An Excellent Permutation Operator for Cryptographic Applications (original) (raw)
Secure and fast encryption using chaotic Kolmogorov flows
1998 Information Theory Workshop (Cat. No.98EX131)
We describe a n e w s y m m e t r i c product ciphering a l g o r i t h m t h a t o p e r a t e s iteratively o n a n a r b i t r a r y s q u a r e block of plaintext w i t h the only constraint t h a t t h e blocklength h a s t o be an integral power of 2. P e r m u t a t i o n s are induced b y the highly unstable nonlinear dynamics of chaotic Kolmogorov flows, while substitutions are implemented using addwith-carry or subtract-with-borrow generators. Encryption performance is excellent i n hardand software which is based o n the fact t h a t only additions, subtractions a n d bit-shifts, b u t n o time-consuming operations like multiplication or exponentiation are necessary for implementing t h e cipher.
Cryptography with chaotic mixing
Chaos Solitons & Fractals, 2008
We propose a cryptosystem based on one-dimensional chaotic maps of the form Hp(x) = r −1 p • G • rp(x) defined in the interval [0, 10 p ) for a positive integer parameter p, where G(x) = 10x(mod 10) and rp(x) = p √ x, which is a topological conjugacy between G and the shift map σ on the space Σ of the sequences with ten symbols. There are three advantages in comparison with the recently proposed cryptosystem based on chaotic logistic maps Fµ(x) = µx(1 − x) with 3 < µ ≤ 4: (a) Hp is always chaotic for all parameters p, (b) the knowledge of an ergodic measure allows assignments of the alphabetic symbols to equiprobable sites of Hp's domain and (c) for each p, the security of the cryptosystem is manageable against brute force attacks.
Fast encryption of image data using chaotic Kolmogorov flows
Journal of Electronic Imaging, 1998
To guarantee s e curity and privacy in image transmission and archival applications, adequate e cient bulk encryption techniques are n e cessary which are able to cope with the vast amounts of image data involved. Experience has shown that block-oriented symmetric product ciphers constitute an adequate design paradigm for resolving this task, since t h e y c an o er a very high level of security as well as very high encryption rates. In this contribution we introduce a new product cipher which encrypts large blocks of plain-text images by repeated intertwined application of substitution and permutation operations. While almost all of the current product ciphers use xed prede ned permutation operations on small data blocks, our approach involves parameterizable keyed permutations on large data blocks whole images induced b y s p eci c chaotic systems Kolmogorov ows. By combining these highly unstable dynamics with an adaption of a very fast shift register based pseudo-random number generator we obtain a new class of computationally secure product ciphers which are rmly grounded on systems theoretic concepts, o ering many features that make them superior to contemporary bulk encryption systems when aiming at e cient image data encryption.
2001
This paper discusses mixing of chaotic systems as a dependable method for secure communication. Distribution of the entropy function for steady state as well as plaintext input sequences are analyzed. It is shown that the mixing of chaotic sequences results in a sequence that does not have any state dependence on the information encrypted by them. The generated output states of such a cipher approach the theoretical maximum for both complexity measures and cycle length. These features are then compared with some popular ciphers.
Fast encryption of image data using chaotic Kolmogorov flows
Proceedings of SPIE, 1997
To guarantee s e curity and privacy in image transmission and archival applications, adequate e cient bulk encryption techniques are n e cessary which are able to cope with the vast amounts of image data involved. Experience has shown that block-oriented symmetric product ciphers constitute an adequate design paradigm for resolving this task, since t h e y c an o er a very high level of security as well as very high encryption rates. In this contribution we introduce a new product cipher which encrypts large blocks of plain-text images by repeated intertwined application of substitution and permutation operations. While almost all of the current product ciphers use xed prede ned permutation operations on small data blocks, our approach involves parameterizable keyed permutations on large data blocks whole images induced b y s p eci c chaotic systems Kolmogorov ows. By combining these highly unstable dynamics with an adaption of a very fast shift register based pseudo-random number generator we obtain a new class of computationally secure product ciphers which are rmly grounded on systems theoretic concepts, o ering many features that make them superior to contemporary bulk encryption systems when aiming at e cient image data encryption.
Weak mixing and chaos in nonautonomous discrete systems
Applied Mathematics Letters, 2012
The paper is devoted to a study of chaotic properties of nonautonomous discrete systems (NDS) defined by a sequence f∞ = {f i } ∞ i=0 of continuous maps acting on a compact metric space. We consider such properties as chaos in the sense of Li and Yorke, topological weak mixing and topological entropy, all defined in a way suitable for NDS. We compare these concepts with the case of single map (discrete dynamical system, DS for short) and relate them to recent results in the topic. While previous research of various authors was focusing on analogues to DS case, we show that in general the dynamics of NDSs is much richer and quite different than what is expected from DS case. We also provide a few new tools that can be used for successful investigation of their qualitative behavior.
On the mixing properties of piecewise expanding maps under composition with permutations
Discrete and Continuous Dynamical Systems, 2013
For a mixing and uniformly expanding interval map f : I → I we pose the following questions. For which permutation transformations σ : I → I is the composition σ • f again mixing? When σ • f is mixing, how does the mixing rate of σ • f typically compare with that of f ? As a case study, we focus on the family of maps f (x) = mx mod 1 for 2 ≤ m ∈ N. We split [0, 1) into N equal subintervals, and take σ to be a permutation of these. We analyse those σ ∈ S N for which σ • f is mixing, and show that, for large N , typical permutations will preserve the mixing property. In contrast to the situation for continuous time diffusive systems, we find that composition with a permutation cannot improve the mixing rate, but may make it worse. We obtain a precise result on the worst mixing rate which can occur as σ varies, with m, N fixed and gcd(m, N) = 1.
Global measures of distributive mixing and their behavior in chaotic flows
Two measures of distributive mixing are examined: the standard deviation σ and the maximum error E, among average concentrations of finite-sized samples. Curves of E versus sample size L are easily interpreted in terms of the size and intensity of the worst flaw in the mixture. E(L) is sensitive to the size of this flaw, regardless of the overall size of the mixture. The measures are used to study distributive mixing for time-periodic flows in a rectangular cavity, using the mapping method. Globally chaotic flows display a well-defined asymptotic behavior: E and σ decrease exponentially with time, and the curves of E(L) and σ(L) achieve a self-similar shape. This behavior is independent of the initial configuration of the fluids. Flows with large islands do not show self-similarity, and the final mixing result is strongly dependent on the initial fluid configuration.
Chaotic encryption scheme based on a fast permutation and diffusion structure
Int. Arab J. Inf. Technol., 2017
The image encryption architecture presented in this paper employs a novel permutation and diffusion strategy based on the sorting of chaotic solutions of the Linear Diophantine Equation (LDE) which aims to reduce the computational time observed in Chong's permutation structure. In this scheme, firstly, the sequence generated by the combination of Piecewise Linear Chaotic Map (PWLCM) with solutions of LDE is used as a permutation key to shuffle the sub-image. Secondly, the shuffled sub-image is masked by using diffusion scheme based on Chebyshev map. Finally, in order to improve the influence of the encrypted image to the statistical attack, the recombined image is again shuffle by using the same permutation strategy applied in the first step. The design of the proposed algorithm is simple and efficient, and based on three phases which provide the necessary properties for a secure image encryption algorithm. According to NIST randomness tests the image sequence encrypted by the p...
The geometry of mixing in 2-d time-periodic chaotic flows
Chemical Engineering Science, 2000
This paper demonstrates that the geometry and topology of material lines in time-periodic chaotic #ows is controlled by a global geometric property referred to as asymptotic directionality. This property implies the existence of local asymptotic orientations at each point within the chaotic region determined by the unstable eigendirections of the Jacobian matrix of the n-period PoincareH map associated with the #ow. Asymptotic directionality also determines the topology of the invariant unstable manifolds of the PoincareH map, which are everywhere tangent to the "eld of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the invariant unstable manifolds associated with a PoincareH section to any desired level of detail. Since material lines evolved by a chaotic #ow are asymptotically attracted to the geometric global unstable manifold of the #ow (this concept is introduced in this article), such reconstructions can be used to characterize the topological and statistical properties of partially mixed structures quantitatively. Asymptotic directionality provides evidence of a global self-organizing structure characterizing physically realizable chaotic mixing systems which is analogous to that of Anosov di!eomorphisms, which turns out to represent the basic prototype of a mixing system. In this framework we show how partially mixed structures can be quantitatively characterized by a non-uniform stationary measure (di!erent from the ergodic measure) associated with the dynamical system generated by the "eld of asymptotic unstable eigenvectors.