PRBG Based on Couple Chaotic Systems & its Applications in Stream- Cipher Cryptography Li Shujun, Mou Xuanqin, Cai Yuanlong School of Electronics & Information Engineering Xi’an Jiaotong University, China

Outlines  Chaotic Cryptography (C 2 ): Overview and Problems  PRBG Based on Couple Chaotic Systems (CCS-PRBG)  Cryptographic Properties of CCS- PRBG  Stream Ciphers with CCS-PRBG  Conclusions and Open Topics

Chaotic Cryptography (C 2 ) Two basic ideas about chaotic cryptography have been developed since 1989: Cryptosystems based on discrete-time chaotic systems : 1 st paper was published in 1989, R. Matthews, Cryptologia, XIII(1). We focus on this idea in our paper. Secure communication approaches based on chaotic synchronization technique : 1 st paper was published in 1990, L. M. Pecora, T. L. Carroll, Physical Review Letters, 64(8).

C 2 - Overview Chaotic Stream Ciphers: Most researchers focus their attention on chaotic stream ciphers. General idea is using one chaotic system to generate pseudo-random key-stream. Chaotic Block Ciphers: Two chief ideas have been proposed – inverse chaotic system approach and 2-D chaotic systems approach. Other Chaotic Ciphers: Two special chaotic ciphers are introduced in our paper. Please see sect. 1.1 for more details.

C 2 - Problems Discrete Dynamics : How to improve the dynamical degradation of digital chaotic systems? Chaotic Systems : How to design a general cryptosystem with chaotic-system-free property? Encryption Speed : How to obtain faster speed? Practical Security : How to avoid potential insecurity hidden in single chaotic orbit? Realization Considerations : How to reduce the realization complexity and cost? (see sect. 1.2 for detailed discussions)

CCS-PRBG In this paper, we propose a novel solution to the above problems of C 2 : CCS-PRBG, which is useful to construct chaotic stream ciphers. Generally speaking, we can regard CCS-PRBG as a nearly “perfect” nonlinear PRBG. When we design a new stream cipher, we can use it just like we use LFSR-s or NLFSR-s in conventional stream ciphers. Theoretical and experimental results have suggested that CCS-PRBG should be promising as a kernel part of chaotic stream cipher.

CCS-PRBG - Definition Give a couple of one-dimensional chaotic maps F 1 (x 1,p 1 ) and F 2 (x 2,p 2 ). Iterate the two maps to generate two chaotic orbits x 1 (i) and x 2 (i). Define a pseudo-random bit sequence k(i)=g(x 1 (i),x 2 (i)), where When some requirements are satisfied, the above PRBG is called CCS-PRBG. We will show CCS- PRBG has rather perfect cryptographic properties.

CCS-PRBG - Requirements R1 – F 1 and F 2 are both surjective chaotic maps defined on a same interval I=[a,b]. R2 – F 1 and F 2 are both ergodic on I, with unique invariant density functions f 1 and f 2. R3 – One of the following facts holds: i) f 1 =f 2 ; ii) f 1 and f 2 are both even symmetrical to the vertical line x=(a+b)/2. R4 – The two chaotic orbits {x 1 (i)} and {x 2 (i)} should be asymptotically independent as i goes to infinity.

CCS-PRBG – Realization To avoid the dynamical degradation of digital chaotic systems, we suggest realizing chaotic systems via pseudo-random perturbation. Please see the following figure, where PRNG-3 can be used to determine the output of g(x 1,x 2 ) when x 1 =x 2.

Cryptographic Properties When CCS-PRBG is realized with pseudo- random perturbation, we can show the pseudo- random bit sequence k(i) generated by CCS- PRBG has the following cryptographic properties: Balance on {0,1} Long Cycle-Length High Linear Complexity: About n/2 Desired Auto/Cross-Correlation Chaotic-System-Free Property

Cryptographic Properties We give detailed discussions on the above properties of CCS-PRBG in Sect. 3 of our paper. Balance Linear Complexity Auto-Correlation Cross-Correlation

Stream Ciphers Based on CCS-PRBG (1) Based on CCS-PRBG, we can easily construct some chaotic stream ciphers. Cipher 1 (C1) –The simplest stream cipher with CCS-PRBG. The initial conditions x 1 (0), x 2 (0) and the control parameters p 1,p 2 compose the secret key, k(i) is used to mask plaintext bit by bit. Most chaotic stream ciphers proposed by other researchers before are just like Cipher 1, except that different chaotic PRBG-s are used.

Stream Ciphers Based on CCS-PRBG (2) Cipher 2 (C2) – Give four chaotic maps CS 0 ~CS 3, and five maximal length LFSR-s m- LFSR 0 ~m-LFSR 4. m-LFSR 0 ~m-LFSR 3 are used to perturb CS 0 ~CS 3. m-LFSR 4 is used to generate 2- bit pseudo-random numbers pn1(i) and pn2(i). If pn1(i)=pn2(i), then pn2(i)=pn1(i) XOR 1. Select CS pn1(i) and CS pn2(i) to compose the digital CCS- PRBG to generate k(i). Finally, k(i) is used to mask the plaintext bit by bit just like Cipher 1.

Stream Ciphers Based on CCS-PRBG (3) Cipher 3 (C3) – Choose two piecewise linear chaotic maps (PLCM) defined on I=[0,1] as F 1 and F 2. Then the invariant density functions of F 1 and F 2 will be uniform: f 1 (x)=f 2 (x)=1. When they are realized in finite precision n, each bit of x 1 (i) and x 2 (i) will be approximately balanced on {0,1}. Thus, we can generalize CCS-PRBG to make a n-bit pseudo-random number K(i)=k 0 (i)~k n-1 (i) for each i: j=0~n-1: x 1 (i,j)=x 1 (i)>>j, x 2 (i,j)=x 2 (i)<<j, k j (i)=g(x 1 (i,j), x 2 (i,j)) Finally, K(i) is used to mask n-bit plaintext.

Stream Ciphers Based on CCS-PRBG - Performance | Key Entropy| Encryption Speed | Complexity C1 4n11* C2 8n12 C3 4n about n1 C2+C3 8n about n2 * n is the finite precision and “1” indicates the order of speed and complexity. Note: The speed of C3 approximately equals to most simple stream ciphers based on LFSR-s.

Stream Ciphers Based on CCS-PRBG – Discussions In fact, more different chaotic stream ciphers still can be constructed with CCS-PRBG. We can see CCS-PRBG may be a promising new source to stream-cipher cryptography. In our paper, we also point out CCS-PRBG is immune to all known cryptanalytic methods breaking some other chaotic ciphers. In addition, one trivial security problem in CCS- PRBG is also discussed and remedy is provided. Please see the last paragraph of Sect. 4.2.

Stream Ciphers Based on CCS-PRBG – Solution? Discrete Dynamics : Solve this problem with pseudo-random perturbation algorithm. Chaotic Systems : A large number of chaotic maps obey the four requirements R1~R4. Encryption Speed : Cipher 3 solves this problem. Practical Security : Two chaotic orbits mix each other to avoid the insecurity induced by single orbit. Realization Considerations : Piecewise linear chaotic maps (PLCM) are suggested.

Conclusions & Open Topics CCS-PRBG, a new chaotic PRBG, is proposed in our paper. Its applications in stream-cipher cryptography is demonstrated. There are still some problems about CCS-PRBG have not perfect answers. The open topics include: The strict proof of k(i) is i.i.d. sequence The optimization problems about the hardware and software realization of digital CCS-PRBG and related stream ciphers Possible attacks to CCS-PRBG