Applications of Chaotic Fractals in Cryptography Jeremy Knight Texas A&M - Math 673 Summary The security of a cryptographic algorithm is based on the level of difficulty that is involved in solving the problem at the heart of the algorithm. With the R.S.A. algorithm, the security is found in the difficulty to factor large integers that are products of large primes. The security of other algorithms are based on the difficulty to solve discrete logarithm problems such as in the Diffie-Hellman algorithm, the Digital Signature Algorithm (DSA), and the ElGamal signature scheme[5]. These "hard problems" offer a vehicle to transfer information in numerical format while ensuring that no one who does not have the secret key(s) can view the meaning of the message. In classical cryptography, the problems considered have been based in the field of number theory. However, in the last 50 years, mathematicians have been intrigued by other "hard problems" based on the theory of chaos. Chaos is a characteristic of a system that shows sensitive dependence on initial conditions and forever continues to experience unstable behavior as time progresses[6]. Chaos is often demonstrated by fractal geometry. A fractal is a geometric path characterized by a seemingly random orbit while it has a very certain order that originates from the algorithm which defines it. This combination of fundamental order and unpredictable chaos makes fractals a strong candidate for the basis of a cryptographic algorithm. Chaos and in Cryptography Before we explore the prospect of applying fractals to cryptography, we will discuss current and potential applications of chaos theory in cryptography. In classical cryptography, which is based on number theory concepts, key spaces are usually limited to a finite set of integers that meet given conditions of an algorithm. When applying chaos in cryptography, we are able to work in the field of Real numbers. This allows for larger key spaces which makes a system more secure. Furthermore, the unpredictable nature of chaotic functions provides protection against reconstruction dynamics and resistance from statistical attack [8]. Another beneficial property of chaotic systems is their defining characteristic of sensitive dependence on initial conditions. Most chaotic functions are defined by recursive functions that define a path from one iterate to the next, called the trajectory. In the logistic map of 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, for example, initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEnLjUwMDAwRidGPkY+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEnLjUwMDAxRidGPkY+ have a fixed trajectory (the blue lines) and a chaotic trajectory respectively as can be seen in the plots below. 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image 1: initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEjLjVGJ0Y+Rj4= image 2: initial conditions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRjs2JFEjLjVGJ0Y+Rj4=0001 When using chaos in cryptography the initial conditions are the encryption key [11]. The effects of a function's sensitive dependence on initial condition must be kept in mind to choose values that will produce the desired chaotic result. However, this characteristic is what makes chaotic functions difficult to reproduce without the initial parameters, and hence makes an attack on the chaos based cryptosystem become virtually impossible. One tool for measuring the stability of a system is called the Lyapunov exponent. A Lyapunov number is defined as the average per-step divergence rate of nearby points along an orbit. A Lyapunov exponent is the natural logarithm of the Lyapunov number. Chaotic systems are defined by Lyapunov exponents that are greater than zero [6]. Ergodicity is another important property of chaotic algorithms that can be used in cryptography. This is the property in which a trajectory becomes arbitrarily close to previous states as in the second logistic map above. This gives us a large number of points in the trajectory that are limited to a finite space called an attractor. The density of these points is important in cryptography because it will produce long strings of psuedo-random numbers that can be used to encrypt information [8]. One of the main challenges that must be addressed when applying chaotic algorithms is precision. When working with integers in number theory, we are able to work accurately with finite precision. However, when we are working in the Real numbers, our existing chaos theory applies only to infinite precision numbers, and floating point numbers used by computers behave quite differently [9]. When generating a chaotic sequence for encryption, in most cases we need to be able to recreate the same sequence accurately. When the numbers in a given trajectory surpass the precision capabilities of an application's precision, then accuracy is lost. Hardware also plays an important role in this finite precision as shown by Wagner [9]. When accuracy is lost, then the cryptographic algorithm will break down unless careful consideration is given to account for this finite precision. Symmetric vs. Public Key Cryptography In a symmetric key cryptographic system, both the sender and the receiver have the same key. Symmetric keys are the basis for all classical cryptography cryptosystems designed before 1970[5]. The application of chaos and fractals has much potential for symmetric key cryptography. In 1976, Diffie and Hellman introduced the concept of a public key cryptosystem. In [7], Diffie and Hellman describe a cryptographic system as a family of single parameter transformations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYmLUYjNiQtSSVtc3ViR0YkNiUtSSNtaUdGJDYlUSJTRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRjQ2JVEia0YnRjdGOkY3RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy9GO1Enbm9ybWFsRidGRS8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGRQ== from space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiQtSSNtaUdGJDYlUSJQRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GMTYjUSFGJ0Y6 of plaintexts to space LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNiQtSSNtaUdGJDYlUSJDRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y4USdub3JtYWxGJ0Y6LyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJy1GMTYjUSFGJ0Y6 of ciphertexts. They then presented the idea of a public key cryptosystem that uses several components to communicate a message securely over public channels. Such a system requires a set M of possible messages and a set K of corresponding "keys". Each key k has an encryption function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and decryption function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSIuRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRldGRA== Once we have established these, we have the following requirements: 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 and 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 for every LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomRWxlbWVudDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GUy1GLDYlUSJNRidGL0YyRjk= and every LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomRWxlbWVudDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GUy1GLDYlUSJLRidGL0YyRjk=. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJtRidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRImtGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJtRidGMkY1Rj1GPUY9 are easy to compute for every m and k For almost every LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomRWxlbWVudDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GUy1GLDYlUSJLRidGL0YyRjk=, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn is known, then it is computationally infeasible to find an algorithm to compute LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JJW1yb3dHRiQ2JC1GLDYlUSJrRidGL0YyL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn. Given LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYuUSomRWxlbWVudDtGJ0YyL0Y1USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPy8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUY4Ni5RIn5GJ0YyRjtGPUZARkJGREZGRkhGSi9GTVEmMC4wZW1GJy9GUEZVLUYsNiZRIktGJ0YvRjJGNEYyRjs= , it is easy to compute the functions LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictSSVtc3ViR0YkNiUtRiw2JVEiREYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYsNiVRImtGJ0Y1RjgvRjlRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGQA==. [7,12] The random nature of orbits created by chaotic algorithms is a powerful tool for creating keys that are unpredictable, but have the ability to be easily recreated by a sender and receiver who use the same algorithm. The Fractal Based Digital Signature (GFDS) scheme and Baptista's method described below are examples of the use of chaos and fractals to create symmetric keys cryptosystems. While symmetric key algorithms have been designed for chaos-based cryptography, public key chaos or fractal based algorithms are not prevelent. This is due to the difficulty to create a decryption function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYlUSJrRicvRjNRJXRydWVGJy9GNlEnaXRhbGljRidGPUY/LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGNQ== that satisfies the requirements given above for a Public Key system. We will explore some of the available fractal cryptograhy algorithms and the possibility of applying other fractals to cryptography. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The Logistic Map and Baptista's Approach One application of chaos in cryptography has been the use of a logistic map. A logistic map is a one-dimensional mapping of a function in the family of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImFGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJ0ZILUkjbW9HRiQ2LVEiPUYnRkgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlAvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaW4tRi82JVEjYXhGJ0YyRjUtRkE2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnRkgtRks2LVEoJm1pbnVzO0YnRkhGTkZRRlNGVUZXRllGZW4vRmhuUSwwLjIyMjIyMjJlbUYnL0Zbb0ZbcEZFRkhGSEZI for some Real number a. The parameter a in the logistic map plays in important role in the dynamics of the system. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1GMzYtUSI8RidGL0Y2RjlGO0Y9Rj9GQUZDRkVGSC1GLDYkUSIxRidGL0Yv the trajectory is attracted to 0, which is called a sink. These intervals of initial values are called the basins of attraction for the sink. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIjxGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnRjItRiw2JFEiM0YnRi9GLw==, then we have a sink at 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. When LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5Rjk= we get a period-two sink, which is a two-point attractor towards which all trajectories are drawn. However, when 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 we begin to see much more interesting dynamics[6]. The diagram below shows that as the value of a increases toward 4, we find trajectories with much larger periods that tend toward a chaotic state. MiwmIiIiRiRJJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiJGJEkiYUdGKQ== 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Image 3: Bifurcation diagram of logistic map. [10] LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The random trajectory of logistic maps with a value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFElMy43NUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIjRGJ0YvRi8= is the basis for Baptista's encryption method. In this method, we choose a parameter r for a chaotic trajectory of the logistic function 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 , and an initial value 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. The ergodic property causes the iterates to frequently return to the interval (0,1) causing the density of the points in this interval to be time invariant. The defining element of Baptista's method is the process of discritizing the points in the trajectory. To do this, an interval 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 is chosen and divided into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RJiZsZXE7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSQyNTZGJ0Y5Rjk= sub-intervals of size 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. We then assign a plaintext character to each of these sub-intervals, sometimes the use of ASCII characters are used here. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= To encrypt the plaintext, the trajectory is the generated using the logistic equation with the given parameter r. When the sub-interval corresponding to the first plaintext character is reached, the number if iterates n is recorded as the ciphertext value for the first character. The value of n is then reset to 0 and the trajectory continues until the subinterval corresponding to the second plaintext characters reached. Again the number of iterates n that were required to reach this sub-interval are recorded as the ciphertext of the second character. The process is then continued for each character in the plaintext. Decryption of the ciphertext is done using the same parameters r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y+ and the same sub-intervals defined previously. For each ciphertext value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn, the trajectory is computed for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn iterations, and the plaintext is determined by the final sub-interval in which LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYsNiUtRi82JVEiY0YnRjJGNS1GIzYlLUYvNiVRImlGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGMkY1RkQvRjZRJ25vcm1hbEYn is contained. The trajectory is then continued for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYnLUYvNiVRImlGJ0YyRjUtSSNtb0dGJDYtUSIrRicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIxRidGQUYyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0ZB value until the entire message is decrypted.[8] This method is resistant to brute-force attacks due to the large number of possible initial values. The use of each ciphertext value as a seed for the next prevents statistical attacks to determine the value of common characters. Baptista's method of mapping the real values to integer values has been an important discovery in chaos-based cryptography. We can now consider other chaotic functions and fractals to which this method can apply. JSFH Fractal based Digital Signature In 2007, a digital signature scheme was introduced by Alia and Samsudin [1,2] based on the close link between the Mandelbrot and Julia fractal sets. The Mandelbrot set is the set of all values LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomRWxlbWVudDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GUy1GLDYlUSgmIzg0NTA7RidGL0YyRjk= such that the function 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 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlhGOkY+ is in the basin of 0, which means that the limit of the trajectory as 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 . For each c in the Mandelbrot set, there are orbits of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImNGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn that are bounded and those that are not. The Julia set is the set of all repelling fixed and periodic points with the limit points of the set. 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi1RIj1GJ0Y+LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlgtRi82JVEieUYnRjJGNUY+.[6] In the Fractal Based Digital Signature scheme, the sender Alice chooses LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GNjYuUSgmaXNpbnY7RidGL0YyRjlGPEY+RkBGQkZERkYvRklRLDAuMjc3Nzc3OGVtRicvRkxGUS1GLDYlUSgmIzg0NTA7RidGL0YyL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GNjYuUSgmaXNpbnY7RidGL0YyRjlGPEY+RkBGQkZERkYvRklRLDAuMjc3Nzc3OGVtRicvRkxGUS1GLDYlUSgmIzg0ODQ7RidGL0YyL0YzUSdub3JtYWxGJw== as her private keys. Similarly, the receiver, Bob, will choose LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YvRjIvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GNjYuUSgmaXNpbnY7RidGL0YyRjlGPEY+RkBGQkZERkYvRklRLDAuMjc3Nzc3OGVtRicvRkxGUS1GLDYlUSgmIzg0NTA7RidGL0YyL0YzUSdub3JtYWxGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomRWxlbWVudDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GUy1GLDYlUSgmIzg0ODQ7RidGL0YyRjk= as his private keys. Bob will then generate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82I1EhRicvRjZRJ25vcm1hbEYn using the Mandelfn function below and Alice will generate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjZRJ25vcm1hbEYn using Mandelfn. Mandelfn: 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 Bob then computes his public key 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. Alice likewise computes her public key 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. Alice then uses the Juliafn function below to input Bob's public key LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJkRidGMkY1L0Y2USdub3JtYWxGJ0ZILUkjbW9HRiQ2LVEiPUYnRkgvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlAvJSlzdHJldGNoeUdGUC8lKnN5bW1ldHJpY0dGUC8lKGxhcmdlb3BHRlAvJS5tb3ZhYmxlbGltaXRzR0ZQLyUnYWNjZW50R0ZQLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGaW4tRi82JVEieUYnRjJGNUZI to compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JS1JI21pR0YkNiVRInpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GNjYlUSJuRidGOUY8RjlGPC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiQtRiM2JC1GNjYlUSJkRidGOUY8L0Y9USdub3JtYWxGJ0ZORk5GTi1GIzYlLUY2NiVRImtGJ0Y5RjxGOUY8RkRGTg== and then 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 Juliafn: 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 She then computes her signature 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 and sends her message and signature to Bob. Bob will then obtain Alice's public key LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictRi82JVEiZUYnRjJGNS9GNlEnbm9ybWFsRic= and uses the Juliafn function to compute LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JS1JI21pR0YkNiVRInpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GNjYlUSJrRidGOUY8RjlGPC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiQtRiM2JC1GNjYlUSJlRidGOUY8L0Y9USdub3JtYWxGJ0ZORk5GTi1GIzYlLUY2NiVRIm5GJ0Y5RjxGOUY8RkRGTg== and then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUklbXN1YkdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JS1JI21pR0YkNiVRInpGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JS1GNjYlUSJrRidGOUY8RjlGPC8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiQtRiM2JC1GNjYlUSJlRidGOUY8L0Y9USdub3JtYWxGJ0ZORk5GTi1GIzYlLUY2NiVRIm5GJ0Y5RjxGOUY8RkQtRi82JC1GIzYkLUY2NiVRImRGJ0Y5RjxGTkZOLUkjbW9HRiQ2LVEiPUYnRk4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlxvLyUpc3RyZXRjaHlHRlxvLyUqc3ltbWV0cmljR0Zcby8lKGxhcmdlb3BHRlxvLyUubW92YWJsZWxpbWl0c0dGXG8vJSdhY2NlbnRHRlxvLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGW3AtRiw2JUYuLUYjNidGUi1GZ242LVEoJm1pbnVzO0YnRk5Gam5GXW9GX29GYW9GY29GZW9GZ28vRmpvUSwwLjIyMjIyMjJlbUYnL0ZdcEZmcC1JI21uR0YkNiRRIjFGJ0ZORjlGPEZELUZnbjYtUSgmdGltZXM7RidGTkZqbkZdb0Zfb0Zhb0Zjb0Zlb0Znb0ZlcEZncC1JJW1zdXBHRiQ2JS1GNjYlUSJjRidGOUY8LUYjNiUtRmlwNiRRIjJGJ0ZORjlGPC8lMXN1cGVyc2NyaXB0c2hpZnRHRkZGXHFGWUZO. Finally, Bob can verify the message by computing 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. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ2RidGL0YyRjk=, then the messages is authentic. [1,2] This symmetric algorithm is a unique use of the Mandelbrot fractal set. However, it is quite involved. Generalized Scheme for Fractal Based Digital Signature (GFDS) In the original scheme [1] for fractal based digital signature, only a specific receiver, Bob, can verify the validity of the signature. In this new Generalized Scheme for Fractal Based Digital Signature (GFDS), Alia and Samsudin has modified the original scheme to create a public key algorithm in which any member of the general public can verify the signature [2,3]. The process is the same as described for the fractal based digital signature described above, with the exception that the sender, Alice, will now choose all the private keys LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRiw2JVEiZEYnRi9GMkY1RlAtRiw2JVEia0YnRi9GMkY1RlAtRiw2JVEiZUYnRi9GMkY5 and will compute and publish the public keys LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRImtGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJlRidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRIm5GJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLzYlUSJkRidGMkY1L0Y2USdub3JtYWxGJ0ZIRkg= along with the parameter c, message m, and signature s. Any member of the public can now verify the message by computing verification v as described above. The message is valid if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJ2RidGL0YyRjk=.
<Text-field style="Heading 1" layout="Heading 1" spacebelow="0" linespacing="1.0" spaceabove="0">Mandelfn and Juliafn</Text-field> The following code is a simplified version of the Fractal Based Digital Signature Scheme. This version only works when the keys LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJrRidGL0YyLUY2Ni1RIi5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVkY5 NiZJInZHNiIsJiQiJTQ4ISImISIiKiYkIiM+ISIkIiIiXiNGLkYuRiksJiQiKCN6IVEiISIpRi4qJiQiJyN6IT0hIihGLkYvRi5GKS1JI2luRyUqcHJvdGVjdGVkRzYkLCYkIixIeiF5UzghIzdGLiomJCInVlM4RjdGLkYvRi5GKUkoY29tcGxleEdGOg== NiUiIzUiIiQtSSNpbkclKnByb3RlY3RlZEc2JCIiJ0koaW50ZWdlckdGJw== LUkjaW5HJSpwcm90ZWN0ZWRHNiQsJiQiJEIiISIlIiIiKiYkIiRAJEYpRipeI0YqRipGKkklcmVhbEc2JEYkSShfc3lzbGliRzYi ; restart; f:=proc(c,initial,steps) description "Function for iterating with z(0)=c for Mandelbrot function, or z(0)=y for Julia Function"; local i; global Z; Z:=table[]; Z[0]:=initial; for i from 1 to steps do Z[i]:=Z[i-1]*c; od; return Z; end proc: mandelfn:=proc(c,privkey,privsteps) description "Mandelbrot function for GFDS. c = public complex #; privkey=private key complex #; privsteps=private key integer to determine iterations."; global mandelpubkey; local j,i; #option trace; #iterate to find path of the the mandelbrot function f(c,c,privsteps); #print(Z()); #we now generate the public key using the path and c and the private key mandelpubkey:=Z[privsteps-1]*c^2*privkey; print("Z[n-1]=",Z[privsteps-1]); print(cat("Public Key =")); return(mandelpubkey): end proc: juliafn:=proc(c,pubkey,privkey,privsteps,msg,x) description"This process will calculate the signature and the verification pubkey: z[n]d or z[k]e privkey: users private d or e privesteps: users private n or k msg: Message value x: integer to reduce the final calculation."; local z,j,juliakey; global s; #compute julia function to find iterations of pubkey f(c,pubkey,privsteps): juliakey:=Z[privsteps-1]*c^2*privkey; print("Julia Key =",juliakey); s:=c^(privsteps-x)*juliakey*msg; print("signature/verification=",s); end proc: We will create the example of fractals based digital signature scheme similar to [1]. Digits:=500: c:=-0.01309-0.019*I; m:=0.12+0.0321*I; k:=6; e:=0.01380792-0.0180792*I; n:=6; d:=0.013407807929-0.0134043*I; x:=4; XiQkISU0OCEiJiQhIz4hIiQ= XiQkIiM3ISIjJCIkQCQhIiU= IiIn XiQkIigjeiFRIiEiKSQhJyN6IT0hIig= IiIn XiQkIixIeiF5UzghIzckISdWUzghIig= IiIl We first calculate Bob's public key. mandelfn(c,d,n); NiRRKFpbbi0xXT02Il4kJCI2VE9KV1JwT00mUjghI0kkITM5dU9BeCRmLCVwISNH US1QdWJsaWN+S2V5fj02Ig== XiQkIkc0NzVsZ0Y0JypvKT5XKyg+Q1oiPT4iISNfJCJEX1ZPPTVPU1lfaFQ1REZHa1oqISNd We now use the Bob's public key (mandelpubkey) with Alice's private keysc, and message m to compute the signature in the second output below. juliafn(c,mandelpubkey,e,k,m,x); NiRRLEp1bGlhfktleX49NiJeJCQhW29PWmNwNkMhKiopKio0Y3VyMGEqUmg/TCIqNEIhNHkkcChbPTRROSEjIiokIWluRyhcbURmWlshKWVDJnpWOS06KSlIa2NyOHphJVJbJXooKXo6ISMhKg== NiRROHNpZ25hdHVyZS92ZXJpZmljYXRpb249NiJeJCQiY29HWDAzQXFpZy1mSmYpKVErLSdbIT1KZnMkUlttOUg+WHYyKlsjZlQnISQvIiQhZG9PNjgvKkckeTN6VFB3OFpLT3o+YTcqXCZRT0xZZ1MqeV4vayNHbjQoISQwIg== We now compute Alice's public key. mandelfn(c,e,k); NiRRKFpbbi0xXT02Il4kJCI2VE9KV1JwT00mUjghI0kkITM5dU9BeCRmLCVwISNH US1QdWJsaWN+S2V5fj02Ig== XiQkIkNLIz1vI2U+RnZ2dyM0aDopW286ISNbJCJBR0Y9VWI4OylRYiF6cSpwJXAkKiEjWg== Finally, Bob will use Alice's public key (mandelpubkey in memory from the last process) along with Bob's private keys, c, and message m to verify the signature in the second output below. Observe that this output is equal to the signature above, hence the message is authentic. juliafn(c,mandelpubkey,d,n,m,x); NiRRLEp1bGlhfktleX49NiJeJCQhX29LdTA4REZCN0h2YmZEa1t6P1xvJFtASjlQSl52JHAqKSp5LjwiISMmKiQhXW9vJDRbdyVIJil5MklfY0hpdWI3eUpWVVstPjc8TTM+a3QoKkchIyUq NiRROHNpZ25hdHVyZS92ZXJpZmljYXRpb249NiJeJCQiZ29vUGdgS3dNIykpUk9QMSwsNixYYUFDZV8pb1dZLyxtaGM6eUo+ZyYzJyEkMyIkIWhvS1U8djhpZk00L0I+eDwxNXNSKj0mZTlVIillaS8mW1I5I1tyQ0xzODomISQ0Ig==
Mandelbrot Encryption Algorithm Now we will consider a possible symmetric encryption algorithm that will use a two dimensional version of Baptista's method of discritizing a trajectory in conjunction with the Mandelbrot set. We will need values of the Mandelbrot set that create a dense orbit over a designated region Image 4: Mandelbrot 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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlNVNGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiUrSUYqLSUpQ0hJTERSRU5HNiI= 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Image 5: close up of Mandelbrot set "valley" Assume that a sender Alice and receiver Bob have agreed upon a key LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJjRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJwRidGNEY3RjotRjE2JVEickYnRjRGN0Y+Rj5GPg== such that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEicEYnRi9GMkY1LUYsNiVRInJGJ0YvRjItRjY2LVEifkYnRjlGOy9GP0Y9RkBGQkZERkZGSEZKL0ZORkwtRjY2LVEqJkVsZW1lbnQ7RidGOUY7RllGQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORmluLUYsNiVRKCYjODQ1MDtGJ0YvRjJGOQ==, 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, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJXJlYWxGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI8RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRiw2JS1GLzYlUSJwRidGMkY1RjhGPUZE, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiVRJWltYWdGJ0YyRjVGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI8RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZILyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlctRiw2JS1GLzYlUSJwRidGMkY1RjhGPUZE, and c is in the Mandelbrot Set, and neither the real part or the imaginary parts of p and r are equal. Define 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 and 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 where 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 are the real parts ,and 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 are the imaginary parts of pand r respectively. 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LSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIiISEiIi0lKUJPVU5EU19ZR0YnLSUtQk9VTkRTX1dJRFRIRzYjJCIlP1NGKi0lLkJPVU5EU19IRUlHSFRHNiMkIiVxR0YqLSUpQ0hJTERSRU5HNiI= Image 6: Sample rectangular region for cipher Thus, p, q, r, and s represent the vertices of a rectangular region of the complex plane. We will divide this rectangle into a 16 x 16 array of 256 regions of dimensions a x b such that 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 , L0kibUc2IiwkKiYjIiIiIiIjRigtSSRhYnNHJSpwcm90ZWN0ZWRHNiMsJkkicEdGJEYoSSJxR0YkISIiRihGKA== 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 This gives us a as the length of intervals of the real axis and b as the length of intervals of the imaginary axis. We will then assign an ASCII character to each region. For each (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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIzE2RidGL0Yv and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RJiZsZXE7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkjbWlHRiQ2JVEiakYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJ0YyLUYsNiRRIzE2RidGL0Yv, we will assign the ACII character that corresponds to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzE2RidGOS1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRImlGJ0YvRjItRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZpbi1GUDYkUSIxRidGOUY5RjktRjY2LVEiK0YnRjlGO0Y+RkBGQkZERkZGSEZobkZqbi1GVDYkLUYjNiYtRiw2JVEiakYnRi9GMkZlbkZbb0Y5RjlGOQ== to the region 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. We can also define the corresponding matrix 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 such that each entry 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 corresponds to an ASCII character. To generate the ciphertext, we will start with the first character in the plain text and find its ASCII value. We will initiate a register t to count iterations and then start iterating the Mandelbrot function 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 using the private key value c until the trajectory reaches the region of our array that corresponds to the plaintext. The value of t will then be recorded as the ciphertext and t will be reset to 0. The process is then repeated until the trajectory reaches the region corresponding to the second character in the plaintext. We continue in this manner for each letter of the plaintext. We must choose our initial value and our test region carefully. We need a trajectory that does not quickly converge to 0, but has a large number of iterates within a circle of radius 2. Once an iterate more than 2 units away from the origin, it will diverge to infinity. This means that we should choose an initial value on the border between the basin of infinity (shaded blue in the images 4 and 5) and the basin of 0 (shaded red in images 4 and 5). The values here will have many iterates that do not diverge. To find these values, just locate a point in image 5 in or near a yellow region.
<Text-field style="Heading 1" layout="Heading 1" spacebelow="0" spaceabove="0">mandelencrypt process</Text-field> restart; mandelencrypt:=proc(c,p,r,ptext) description"This process will encrypt message m using the Mandelbrot Fractal function z[n]=z[n+1]^2+c. Parameters p and r will define the encryption array. p,r = complex numbers such that Re(r)<Re (p) and Im(r)<Im(p). Parameters |p|<1, |r|<1 and c must be an element of the Mandelbrot set. "; #option trace; local i,j,k,Z,a,b,P,z,q,s,M,t; global ctext,C; interface(rtablesize=100): Digits:=100: P:=Array(): C:=Array(1..length(ptext),fill=0): p,r; z:=0; "Translate the plaintext into ASCII numerical format"; for k from 1 to length(ptext) do P(k):=StringTools[Ord](substring(ptext,k)); od; print(P()); "Define key region"; Re(p);Im(p);Re(r);Im(r); q:=Re(p)+Im(r)*I; s:=Re(r)+Im(p)*I; a:=abs(p-s)/16; b:=abs(p-q)/16; print(p,q,r,s); #Define ASCII matrix M:=Matrix(16,16); for i from 1 to 16 do for j from 1 to 16 do M(i,j):=StringTools[Char](16*(i-1)+(j-1)); od; od; M(); #"Loop to encrypt message one letter at a time."; for k from 1 to ArrayTools[Size](P,2) do i:=ceil(P(k)/16); j:=P(k)-16*(i-1)+1; print(i,j); print(M(i,j)); t:=0; while C(k)=0 do if (Re(s)+a*(i-1))<=Re(z) and Re(z)<(Re(s)+a*i) and (Im(s)-b*(j-1))>=Im(z) and Im(z)>(Im(s)-b*j) then C(k):=t; print (`target reached`,C(k)); t:=0; break; fi; z:=z^2+c: #print(z);#uncomment to see orbit numerically t:=t+1; #if |z|>2, the iterates will diverge, so we can stop checking these values, the process has failed. We must choose a different initial value. if abs(z)>2 then print(`diverges to infinity after`,t,steps); C(k):=0; break; fi; od: od; C(); end proc: mandelencrypt(-.75+.0001*I,-.05+0.5*I,-0.5-.5*I,"Hi"); LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKi9wXWcj LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKldSTFwj NiZeJCQhIiYhIiMkIiImISIiXiRGJCRGJUYpXiRGK0YrXiRGK0Yn NiQiIiYiIio= USJINiI= NiRJL3RhcmdldH5yZWFjaGVkRzYiIiNB NiQiIigiIzU= USJpNiI= NiVJO2RpdmVyZ2VzfnRvfmluZmluaXR5fmFmdGVyRzYiIiYmUkpJJnN0ZXBzR0Yk LUkmQXJyYXlHJSpwcm90ZWN0ZWRHNiMvSSQlaWRHNiIiKi9wXWcj
SSRkZmRHNiI= To decrypt, we calculate the iterates of 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 to the numbers given in the cyphertext. This algorithm clearly has some limitations that need much more investigation. However, there is definitely much potential in the use of fractals in cryptography. If we carefully choose our keys and parameters, then we can have potential to have a highly secure cryptosystem. As it stands now, after many variations on the parameters, we were only able to encrypt one letter of the word "Hi". The trajectory then diverged after 31395 steps. Density is an issue that must be addressed when considering where to place the encryption grid. Conclusion The integration of fractals and chaos with cryptography is still in its infancy. This is an exciting opportunity for both fields of chaos and cryptography to expand into uncharted territory. While the defining characteristics of chaos such as sensitive dependence on initial conditions and ergodicity appear to offer valuable tools for cryptosystems, they bring with them a new set of challenges to overcome. However, this in itself is a good sign when chaos and fractals present "hard problems" which are the key to a good cryptosystem. We must also keep in mind that chaos based cryptography may not parallel the methods and concepts that have defined traditional cryptography [8]. As modern cryptography challenges change at the speed of technology, we may just find the key to solve these modern problems with modern mathematics such as chaos theory. References M. Alia, and A. Samsudin, "A New Digital Signature Scheme Based on Mandelbrot and Julia Fractal Sets." American Journal of Applied Sciences vol. 4, no. 11: 848-856. ISSN 1546-9239. 2007. M. Alia and A Samsudin, " New Key Exchange Protocol Based on Mandelbrot and Julia Fractal Sets." International Journal of Computer Science and Network Security, vol. 7, no. 2, February 2007 M. Alia and A. Samsudin, "Generalized Scheme for Fractal Based Digital Signature (GFDS)." International Journal of Computer Science and Network Security, vol. 7, no. 7, July 2007. M. Alia, " A New Approach to Public-Key Cryptosystems Based on Mandelbrot and Julia Fractal Sets". University Sains Malaysia, Doctoral Thesis. 2008. http://eprints.usm.my/8981/1/A_NEW_APPROACH_TO_PUBLIC_KEY.pdf W. Trappe and L. C. Washington, Introduction to Cryptography with Coding Theory, Second Edition. Prentice Hall. 2006. K.T. Alligood, T.D. Sauer, J.A. York, Chaos, An Introduction To Dynamical Systems, Springer-Verlag, 1996. W. Diffie and M. Hellman, "New Directions in cryptography," IEEE Trans. inInformation Theory, 22, 1976, 644-654. http://www.cs.rutgers.edu/~tdnguyen/classes/cs671/presentations/Arvind-NEWDIRS.pdf Q.V. Lawande, B.R. Ivan, and S.D. Dhodapkar, "Chaos Based Cryptography: A New Approach to Secure Communications," BARC Newsletter. No. 258, July 2005. N.R. Wagner, "The Logistic Lattice in Random Number Generation." Proc. 30th Annual Allerton Conf. on Comm., Control, and Computing, 1993, pp. 922-931. http://www.cs.utsa.edu/~wagner/pubs/logistic/logistic.pdf Bifurcation Image. Dr. Michael Pilant. http://www.math.tamu.edu/~mpilant/math614/ B. Cristea, C. Cehan, A. Serbanescu, and D. Grecu, ``Applications of chaos theory in cryptography,'' in METRA International Conference, 2002, Bucharest, Romania.