and L.Adleman) 376-803 338 sr(n) S,W Signal ~ r(n+l) =FR (r(n),s, (n» t(n+l) =FT(t(n),sr (n),m) 5r(n) =GR (r(n» s, (n) =GT (t(n» Receiver Transmitter Receiver Transmitter Data Figure 1. Nonlinear dissipative dynamical system with an Figure 2. Distributed nonlinear dynamics.Private: attractor is split into two parts: Transmitter (public dynam- F R(. ),GR(. ),r(n). Public: FT(. ),GT(. ),sr(n),St(n). ics) and receiver (secret dynamics). Bi-directional signal Known only to transmitter: t(n),m. (public) is used to couple between transmitter and receiver. with multi-dimensional coupling signals. 2 DDE General overview The structure of DDE is illustrated In Fig. 2. The receiver state r(n) = h(n),..., rDR(n)]hasdimension DR and is controlled by the map The principle of DDE is illustrated in Fig. 1. A nonlinear dissipative dynamical system is split into two parts: one r(n + 1) = FR(r(n),st(n)). (1) part is placed in a receiver and the other in a transmitter. The transmitter and receiver are coupled through a channel The receiver state r( n) and the receiver dynamics using bi-directional signal. F R(.) are kept secret (known only to the authorized re- The dynamical systemis represented by a state which ceiver) and serve as a part of the secret key. The scalar sig- is a complete set of system variables, and by the dynamics nal Sr(n) transmitted from the receiver to the transmitter is which specify the roles governing the transition from one defined as state to another (differential equations or maps). Data is transmitted by modulating one parameter of the transmitter (2) part and thereby shifting the attractor of the overall sys- tem. For simplicity we only consider binary information The signal sr(n) can be observed by both authorized signals, so the parameter switches between two possible and unauthorized receivers and is a part of the public key, values. The method relies on the assumption that the dis- however the function GR(.) is kept secret (known only to tributed dynamical system which is comprised of the trans- the authorized receiver) and is a part of the secret key. The mitter, receiver, and the coupling bidirectional signals, con- transmitterstate t(n) = [t1(n),..., tDT(n)}hasdimen-. verges to a single attractor for each value of the modulation sion DT and is given by parameter, independent of the initial conditions. Shifting chaotic attractors as a method of information transmission t(n + 1) = FT(t(n), sr(n), m(n)). (3) has been proposed by Dedieu et al [12] as a method for The transmitter state t(n) and the transmitted bit m(n) are symmetric (secret key) communication. Here we show that explicitly known only to the transmitter. Both authorized a similar principle can be used for the asymmetric (public and unauthorized receivers need to estimateboth quantities. key) encryption as well. The dynamics of the transmitter state, FT(.)' is known to The key idea is that the receiver has the full knowl- all and is a part of the public key. Data is sent by switch- edge of the dynamics and therefore knows the positions of ing the parameter m( n) at the transmitter which results in the attractors corresponding to the two values of the modu- a change in the position of the attractor of the dynamical lation parameter, soit can decode the message. Meanwhile, system. Without losing generality,in this paper we assume unauthorized recipients only know the transmitter part of binary transmission where the parameter m( n) can have the full dynamics, and the protocol of communication is values '0' or '1'. chosen in such a way that the attractor can not be recon- The scalar signal St(n) transmitted from the transmit- structed based on the transmitted signals. The attractor can ter to the receiver is given by take the form of a limit cycle, a high dimensional hyper- surface, or a chaotic attractor. The nonlinear dynamical (4) system is continuous in the state space and can be either continuous or discrete in time. The public key encryption Both the signal St(n) and the function GT (.) are known to scheme discussed in this Letter is discrete in time and the all and are a part of the public key. An authorized receiver coupling signals are scalars, however the concept can be who knows the full dynamics of the system can simulate applied to continuous time dynamical systems and systems the system off line and find the position of the attractors 339 sin) i ----- + /' 1 random initial I state '1' attractor I I ,/ I I I Transmitter I 1 1 1_____-. Figure 3. A trajectory in the reconstructed embedding Figure 4. Eavesdropper attempts to estimate message m phase space E starts at a random initial state and converges and hidden initial state of transmitter t (0) without knowing to one of two attractors that correspond to transmitted '0' secret dynamics and secret state of receiver.