New Results on Scrambling Using the Mesh Array

Abstract. This paper presents new results on randomization using Kak’s Mesh Array for matrix multiplication. These results include the periods of the longest cycles when the array is used for scrambling and the autocorrelation function of the binary sequence obtained from the cycles.
INTRODUCTION The mesh array of matrix multiplication was introduced by Kak in 1988 [1],[2]. It is able to multiply the matrices in only 2n-1 steps for two n×n matrices. In a new paper, this array has been proposed as a scrambling transformation [3]. Figure 1 presents the mesh array for multiplying two 4× 4 matrices. Here we investigate some additional scrambling properties of the array and also consider triple matrix multiplication.
PRELIMINARIES When multiplying two matrices A and B (C=AB), the components of C are obtained in the following arrangement:
11 22 33 44 12 31 24 43 32 14 41 23 34 42 13 21
As shown in [3], the items of both standard array and mesh array will be written in an array as follows: 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 (

) 11 22 33 44 12 31 24 43 32 14 41 23 34 42 13 21 By writing the above arrays into cycles, we can get period of the matrix of order 4. The period is nothing but the maximum of lengths of the cycles. The cycles of the matrix of order 4 are as follow: = (11) (42) (12 22 31 32 14 44 21) (13 33 41 34 23 24 43) Here the lengths of the cycles are {1,1,7,7}, and the period of the scrambling transformation = 7. We will now consider the longest cycle in each scrambling matrix. The period of the scrambling 2    transformation will be the lcm of the cycles associated with the scrambling. Since the periods increase very rapidly, we shall consider only the longest cycles.

b41 a14 a24 b42 b43 a34 a44 b44 b31 a13 a23 b32 b33 a33 a43 b34 b21 a12 a22 b22 b23 a32 a42 b24 b11 a11 a21 b12 b13 a31 a41 b14 11

22

33

44

1 2 31 24 43

32 14 41 23

34 42 13 21

Figure 1: Mesh Architecture for multiplication of matrices A and B and store the result in C from [1]

We now consider further properties of the array for scrambling [4],[5], which has many applications in signal processing.

Table 1: Longest cycles for the matrices from order 2 to 100 ORDER LONGEST CYCLE 2 3 3 7 4 7 5 20 6 23 7 19 8 27 9 79 10 31 11 88 12 46 13 150 14 180 3    15 103 16 197 17 242 18 270 19 121 20 220 21 438 22 402 23 367 24 455 25 478 26 362 27 667 28 514 29 262 30 678 31 697 32 414 33 507 34 620 35 512 36 492 37 1357 38 687 39 751 40 1110 41 1065 42 824 43 813 44 1221 45 912 46 1435 47 1347 48 877 49 2015 50 1391 51 1341 52 1090 53 2370 54 2182 55 974 56 2508 57 2064 58 2955 59 2146 4    60 2392 61 2452 62 2171 63 1448 64 2687 65 1957 66 4046 67 3069 68 1116 69 1501 70 3539 71 2219 72 2064 73 2542 74 3191 75 3194 76 5085 77 5329 78 2831 79 6060 80 3140 81 5390 82 3007 83 4786 84 6970 85 4012 86 3213 87 5143 88 7488 89 7685 90 5941 91 3383 92 6903 93 2521 94 4930 95 5869 96 6214 97 4419 98 3173 99 5150 100 7984

5    Prime orders The number of primes in the list of longest cycles has the following distribution:
001 – 100 —- 16 101 – 200 —- 15 201 – 300 —- 10 301 – 400 —- 11 401 – 500 —- 5 501 – 600 —- 3 601 – 700 —- 8 701 – 800 —- 5 801 – 900 —- 4 901 – 1000—- 12

Figure 2: The graph for the number of prime orders from 0 to 1000

This in itself does not tell us how good are the randomness properties of the sequences of cycles associated with the mesh array. For this we will look at the autocorrelation function derived from the sequence. BINARY SEQUENCE FOR THE CYCLES We can create a binary sequence of the longest cycles in terms of 1s and 0s where the even cycle is represented as 1 and odd cycle is represented as 0. The binary sequence for the cycles of orders 2 to 1000 is as follows: 0 2 4 6 8 10 12 14 16 18 1 2 3 4 5 6 7 8 9 10 11 Number of prime orders Number of prime orders 6    111011011000011000000100000000000000010000000010000000000000001000001000000000 000000000000010100000110000000001000100000001010110000010000000000000000000000 101000000100000000000000001000000100000001000001100000000010000000010000100010 000000000000000000000000001000100000000000000000000000100000010000000000010000 000010000000000011000000000000010000000000010000000000000000000000000000000000 001101110000000000000000000000010100000000000000000001000000000000000000100000 000010000000000000000000000000000000000000000100000000000000000000000000000000 000000000