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CASC 3N Vs. 4N: Effect of Increasing Cellular Automata Neighborhood Size on Cryptographic Strength

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CASC 3N Vs. 4N: Effect of Increasing Cellular Automata Neighborhood Size on Cryptographic Strength (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 11, No. 4, 2020 CASC 3N vs. 4N: Effect of Increasing Cellular Automata Neighborhood Size on Cryptographic Strength Fatima Ezzahra Ziani1, Anas Sadak2, Charifa Hanin3, Bouchra Echandouri4, Fouzia Omary5 Computer Science Department University Mohammed V Rabat, Morocco Abstract—Stream ciphers are symmetric cryptosystems that 3Neighborhood configurations [4]. Accordingly, this article rely on pseudorandom number generators (PRNGs) as a primary presents two versions of Cellular Automata-based Stream building block to generate a keystream. Stream ciphers have Cipher (3-CASC and 4-CASC). These versions were analysed been extensively studied and many designs were proposed and investigated to identify differences between their throughout the years. One of the popular designs used is the cryptographic properties and statistical analysis as well as combination of linear feedback shift registers (LFSRs) and their resistance against attacks targeting stream ciphers. nonlinear feedback shift registers (NFSRs). Although this design is suitable for both software and hardware implementation and The study of the 4Neighborhood 1-dimensional CA rules provides a good randomness behavior, it is still subject to attacks is a challenging task. The authors chose the rules according to such as fault attacks and correlations attacks. Cellular automata the recommendations in [5] for the 3-CASC version. Then, (CAs) based stream ciphers are another design class that has these rules are combined with a new variable to get the been proposed. CAs display good cryptographic properties as 4Neighborhood 1-dimensional CA rules. Section 2 details this well as a good randomness behavior, also high computational step. The N-CASC design, which was inspired by grain-like speed and a higher level of security. The use of CAs as CA-based ciphers, consists of three building blocks: a linear cryptographic primitives is not recent and has been thoroughly block, a nonlinear block, and a mixing block. For the linear investigated, especially the use of three-neighborhood one- block and nonlinear block, only linear rules and nonlinear dimensional cellular automata. In this article, the authors rules are used respectively. For the mixing block, a hybrid investigate the impact of increasing the neighborhood size of CAs ruleset with both linear and nonlinear rules is adopted. on the security level and the cryptographic properties provided. Thereafter, four-neighborhood one-dimensional CAs are studied The goal of the article is to look at the effect of and a stream cipher algorithm is proposed. The security of the transitioning from the 3N version to the 4N version on the proposed algorithm is demonstrated by using the results of cryptographic properties as well as the statistical features of standard tests (i.e. NIST Test Suite and Dieharder Battery of the stream cipher proposed. Tests), particularly by computing the cryptographic properties of the used CAs and by showing the resistance of the suggested The rest of this article is organized as follows: algorithm to mostly known attacks. Section II presents cellular automata and cryptographic Keywords—Stream ciphers; cellular automata; neighborhood properties. Section III provides related works. Section IV size; dieharder; NIST STS; cryptographic properties; attacks on details the design of the proposed scheme. Section V displays stream ciphers the results including the statistical test, the avalanche effect, and the cryptographic properties. Section VI shows the I. INTRODUCTION security analysis of the proposed scheme. In stream ciphers design, fast encryption and simplicity are II. BACKGROUND particularly essential criteria. To get a ciphertext, a stream cipher processes by applying the XOR operation to the A. Cellular Automata plaintext with the keystream. This latter is generated by a Cellular automata are dynamic systems that were first PRNG that should provide good randomness and a good introduced in the 1950s by John von Neumann and later security level. The strength of a stream cipher resides in the popularized by Stephen Wolfram in the 1980s [6]. They were robustness of the strength of the PRNG [1]. The outstanding first studied for the modeling of biological self-reproduction primitive recommended to use for the design of a PRNG is by von Neumann upon Stanislas Ulam recommendations [7]. Cellular Automaton. Since then, they were used in different fields such as physics, Thanks to the simplicity producing the complex behaviour chemistry, mathematics, biology etc. … to model and solve of cellular automata (CA), especially the one-dimensional physical, natural and real-life problems [6]. Researchers took 3Neighborhood CAs which are widely used in the field of interest in cellular automata because of the complex global cryptography. They were studied [2-4] to ensure a good behavior that stems from simple interactions and computations security level. However, some attacks are inevitable in at the cellular level. Moreover, global properties such as 308 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 11, No. 4, 2020 4 universality in computation and randomness explains the 22 = 65535 total rules. Table II shows an example of a linear attraction of the scientific community [8]. and a non-linear rule (left skewed). A cellular automaton is a finite set of n cells arranged as a One way to visualize the evolution of a cellular automaton network that evolve in discrete space and time. Formally, a is to use a space/time diagram. In a space/time diagram the cellular automaton is a tuple (L, S, N, f, R) [6], where: cellular space lies on the x-axis, with different colors for each L is the d-dimensional cellular space. state, while time is represented by the y-axis. Space/time diagrams are a good tool to visualize the global behavior of a S is the finite state set. cellular automaton and the rule(s) associated with it. Fig. 1 represents the space time diagram of rule 90 for a N is the neighborhood vector linking each cell to its configuration of 256 cells and 100 time steps neighbors and represented by a radius r representing (https://www.wolframalpha.com/input/?i=rule+90). the number of consecutive cells a cell depends on. f is the local update rule or simply the rule that gives TABLE I. EXAMPLES OF 1-DIMENSIONAL, 2-STATE, 4N EXAMPLE OF ECA RULES the next state of each cell. Neighborhood Rule 120 (nonlinear) Rule 150 (linear) R is the rule vector consisting of the rule(s) applied to configuration 푥 ⊕ 푥 ⋅ 푥 푥 ⊕ 푥 ⊕ 푥 each cell. 푖−1 푖 푖+1 푖−1 푖 푖+1 111 0 1 The L, S and N parameters can be varied to define different 110 1 0 types of CAs. For example, von Neumann studied 2- dimensional, 5-neighborhood, 29-state cellular automata. If 101 1 0 the rule f is a linear Boolean function including only XOR 100 1 1 logic, then the CA is called a linear CA. Otherwise, if f 011 1 0 comprises also AND or OR logic, then the CA is called a non- 010 0 1 linear CA. The rule vector R can consist of a single rule applied to all the cells (uniform CA) or a set of rules assigned 001 0 1 to each cell (hybrid CA). 000 0 0 Despite the fact that multi-dimension cellular automata can TABLE II. EGHBORHOOD RULES display a more complex behavior, effectively characterizing them and mathematically analyzing them is much difficult Neighborhood Rule 32640 (nonlinear) Rule 27030 (linear) than their 1-dimensional counterpart. This explains that much configuration 푥푖−2 ⊕ 푥푖−1. 푥푖. 푥푖+1 푥푖−2 ⊕ 푥푖−1 ⊕ 푥푖 ⊕ 푥푖+1 of the studies conducted on cellular automata and their 1111 0 0 application in cryptography has been done on 1-dimensional 1110 1 1 cellular automata, particularly a special kind of cellular automata introduced by Stephen Wolfram [2]. These cellular 1101 1 1 automata are called Elementary Cellular Automata (ECA) [8]. 1100 1 0 They are 1-dimensional, 3Neighborhood and 2-state cellular 1011 1 1 3 automata. For ECAs, there are 2 = 8 neighborhood 1010 1 0 23 configurations and 2 = 256 total rules. Table I shows an 1001 1 0 example of a linear and a non-linear rule. 1000 1 1 In Table I, 푥푖−1, 푥푖and 푥푖+1are the left neighbor, the cell 0111 1 1 and the right neighbor respectively. The rule name (e.g. Rule 0110 0 0 120) is the decimal representation of the binary rule read from left to right. This naming convention was introduced by 0101 0 0 Wolfram [9]. 0100 0 1 For 4Neighborhood cellular automata, two possible 0011 0 0 neighborhood arrangement are possible [5]: 0010 0 1 0001 0 1 Left skewed: each cell ( 푥푖 ) depends on two left 0000 0 0 neighbors ( 푥푖−2 and 푥푖−1 ) and one right ( 푥푖+1 ) neighbors. Right skewed: each cell ( 푥푖 ) depends on one left neighbors ( 푥푖−1 ) and two right ( 푥푖+1 and 푥푖+2 ) neighbors. For 1-dimensional, 2-state, 4Neghborhood cellular automata, there are 24 = 16 neighborhood configurations and Fig. 1. Rule 90 Space Time Diagram. 309 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 11, No. 4, 2020 For practical reasons, cellular automata are studied for a were presented as good candidates to solve the problem of finite cellular space L. In this case, boundary conditions attacks related to LFSR/NFSR based stream cipher designs. should be specified. Two broad categories of boundary conditions exist [6] for 1-dimensional cellular automata: open In [2] and [3], Stephen Wolfram was the first to propose boundary conditions and periodic conditions. For open the use of cellular automata as a keystream generator using boundary conditions, the neighbors of the leftmost and rule 30.
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