PRBG Based on Couple Chaotic Systems & its Applications in Stream-Cipher Cryptography

1. 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

2. Outlines • Chaotic Cryptography (C2): 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

3. Chaotic Cryptography (C2) Two basic ideas about chaotic cryptography have been developed since 1989: • Cryptosystems based on discrete-time chaotic systems: 1st 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: 1st paper was published in 1990, L. M. Pecora, T. L. Carroll, Physical Review Letters, 64(8).

4. C2 - 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.

5. C2 - Problems (see sect. 1.2 for detailed discussions) • 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?

6. CCS-PRBG In this paper, we propose a novel solution to the above problems of C2: 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.

7. CCS-PRBG - Definition Give a couple of one-dimensional chaotic maps F1(x1,p1) and F2(x2,p2). Iterate the two maps to generate two chaotic orbits x1(i) and x2(i). Define a pseudo-random bit sequence k(i)=g(x1(i),x2(i)), where When some requirements are satisfied, the above PRBG is called CCS-PRBG. We will show CCS-PRBG has rather perfect cryptographic properties.

8. CCS-PRBG - Requirements • R1 – F1 and F2 are both surjective chaotic maps defined on a same interval I=[a,b]. • R2 – F1 and F2 are both ergodic on I, with unique invariant density functions f1 and f2. • R3 – One of the following facts holds: i) f1=f2; ii) f1 and f2 are both even symmetrical to the vertical line x=(a+b)/2. • R4 – The two chaotic orbits {x1(i)} and {x2(i)} should be asymptotically independent as i goes to infinity.

9. 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(x1,x2) when x1=x2.

10. 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

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

12. 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 x1(0), x2(0) and the control parameters p1,p2 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.

13. Stream Ciphers Based on CCS-PRBG (2) • Cipher 2 (C2) – Give four chaotic maps CS0~CS3, and five maximal length LFSR-s m-LFSR0~m-LFSR4. m-LFSR0~m-LFSR3 are used to perturb CS0~CS3. m-LFSR4 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 CSpn1(i) and CSpn2(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.

14. Stream Ciphers Based on CCS-PRBG (3) • Cipher 3 (C3) – Choose two piecewise linear chaotic maps (PLCM) defined on I=[0,1] as F1 and F2. Then the invariant density functions of F1 and F2 will be uniform: f1(x)=f2(x)=1. When they are realized in finite precision n, each bit of x1(i) and x2(i) will be approximately balanced on {0,1}. Thus, we can generalize CCS-PRBG to make a n-bit pseudo-random number K(i)=k0(i)~kn-1(i) for each i: j=0~n-1: x1(i,j)=x1(i)>>j, x2(i,j)=x2(i)<<j, kj(i)=g(x1(i,j), x2(i,j)) Finally, K(i) is used to mask n-bit plaintext.

15. Stream Ciphers Based on CCS-PRBG - Performance | Key Entropy | Encryption Speed | Complexity C1 4n 1 1* C2 8n 1 2 C3 4n about n 1 C2+C3 8n about n 2 * 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.

16. 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.

17. 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 requirementsR1~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.

18. 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

19. Thanks! Welcome to contact us via hooklee@mail.com.