Synthesis of all Maximum Length Cellular Automata of Cell Size up to 12

Synthesis of all Maximum Length Cellular Automata of Cell Size up to 12

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Abstract. Maximum length CA has broad scope of applications in design of additive block codification, cryptanalytic primitives and VLSI proving partic- ularly in Built-In-Self-Test. In this paper, an algorithm to calculate all n-cell maximal length CA-rule vectors is proposed. Besides rule vectors for each crude multinomial in GF ( 22 ) to GF ( 212 ) have been computed by simulation and they have been listed.Programmable regulation vectors based maximal length CA can be used to plan cryptanalytic primitives. Keyword Linear intercrossed upper limit length CA, Rule vectors, crude multinomial

1 Introduction

A cellular zombi ( CA ) consist of a figure of cells arranged in a regular mode. Each cell consists of a storage component ( D reversal ) and a combinable logic implementing the next-state map. CA is universally accepted as a really good generator of pseudo random sequences. It is besides really good suited for VLSI design due to its regular structure.If the combinable logic of a CA cell merely involves XOR logic, so it is called a additive CA. For a three vicinity one dimensional CA, the combinable logic implementing the following province is si ( t + 1 ) = degree Fahrenheit ( si?1 ( T ) , si ( T ) , si+1 ( T ) ) . Where Si ( T ) is the end product province of the ith cell at tth clip measure. si?1 ( T ) and si+1 ( T ) are the end product provinces of left and right neighbours of ith cell and degree Fahrenheit denotes the local passage map realized with a combinable logic and is known as a regulation of the CA. A CA is said to be intercrossed if the regulations of different cells vary. An n-cell upper limit length CA is characterized

by the presence of a rhythm of length 2n ? 1 with all non-zero provinces. In instance of

a maximal length CA, it has a characteristic multinomial which is crude.

CA-rules 90 and 150 have been considered. The combinable logic for regulation 90 and govern 150 are as follows.

Rule 90: Si ( t + 1 ) = si?1 ( T ) ? si+1 ( T )

Rule 150: Si ( t + 1 ) = si?1 ( T ) ? Si ( T ) ? si+1 ( T )

where Si ( T ) is the end product province of the i-th cell at clip T.

Efficient word picture of 1D CA based on matrix algebra and its applica-

tion in mistake correcting codifications, cryptanalysis [ 1 ] and VLSI testing is available in [ 3 ] . The characteristic matrix of a additive CA operating over GF ( 2 ) is a matrix that describes the behaviour of the CA. We can cipher the following province of the CA by multiplying the characteristic matrix by the present province of the CA. Angstrom

characteristic matrix is constructed as: T [ I, J ] = 1, if the following province of the ith

cell depends on the jth cell and T [ I, J ] = 0, otherwise.

Merely one regulation vector for each n-length CA has been provided in [ 3 ] . A new architectural design of CA-based codec based on additive maximal length CA has been proposed in [ 5 ] . In [ 2 ] writers proposed an algorithm for finding minimum cost n-cell upper limit length CA of grade up to 500. Programmable regulation vectors based additive maximal length CA has many applications in the design of cryptanalytic primitives. In [ 4 ] one such application has been mentioned, where programmable additive upper limit length CA has been used to plan an incorporate strategy for both error rectification and message hallmark. Therefore, interior decorator demands list of maximal length CA-rule vectors for a peculiar cell size.

Method and Result The algorithm of finding whether a given n- cell

CA has a maximal length rhythm is as follows.

1. Take n ? n tridiagonal matrix with all non-zero elements are 1

2. Change chief diagonal consecutive by one of the 2n combinations

3. Calculate the characteristic multinomial matching to the n ? n con- structed matrix

4. Calculate the figure of non-zero coefficient in the characteristic multinomial and if figure of coefficients is even so travel to step 2

5. Check the coefficients of xn and x0, if they are zeros so go to step 2.

6. Check if the characteristic multinomial lucifers with any one of the list of crude multinomials.

7. If lucifers so matching chief diagonal of the matrix represents the maximal length CA-rule vector

In Table under the caption ’CA-rule vector’ , ‘0? and ‘1? correspond to govern

90 and 150 severally. Under caption ’Primitive poly.’ the entries represent crude multinomial in binary format. It has been observed that mirror image of each regulation vector corresponds to same crude multinomial. For illustration in

8-cell CA, 00000110 and 01100000 are two regulation vectors for crude multinomial x8 + x4 + x3 + x2 + 1 ( 100011101 ) , where regulation vectors are mirror image of each other

# cells

Crude Poly.

CA-rule vector

2

111

10

3

1011

110

1101

100

4

10011

1010

11001

1101

5

100101

11100

101001

10000

101111

01100

110111

10011

111011

11000

111101

11110

6

1000011

000110

1011011

101110

1100001

011010

1100111

100101

1101101

101010

1110011

100000

7

10000011

1011001

10001001

0111010

10001111

1110001

10010001

1110100

10011101

1101010

10100111

0010010

10101011

1101111

10111001

1001000

10111111

1000010

11000001

0010000

11001011

1011011

11010011

0110111

11010101

1011110

11100101

1010100

11101111

1101000

11110001

1000101

11110111

0001110

11111101

0100110

# cells

Crude Poly.

CA-rule vector

9

1011010001

010000001

1011011011

101011110

1011110101

001111011

1011111001

001011111

1100010011

101100011

1100010101

100011011

1100011111

100010111

1100100011

110010101

1100110001

011100110

1100111011

010011110

1101001111

010110011

1101011011

011001101

1101100001

000101111

1101101011

011110001

1101101101

000110111

1101110011

000000001

1101111111

010001111

1110000101

000001011

1110001111

001000011

1110110101

010100001

1110111001

011111101

1111000111

111011011

1111001011

010000110

1111001101

001001100

1111010101

000011010

1111011001

000110010

1111100011

000001110

1111101001

100111111

1111111011

101000001

10

10000001001

1100001111

10000011011

1001110101

10000100111

1010100111

10000101101

1001010111

10001100101

1000111011

# cells

Crude Poly.

CA-rule vector

10

11010110101

1011101110

11011000001

1000001010

11011010011

0010111111

11011011111

0100010001

11011111101

0111110011

11100010111

0011000111

11100011101

0011100011

11100100001

1101001010

11100111001

0110000111

11101000111

1100100110

11101001101

0001101101

11101010101

0011011001

11101011001

0100100111

11101100011

0101010101

11101111101

0110110001

11110001101

1100000111

11110010011

0101010110

11110110001

0011100110

11111011011

1100011001

11111110011

0001111100

11111111001

0110010110

11

100000000101

01000011010

100000010111

11110101011

100000101011

01000110010

100000101101

01101111110

100001000111

00110010010

100001100011

10001000011

100001100101

00110100010

100001110001

00010110010

100001111011

00100010110

100010001101

11101001111

100010010101

00110011000

100010011111

10100001001

100010101001

11100110111

100010110001

00011100100

# cells

Crude Poly.

CA-rule vector

11

101001101101

00110101110

101001111001

01011011100

101001111111

10000111011

101010000101

00000011000

101010010001

01011010110

101010011101

00000001100

101010100111

10100011101

101010101011

01010111010

101010110011

10111010001

101010110101

10011001101

101011010101

11011001001

101011011111

11010011001

101011101001

11010001101

101011101111

10110100011

101011110001

11010110001

101011111011

00000000110

101100000011

01010010111

101100001001

00110101101

101100010001

01111111111

101100110011

00001111101

101100111111

01011010011

101101000001

01100101101

101101001011

00001011111

101101011001

01011001101

101101011111

00101010111

101101100101

10100101110

101101101111

00101101011

101101111101

01010011101

101110000111

00011101101

101110001011

00111101001

101110010011

00000100001

101110010101

00010000001

101110101111

01010101011

101110110111

11000101110

101110111101

01000110111

# cells

Crude Poly.

CA-rule vector

11

110101011001

01001011001

110101100011

00101001011

110101101111

01011010001

110101110001

10101100010

110110010011

00001101011

110110011111

00001011011

110110101001

00011010011

110110111011

00100011011

110110111101

00001010111

110111001001

00101010101

110111010111

00111010001

110111011011

00010011101

110111100001

01111011111

110111100111

01111110111

110111110101

11100010010

111000000101

10000100010

111000011101

00110011111

111000100001

01011101101

111000100111

00011111011

111000101011

01111110001

111000110011

00011011111

111000111001

01101011101

111001000111

11110001110

111001001011

00011111101

111001010101

00010001001

111001011111

00001001001

111001110001

01101010111

111001111011

10011111100

111001111101

00001000101

111010000001

01001101111

111010010011

01110100111

# cells

Crude Poly.

CA-rule vector

12

1000001010011

011011000110

1000001101001

100101100101

1000001111011

011100110010

1000001111101

000001000100

1000010011001

100101010011

1000011010001

001100111010

1000011101011

001110100110

1000100000111

101001010101

1000100011111

110000101011

1000100100011

100100011101

1000100111011

000111101100

1000101001111

111101110111

1000101010111

100111001001

1000101100001

111011111011

1000101101011

101100110001

1000110000101

110001011001

1000110110011

101100100011

1000111011001

000011101110

1000111011111

010101101100

1001000001101

000100011111

1001000110111

010101010101

1001000111101

001000011111

1001001100111

011100101001

1001001110011

011010110001

1001001111111

000011110011

1001010111001

001101001101

1001011000001

010011010011

1001011001011

000101101101

1001100001111

001101000111

1001100011101

000101110011

1001100100001

110111000100

1001100111001

001011100011

1001100111111

110100011100

1001101001101

010010101101

1001101110001

100101101010

# cells

Crude Poly.

CA-rule vector

12

1011110111111

010010010001

1011111000001

011100000001

1100001010111

100111001101

1100001011101

111010010011

1100010010001

000101000010

1100010010111

000010001010

1100010111001

001010000010

1100011101111

100000111111

1100100011011

111000111001

1100100110101

001111101010

1100101000001

000000010110

1100101100101

001110101110

1100101111011

101011010011

1100110001011

001000001100

1100110110001

110010001111

1100110111101

000111111100

1100111001001

011111001010

1100111001111

101101101001

1100111100111

110101001011

1101000011011

000100000101

1101000101011

010010101111

1101000110011

000101000001

1101001101001

010111001011

1101010001011

101011110010

1101011010001

001101011011

1101011100001

001111000111

1101011110101

010011011101

1101100001011

001011110011

1101100010011

011100011101

1101100011111

000001111111

1101101010111

011100111001

1101110010001

010010011111

1101110100111

011010001111

1101110111111

011100010111

1101111000001

000110111101

Decision In this paper a simple algorithm to calculate regulation vectors for n-cell maximal length CA has been introduced. Besides, all maximal length CA regulation vectors for cell size 2 to 12 have been computed by using proposed algo- rithm and they have been tabulated. Programmable regulation vectors based maximal length CA can be used to plan cryptanalytic primitives. Since the list of all regulation vectors are available to a interior decorator so it will surely cut down design rhythm clip

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