A STUDY OF DES ALGORITHM WITH CELLULAR AUTOMATA

International Journal of Innovative Management, Information & Production ISME International c2013 ISSN 2185-5439 Volume 4, Number 1, June 2013 PP. 10-16 A STUDY OF DES ALGORITHM WITH CELLULAR AUTOMATA RAMA R 1, BALA SUYAMBU J 2, ANDREW AROKIARAJ 3, SHANMUGAM SARAVANAN 4 Department of Mathematics 1, 2 Indian Institute of Technology Madras Chennai, 600053, India ramar@iitm.ac.in 1 paulsuyambu.jesus@gmail.com 2 Department of Mathematics Loyola College 3 Chennai, 600053, India andrewarokiaraj@gmail.com School of Computer Science and Engineering VIT University 4 Vellore,632012, India shaanmugam@gmail.com ABSTRACT. Data Encryption Standard (DES) was accepted as the standard encryption algorithm in 1976. Since then the development and usage of DES is widely recognized. The concept of Cellular Automata (CA) introduced by John von Neumann (1950s) is the first model which exhibited the capability of producing complex and random behavior. Cellular Automata is a parallel processing machine and it involves context depent behavior of all the value at time t. The literature has witnessed the usage of Cellular Automata rules which are simple in nature for the operation of DES and Advanced Encryption Standard (AES) algorithms. In this paper, we ext the usage of Cellular Automata for the key generation in DES. The proposed CA-based key generation methodology uses the elementary Cellular Automata rule 30 which possess more randomness. The initial seed involved is 15 bit multi-seed. Also we presented the implemented DES with key generated using Cellular automata in Java. Keywords: Cryptography; DES; Cellular Automata; Random key; Rule 30; Java 1. Introduction. Cryptography is the science of information security. In modern age, the codes and ciphers play a vital role in communication and commerce secrecy. One of the major tools for the development of cryptology is Mathematics. During 1970s, the astounding idea of public key cryptography burst upon the scene. Two basic encryption techniques are Symmetric and Asymmetric Encryptions. DES and AES are the effective modern symmetric algorithms (Hoffstein et al., 2011). A Cellular Automata is an array of identically programmed automata, or cells, which interact with one another in a neighborhood and have definite state. The elementary CA is a one dimensional CA with two states 0 and 1. Each cell is updated with respect to its old state and the state of its nearest neighbor. There are 2 8 = 256 elementary CA rules. In Cellular Automata extremely simple rules have the capability of producing more complex

A STUDY OF DES ALGORITHM WITH CELLULAR AUTOMATA 11 and random behavior. In this paper, Section 2 deals with DES and its operation, also the development of DES. Section 3 describes Cellular Automata and mainly the randomness of the Rules. Section 4 deals with usage of Cellular Automata in DES. We propose a method in this section which uses Rule 30 for key generation in DES. 2. Data Encryption Standard. In 1972, National Bureau of Standards (NBS) led up a project for the computer data security. The ICST is one of the important functioning units of NBS. It established new standards and rules for improving the usage of computer due to Federal law which is known as Brooks Act. In 1975, DES Algorithm was published in the Federal Register for public comment. After which several workshops sprung up in DES. In the workshop held on September 1976, mathematical basis of DES was discussed. In 1976, DES was approved as a standard crypt-algorithm. After the standard approval, more enhancement work has been done in DES. The disclosure of Biham-Shamir on differential cryptanalysis was implemented as a 15 round DES-like cryptosystem and they made the first theoretical attack with less complexity than brute force. Shortly, Triple DES has emerged into cryptosystem (Stallings, 2005; Smld and Branstad, 2005). 2.1. Components of DES. Initially plain text is converted to bits and segmented into blocks of each 64-bit. In case the block has less than 64-bit then 0 s are apped. There are various stages for encryption and reversals of those stages are used for decryption. The following are the operations involved in DES: Initial Permutation does the transposition of each bit over 64-bits based on the initial permutation table. The outcome of this would be 64-bits. Inverse Initial Permutation does the reversal operation of Initial Permutation. Permuted Choice makes a transposition, accepts 64bits and produces 56-bits which are used in key generation. Permuted Choice 2 makes a transposition, accepts 56-bits and produces 48-bits which are used as keys. In Left Circular Shift, every bit is shifted left. Expansion is a rearrangement process accepting 32-bits and produces 48bits.In S-box, substitution consists of a set of eight S-boxes, each of which accepts 6 bits as input and produces 4 bits as output. (Stallings, 2005) FIGURE 1. General Flow structure of DES

RAMA R, BALA SUYAMBU J, ANDREW AROKIARAJ AND SHANMUGAM SARAVANAN 12 2.2. How Strong is DES? The attack of DES is mainly depent upon the construction of S-boxes. On increasing the number of rounds, the resistance of DES is more enhanced. The main aspects in DES are: Diffusion Breaks up the structure of plaintext over majority of cipher text. Confusion interlaces the key and cipher text as complex as possible. The Key in DES is a 56-bit key having 2 56 = 7.2 x 10 16 values. Thus tracking the key seems hard by brute force search. On further strengthening the key size, the key attack becomes impossible. The timing attack of any cryptosystem is based on several attributes such as system architecture, plaintext size, the execution time, complexity of the algorithm, application details. In order to perform the analytical attack, one should know basic information about encryption and keys used. Normally, cryptanalyst uses tools like Linear, Differential, Integral and related-key cryptanalysis. (Tingyuan and Teng, 2009) 3. Cellular Automata. The origin of CA dates to 1950s, when John von Neumann worked on the concept of self-replicating machine, the Kinematon. Ulam suggested him the model of cells for the constructions. Then von Neumann worked on cells with four nearest neighborhood (called the von Neumann neighborhood) and developed the complex universal self-replicating automata with 29 states. Later in 1968, E. F. Codd reduced the number of states required for self-replicating automata from 29 to 8. After which, in 1970 John Horton Conway introduced the well known Game of Life Sipper (1998). Stephen Wolfram in his book A New Kind of Science published in 2002, explains how in Cellular Automata, very simple rules would lead to great complexity. He also categories the rules of CA into four classes known as Wolfram classes : Class I-Homogeneous, Class II-Regular, Class III-Chaotic and Class IV-Localized structure. A Cellular Automata is an array of identically programmed automata, or cells, which interact with one another in a neighborhood and have definite state. The elementary CA is a one dimensional CA with two states 0 and 1. Each cell is updated with respect to its old state and the state of its nearest neighbors (left and right neighbors). There are 2 8 = 256 elementary CA rules. Stephen Wolfram designed a simple naming scheme for these Wolfram and Stephen(2002). The local rules get specified giving the next state for all 8 possible triplets. The rule number for the CA is the integer whose binary expansion is b7b6b5b4b3b2b1. For example, CA Rule 30 has the following rule: Consider series of states in one dimensional CA. What happens to the cells at the rear and front? From a topological point of view, this results in wrap around horizontally, leading a cyclic grid structure.

A STUDY OF DES ALGORITHM WITH CELLULAR AUTOMATA 13 FIGURE 2. Cyclic grid structure and Iteration in Rule 30 3.1. Randomness. All these cellular automata patterns can always be allotted with ease to one of the four basic classes irrespective of the initial seed. In Class I (Homogeneous) very simple patterns are produced. Simple and random initial condition always leads to a consistent final state. Example in elementary CA: Rule 0, 4, 16, 32, 36, 48, 54, 60 and 62. In Class II (Regular) varying final state are possible, but all of which leads to a set of separated simple stable or periodic structures. Example, CA: Rule 8, 24, 40, 56 and 58. In Class III (Chaotic) the pattern is more complicated, and seems random in many respects. Example, CA Rule: 2, 10, 12, 14, 18, 22, 26, 28, 30, 34, 38, 42, 44, 46 and 50. In Class IV (Localized structure) the pattern produced though seems to be fairly simple, but these pattern when creeping around each other forms complicated behavior. Example, CA Rule 52 and Rule 110. FIGURE 3. State-time behavior of Rule 30 and Rule90 with limited width Figure3 pattern produced is repetitive, but the period of repetition is often quite large. An increase in the size results to Class III behavior. The above observation shows that extremely simple rules have the capability of producing more complex and random behavior. A New Kind of Science tells that Rule 30 possesses a good deal of randomness. Henceforth rule 30 has an effective usage in cryptology, where random property plays a vital role. Investigation proved using statistical test and characteristic test that rule 30 shows enough randomness for a high level of security Gage et al. (djp). 4. DES Operations with CA. The literature has witnessed the usage of Cellular Automata rules which are simple in nature for the operation of DES and AES algorithms. The permutation operations in DES reveal the diffusion property. Hence these permutations can be evidently replaced by linear Cellular Automata rules. The substitution operation in DES reveals the confusion property. Therefore this substitution can be replaced by non-linear Cellular Automata rules. Rules 1, 2, 4, 8, 16, 32, 64 and 128 are called primary cellular automata rules having linear property. It is shown that the initial permutation, expansion

RAMA R, BALA SUYAMBU J, ANDREW AROKIARAJ AND SHANMUGAM SARAVANAN 14 permutation, inverse initial permutation and left circular shift operations are linear functions, providing diffusion property of cryptography. Rules 2 and 32 are inverses to each other. So, these rules can be used in encryption and decryption respectively. Similarly, Rule 4 and 64, Rule 8 and 128, Rule 16 and 256 are inverses of each other. (Panda et al., 2011) The substitution boxes are the most important component in DES Algorithm, which does a non-linear transformation of the data. M.Szaban and F.Seredynski Szaban and Seredynski(2010) proposed a dynamic cellular automata based S-boxes satisfying non-linearity criteria. This CA based S-boxes are dynamic flexible in structure and fully functionally realized by non-linear CA rules such as Rules 30, 86, 135 and 149. Here, we present a new idea to generate key in DES using Cellular Automata (Rule 30). As discussed in previous section that Rule 30 is good in randomness, we use this rule for key generation. In DES we require 16 keys of 48-bit each; in total we need 16 x 48 random bits. Thus it is enough to choose Rule 30 with the width (size) 15 which has the periodicity of 1455. A multiple seed of 15-bits is set initially. Onto which Rule 30 is iterated for time length t= 16 x 48. It is examined Wolfram and Stephen(2002) that the center column in Rule 30 pattern does not show repetition for longer period. Therefore in our generated pattern (16 x 48 rows and 15 columns), the center column is focused. The first 48-bits in the center column are fetched as the key for Round-1, the next 48bits in the center column is the key for Round-2 and so on. FIGURE 4. CA-based Key generation using Rule 30 (width 15, single seed) The entire implementation of DES with the proposed CA based Key generation is programmed in Java language. The Algorithm for CA-based key generation is as follows: Algorithm for generating keys: procedure KEYGEN (maxrow, initialval) len LENGTH (intialval) //finding the length of the initial value for j 0 to len-1 do

A STUDY OF DES ALGORITHM WITH CELLULAR AUTOMATA 15 CA[0][j] initialval [j] // Initializing the seed as an integer array for i 1 to maxrow-1 do for j 0 to len-1 do CA[i][j] CA[i-1][(j-1+len)modlen] (CA[i-1][j] CA[i-1][(j+1)mod len]) // Boolean Expression for Rule 30 is p (q r) k 0 for i 0 to maxrow-1 by +48 do // Fetching the keys into array KEY[k] = NULL for j i to (KEYSIZE+i) do KEY[k] KEY[k] ++CA[j][ len/2 ] k k +1 KEYGEN Remarks. The source code can be obtained from any one of the authors via email. 5. Conclusion. This paper gives an overview of the status of DES, the concept of Cellular Automata and its usage in DES is reviewed. The new idea of Cellular Automata for key generation in DES is implemented in Java. The proposed methodology gives 48-bit key in direct instant and avoiding the operations (Permuted choice1 and 2, Left Circular Shift) in the usual DES key generation, and also achieving the high randomness. Thus the usage of Cellular Automata in random number generation becomes explicit and useful. One can venture to explore the usage of Cellular Automata rules in existing paradigms which require randomness. This will open up a new avenue of research in algorithms. Acknowlegement. Andrew Arokiaraj and Shanmugam Saravanan would like to thank the Department of Mathematics, IIT Madras for providing Summer Internship and Fellowship. REFERENCES [1] Dustin Gage, Elizabeth Laub and Briana Mcgarry, Dr. Ken Smith -Faculty Adviser, Cellular Automata: Is Rule 30 Random?, Central Michigan University. [2] Jeffrey Hoffstein, Jill Pipher and Joseph H. Silverman (2011), An Introduction to Mathematical Cryptography, Springer Publication. [3] Miles E. Smld and Dennis K. Branstad (1988), The Data Encryption Standard: Past and Future, Proceedings of the ieee, vol. 76, no. 5, pp.550-559. [4] Miroslaw Szaban and Franciszek Seredynski (2010), CA-based Generator of S-boxes for Cryptography Use, IEEE, pp.1-8. [5] Moshe Sipper (1998), Fifty Years of Research On Self-Replication: An Overview, Massachusetts Institute of Technology, Artificial Life, vol.4, no.3, pp. 237-257. [6] Sambhu Prasad Panda, Madhusmita Sahu, Umesh Prasad Rout and Surra Kumar Nanda(2011),

RAMA R, BALA SUYAMBU J, ANDREW AROKIARAJ AND SHANMUGAM SARAVANAN 16 Equivalence of DES and AES Algorithm with Cellular Automata, International Journal of Communication Network & Security, vol.1, no.1. [7] Tingyuan Nie and Teng Zhang (2009), A Study of DES and Blowfish Encryption Algorithm, IEEE, pp1-4. [8] William Stallings (2005), Cryptography and Network Security Principles and Practices, 4 th Edition, Prentice Hall, November 16. [9] Wolfram and Stephen (2002), A New Kind of Science, Illinois: Stephen Wolfram LLC.