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