New Results on Scrambling Using the Mesh Array

1   NEW RESULTS ON SCRAMBLIMG USING THE MESH ARRAY Sandhya Rangineni Abstract. This paper presents new results on randomi zation using Kak’s Mesh Array for m atrix 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 intr oduced by Kak in 1988 [1],[2]. It is able to multiply the matrices in only 2n-1 steps for two n×n m atrices. 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 addition al scrambling properties of the array and also consider triple matrix mu ltiplication. PRELIMINARIES When multiplying two matrices A and B (C=AB), the com ponents 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 ge t period of the matrix of order 4. The period is nothing but the maximum of lengths of the cycles. The cycles of the m atrix 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 th e period of the scrambling transformation = 7. We will now consider the longest cycle in each scrambling m atrix. The period of the scramb ling 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. b 41 a 14 a 24 b 42 b 43 a 34 a 44 b 44 b 31 a 13 a 23 b 32 b 33 a 33 a 43 b 34 b 21 a 12 a 22 b 22 b 23 a 32 a 42 b 24 b 11 a 11 a 21 b 12 b 13 a 31 a 41 b 14 11 22 33 44 1 2 31 24 43 32 14 41 23 34 42 13 21 Figure 1: Mesh Architecture for multiplication of m atrices A and B and store the result in C from [1] We now consider further properties of the a rray for scrambling [4], [5], which has m any 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 follo wing 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 prim e 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 auto correlation function derived from the sequence. BINARY SEQUENCE FOR THE CYCLES We can create a binary sequence of the longest cycl es in terms of 1s and 0s where the even cycle is represented as 1 and odd cycl e is represented as 0. The bina ry sequence for the cycles of orders 2 to 1000 is as follows: 0 2 4 6 8 10 12 14 16 18 123456789 1 0 1 1 Number  of  prime  or der s Number  of  prime  orders 6   111011011000011000000100000000000000010000000010000000000000001000001000000000 000000000000010100000110000000001000100000001010110000010000000000000000000000 101000000100000000000000001000000100000001000001100000000010000000010000100010 000000000000000000000000001000100000000000000000000000100000010000000000010000 000010000000000011000000000000010000000000010000000000000000000000000000000000 001101110000000000000000000000010100000000000000000001000000000000000000100000 000010000000000000000000000000000000000000000100000000000000000000000000000000 000000000000000000000000000000000000010001000000000000000000001000001100100000 000000000000000000000000000100000000000000000000000110000100000000000000000000 000000000000010000000000000000000000000000110000000000000001000000000000000000 000000000000000100000000000000000000000010000001000000000001000000000010000000 000000000000000000000000000000000000000010000000100100000000001000000000000000 000000000110000000000001000000101000000000100000000000000001000 AUTOCORRELATION FUNCTION Autocorrelation function is used to show the simila rity between the observations as a function according to time. Here autocorrelation function is used to represent the variations of the cycles as a single function. C(k) = 1/999 ∑ A(i)*A(i+k) where i=1 to 999 Here, A(i) is the polar sequence of the cycles where 0 is converted as -1 in binary sequence and 1 remains same. The autocorrelation function for k ranging from 0 to 100 is shown in Figure 3. Figure 3: Autocorrelation function C(k) where k ranges from 0 to 1000 The autocorrelation function is e ffectively two valued which dem onstrates that the sequence of orders is random. ‐ 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1 49 97 145 193 241 289 337 385 433 481 529 577 625 673 721 769 817 865 913 961 C(K) C(K) 7   TRIPLE MATRIX MULTIPLIC ATION ON A MESH ARRAY Now we consider the multiplication of three m atrices of the same order in the m anner of [6],[7]. Let A, X, B be the matrices to be multip lied and le t us store the result in another matrix Y i.e., Y = A X B. The computation of Y = A X B is decomposed into • Z = XB • Y = AZ b 41 x 14 x 24 b 42 b 43 x 34 x 44 b 44 b 31 x 13 x 23 b 32 b 33 x 33 x 43 b 34 b 21 x 12 x 22 b 22 b 23 x 32 x 42 b 24 b 11 x 11 x 21 b 12 b 13 x 31 x 41 b 14 11 22 33 44 12 31 24 43 32 14 41 23 34 42 13 21 Figure 4: Mesh Architecture for Z = XB where 11 in the node represent Z 11 At time t=n the first row of the product XB is completed and the re sults of Z are stored in the nodes of first row and then we switch the X and B values to the second row transm itting downwards in the array. Now immediately after t=n the elements of the matrix, A follows the same path as X has passed. When t=2n, the pro duct of XB in the second row and the product of AZ in the first row are done in parallel and the result of Z is stored in the nodes of second row and the result of AZ is stored in Y ij (1) respectively. 8   Figure 5: The triple matrix m ultiplication at tim e t=2n Then at time t=2n, the results obtained are the Z values of second row and the Y values of first row. The process is continued t ill all the results are obtained. Y ij = Y ij (1)+ Y ij (2)+ Y ij (3)+ Y ij (4) At time t=4n, the mesh architecture for tr iple matrix multiplication is as f ollows: Figure 6: The mesh array representation for Y = AXB 9   DISCUSSION This article represents new results on scra mbling using Kak’s m esh array for matrix multiplication. These results inc ludes the periods of the matrices multiplied, the binary sequence of the longest cycles for even and odd numbers, and the autocorrelation f unction for an even and odd sequence obtained from the order. The struct ure for triple matrix multiplication on a m esh array has also been presented. REFERENCES [1] S. Kak, Multilayered array computi ng. Information Sciences 45, 347-365, 1988. [2] S. Kak, A two-layered mesh array for ma trix multiplicatio n. Parallel Computing 6, 383-385, 1988. [3] S. Kak, On the mesh array for matrix m [4] S. Kak, An overview of analog encrypti on. Proceedings IEE 130, Pt. F, 399-404, 1983. [5] S. Kak and N.S. Jayant, On speech encr yption using waveform scrambling. Bell System Technical Journal 56, 781-808, 1977. [6] A.Benaini and Y.Robert, An even faster systolic array for matrix multiplication. Parallel Computing 12, 249-254, 1989. [7] J.L.Aravena, Triple matrix product architectures for fa st signal processing. IEEE Trans. Circuits Syst. 35, 119-122, 1988.