PRBG Based on Couple Chaotic Systems

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

• Li Shujun, Mou Xuanqin, Cai Yuanlong
• School of Electronics Information Engineering

Xian Jiaotong University, China
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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

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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).

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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.

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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?

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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.
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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.
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CCS-PRBG - Requirements

• R1 F1 and F2 are both surjective chaotic maps
  defined on a same interval Ia,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) f1f2
  ii) f1 and f2 are both even symmetrical to the
  vertical line x(ab)/2.
• R4 The two chaotic orbits x1(i) and x2(i)
  should be asymptotically independent as i goes to
  infinity.

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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
x1x2.
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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

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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
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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.

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Stream Ciphers Based on CCS-PRBG (2)

• Cipher 2 (C2) Give four chaotic maps CS0CS3,
  and five maximal length LFSR-s m-LFSR0m-LFSR4.
  m-LFSR0m-LFSR3 are used to perturb CS0CS3.
  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.

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Stream Ciphers Based on CCS-PRBG (3)

• Cipher 3 (C3) Choose two piecewise linear
  chaotic maps (PLCM) defined on I0,1 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

j0n-1 x1(i,j)x1(i)gtgtj, x2(i,j)x2(i)ltltj,
kj(i)g(x1(i,j), x2(i,j))
Finally, K(i) is used to mask n-bit plaintext.
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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 C2C3
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.
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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.
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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 R1R4.
• 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.

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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

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• Thanks!
• Welcome to contact us
• via hooklee_at_mail.com.