A New Public Key Cryptography Algorithm Using Chaotic Systems and Hyperelliptic Curves. Rodrigo Abarz´ ua, Ivan Jir´on, Miguel Alfaro, Ismael Soto. Departamento de Ingenier´ ıa Industrial. Universidad de Santiago de Chile. Av. Ecuador 3769, Santiago. CHILE. roabarzua@gmail.com, ivanjiron.araya@gmail.com, malfaro@lauca.usach.cl, ismael.soto@gmail.com. Abstract The aim of this paper is to make a contribution to the development of the new stronger cryptographic algorithm using chaotic systems and hyperelliptic curve. In this context, the Diffie-Hellman scheme is imple- mented with chaotic systems and ElGamal scheme is constructed with hyperelliptic curves. Futhermore, the complexity algorithm is determinated for proposed algorithm. Also, this algorithm is compared with other system and BER v/s SNR curves are obtain in developed experiments. Key-Words: -Chaos, -Synchronization of Chaotic Systems, -Hyperelliptic Curves. 1 Introduction [1]Modern telecommunication networks, and especially Internet and Mobile-phone networks, have tremendously extended the limits and possibilities of communications and information transmissions. Associated with this rapid development, there is a growing demand of cryp- tographic techniques, which has spurred a great deal of intensive research activities in the study of cryptogra- phy [2], [3]. Since 1990s, many research have noticed that there exists and interesting relationship between chaos and cryptography, many properties of chaotic systems have their corresponding counterparts in traditional cryp- tosystems. There exist two main approaches of designing chaos- based cryptosystems: analog and digital. Most analog chaos-based cryptosystems are secure com- munication schemes designed for noisy channels based on the technique of chaos synchronization [4]. Chaos synchronization is a technique developed since 1990. Roughly speaking, it means that two identical dynam- ical systems, starting from different initial conditions, can be synchronized by common external signal which is coupled to the two systems [5]. It has been shown that even chaotic systems can be synchronized although the correlation between the external signal and the com- mon dynamics still remains chaotic [5], [6], [7]. This phe- nomenon has been applied to private key: If two partner A and B want to exchange a secret message, A adds its message to a synchronized signal while B obtains it. Of course, A and B need a common secret (private key), namely, the algorithm and the parameters of the used identical chaotic systems. Other side Hyperelliptic curves to be used in cryptosys- tems of public key [8]. This curve form a special class of algebraic curves. The most important algebraic struc- ture is given by a quotient group called Jacobian, of the hyperelliptic curve over a finite field, this quotient group was suggested by Neal Koblitz for an cryptosystems of public key.[8], [9]. The represented idea is the following: If two partners A and B want to exchange a secret message. Of course, A and B, need a common secret key, they are obtained. Given a hyperelliptic curves C, the [10] Diffie-Hellman algorithm is used for private key generating β, this is to transforms a parameter of the chaotic system. The syn- chronization of chaotic systems is implemented using a common external signal [4], two identical dynamical sys- tems the A (Master system) and B (Slave system) are synchronized and γ parameter is obtained. Then, a new type of private key is produced (β, γ ), where β is gener- ated on an Abelian Jacobian group of the hyperelliptic curves Diffie-Hellman algorithm and γ it is generated on synchronization of chaotic systems. 1 Proceedings of the 10th WSEAS International Conference on SYSTEMS, Vouliagmeni, Athens, Greece, July 10-12, 2006 (pp771-774)