Chaotic maps are a promising new direction for image encryption compared to conventional
cryptosystems. Advantages include highly sensitive keys (e.g. initial conditions), greater run-time efficiency, and tractability within embedded imaging devices such as satellites. This investigation describes the design and justi.cation of a simple chaotic cryptosystem based upon a revised logistic map. Two primary design objectives, which we call diffusion and confusion, are articulated. A shuffle-and-mask algorithm based on the revised logistic map is then presented which fulfills diffusion goals. Later we present a plaintext attack that breaks the presented scheme and highlights how confusion design weaknesses can be exploited.
To fulfill my final mathematics requirement at Olin, I took a class on Nonlinear Dynamics and Chaos with Prof. John Geddes. The class focused on interactively exploring basic concepts in the field, covering discrete and continuous theory. The course's only deliverables were four-page research reports in the style of Physical Review Letters highlighting results of a multi-week investigation on a topic of the student's choosing (related to the current topic, of course). This page highlights my second such report, which focused on the topic of advanced topics in discrete maps. I conducted a literature review to find an exciting application that used 1D maps and then explored ideas of my own via Matlab simulations. This was a highly rewarding project, since I discovered a sweet application of chaos theory relevant to my own interests in computing and security. I also felt very accomplished as a security analyst, as I went from zero understanding at the beginning to discovering a complete break of a simple encryption scheme I had proposed.
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Catalyzed by dramatic increases in both the availability of image capture devices and the bandwidth of information transfer, digital photos and video .nd utility in a growing number of applications. Many of these applications require con.dentiality of image content, whether for medical privacy, discreet surveillance, or other uses, and so research in image encryption remains a widely pursued and well-funded research .area.
The increasing number of embedded cameras within remote sensing devices (e.g. satellites and UAVs) require special research attention due to limited computational resources. Many in-practice cryptographic schemes that are widely used in e-commerce have signi.cant shortcomings when applied the image processing domain. RSA, DES, and other systems that secure online banking rely on desktop computing power to perform complex mathematical operations such as modular exponentiation, primality testing, and so on. Furthermore, even when processing power is not a concern these schemes o.er poor run-time performance on the order of several seconds per image, which precludes real-time transfer of high-frame rate video. Additionally, conventional algorithms often cannot adequately disperse the highly correlated information found in images to maintain adequate con.dentiality .
These disadvantages of conventional cryptography have inspired a recent wave of research applying chaotic maps to cryptographic applications. A chaotic mapping is a parameterized system of nonlinear equations that iteratively forces system state through a pseudo-random trajectory across bounded state space in a way that is highly sensitive to initial conditions. Many researchers have observed that this behavior is highly analogous to conventional symmetric key cryptographic schemes. That is, if two communicating parties agree on a specific map function as well as a secret set of parameters and initial conditions, the sender can use the function as well as secret parameters (keys) to produce an arbitrarily long source of pseudo-random numbers with which to scramble a secret message. Later, the receiver can consume this scrambled message and, after generating the same exact source of pseudo-randomness, invert the scrambling operation to recover the secret message.
Chaotic encryption systems have many advantages over conventional schemes. First, although non-linear the typical mathematical operations required have considerable performance advantages compared to the algorithmic complexities and often specialized hardware required for AES and other desktop standards, making chaotic encryption favorable for cost-efficient embedded devices. Second, the sensitivity of chaotic systems to parameter values and initial conditions allow con.dence in using these values as secret keys, as near-miss values will yield dramatically di.erent pseudo-random trajectories. Finally, mathematical tools for analyzing chaotic behavior (e.g. Lyaponov numbers) are well-studied and helpful in designing an effective protocol.
Many di.fferent proposals for chaos-based encryption have been made (e.g , ,  ), and each one diff.ers in the type, number, and dimensionality of the map used. This paper provides a modest tour of the design considerations, algorithmic execution, and security analysis of a simple one-dimensional cryptosystem based on the logistic map. Although advanced multi-dimensional systems have been proposed and analyzed (e.g. , ), high-dimensionality is not necessary and may place additional burdens on performance. We thus restrict ourselves for the sake of clarity to one-dimension, so that we may provide detailed justi.cation of design choices for the map and parameter space while avoiding unnecessarily sophisticated mathematics.
SHUFFLE AND MASK PROCEDURE OVERVIEW
Illustration of the two stage image encryption process used here. First, we shuffle pixels around the image based on pseudo-random number stream generated by the logistic map with stage one parameters. Next, we mask that shuffled image with a separate pseudo-random number stream generated by the logistic map with stage two parameters. This simple process can achieve diffusion objectives, as it makes it difficult for the encrypted image to yield information about the original.
EXAMPLE IMAGE ENCRYPTION
This figure shows original, encrypted, and decrypted images using the shuffle and mask process. We can see from the histograms of gray values in the encrypted case that diffusion objectives have been achieved, since information about the gray values of the original have been disguised so that an attacker with just the encrypted image stands little chance of recovering any information about the original. The procedure is still vulnerable to other attacks, as the report discusses.
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