## Ranara Damasceno

*Federal Institute of Education, Science and Technology of Ceara*

Telecommunications Engineer, master degree and doctor degree student in Quantum Information area.Teacher at Federal Institute of Education, Science and Technology of Ceara.

In this work, we propose a new scheme for quantum-haotic key distribution using two synchronized logistic maps and the Sagnac interferometer. The synchronized chaotic dynamic is given by the equations:

* *

z_{n}=QRNG

x_{n+1}=λx_{n}(1-x_{n})+c[k-λ(1-x_{n}- dz_{n})](x_{n}- dz_{n})

y_{n+1}=λy_{n} (1-y_{n} )+c[k-λ(1-y_{n}- dz_{n})](y_{n}- dz_{n})

Thus, in order to achieve the synchronism between the two chaotic systems (X belongs to Alice while Y belongs to Bob) the value of z_{n} has to be sent from Alice to Bob. The bits of the key are obtained from discretization of the output chaotic variables, x_{n} and y_{n}. The optical scheme used to send z_{n} from Alice to Bob is shown in Fig. 1.

Thus, the information used to keep the chaotic systems synchronized is carried by weak coherent states. In Fig. 1 the function *f* is the logistic map *f*_{m}_{+1}= *Xf*_{m}(1-*f*_{m}). The input value *f*_{0} is *x*_{n}_{-1} (*y*_{n}_{-1}) for Alice (Bob) and the output value is *f*_{t}. The parameters *X* and *t* are only known by Alice and Bob. The probabilities of detection in D_{0} and D_{1} are given by:

p_{0}=[1-e(-|a|^{2}ղ)*(1-p_{d})]*cos^{2}[π(z_{k}⊕a+f(x_{n-1},X,t)-f(y_{n-1},X,t))/2]

p_{1}=[1-e(-|a|^{2}ղ)*(1-p_{d})]*sin^{2}[π(z_{k}⊕a+f(x_{n-1},X,t)-f(y_{n-1},X,t))/2].

Where |a|^{2} is the mean photon number after attenuation, t is the channel transmissivity and p_{d} is the probability of detection of the avalanche photodiode. For Bob, detection in D_{0} means z_{n}=0 while detection in D_{1} means z_{n}=1. Hence, the probabilities of detection depend on the synchronism and the synchronism depends on the quantum probabilities. In other words, the chaotic and quantum parts are integrated in such way that it is impossible their separation. In this system, the bits of the key do not travel along the channel. An eavesdropper will have to attack the signals of synchronism. Disturbing the synchronism, the error rate increases revealing the attack. Thus, its security is based on chaotic and quantum rules and the scheme can be easily implemented with today technology.

Figure 2 shows a simulation of the synchronization obtained using the optical setup in Fig. 1. The parameters values used are λ = 3.9, k = 0.2, c = 0.5, d = 0.5, while the initial values of the dynamic variables are x(1) = 0.8, y(1) = 0.16.

Quantum information processing and computing , Quantum communication