An algorithm for multiplication of trigintaduonions

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An algorithm for multiplication of trigintaduonions Article · October 2014

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Journal of Theoretical and Applied Computer Science ISSN 2299-2634 (printed), 2300-5653 (online)

Vol. 8, No. 1, 2014, pp. 50–75 http://www.jtacs.org

An algorithm for multiplication of trigintaduonions Alexandr Cariow, Galina Cariowa Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin, Poland {atariov, gtariova}@wi.zut.edu.pl

Abstract:

In this paper we introduce efficient algorithm for the multiplication of trigintaduonions. The direct multiplication of two trigintaduonions requires 1024 real multiplications and 992 real additions. We show how to compute a trigintaduonion product with 498 real multiplications and 943 real additions. During synthesis of the discussed algorithm we use a fact that trigintaduonion multiplication may be represented by a vector-matrix product. Such representation provides a possibility to discover repeating elements in the matrix structure and to use specific properties of their mutual placement to decrease the number of real multiplications needed to compute the product of two trigintaduonions.

Keywords:

trigintaduonion, multiplication of trigintaduonions, fast algorithm, matrix notation

1. Introduction Science and technology has generated new knowledge, such as discoveries of new methods of data processing, and has dramatically contributed development and progress in cybernetics and computer science. This progress has stimulated the emergence of new problems that can not be solved with old traditional approaches. Implementation and solution of the mentioned problems requires the use of new mathematical abstractions and formalisms. The above was the reason for the emergence and widespread methods of processing data using Clifford algebras and hypercomplex number systems [1, 21]. Currently, hypercomplex numbers are used in physics [18, 20, 48, 49] digital signal and image processing [2, 3, 4, 5, 6, 7, 14, 19, 25, 29, 30, 31, 32, 33, 34, 35, 36, 40, 45, 46], computer graphics [17, 22, 38], cryptography [27], watermarking [41], wireless data transmission [8, 16, 24, 37] and navigation of mobile vehicles [23, 28]. At the hypercomplex algebra the most time-consuming operation is the multiplication of two hypercomplex numbers. This is because the multiplication of two N -dimensional hypercomplex numbers requires N 2 real multiplications and N (N − 1) real additions. In low-power digital design, optimization must be done at both algorithmic and logic-circuit levels. Multiplying of hypercomplex numbers involves a large number of real multiplications, which require much more intensive hardware than real addition operations. Therefore in this case, the hypercomplex numbers multipliing algorithms which contain as little as possible of real multiplications are preferable. Efficient algorithms for the multiplication of quaternions, octonions, and sedenions with reduced number of real multiplications is already exist [9, 10, 11, 12, 26, 42, 43, 44]. No such algorithms for the multiplication of trigintaduonions have been proposed. The aim of the present paper is to suggest the efficient algorithm for purpose.

51

An algorithm for multiplication of trigintaduonions

A trigintaduonion is defined as follows [13, 15, 47]:

t = a0 +

31 X

ai e i

i=1

where {ai }, i = 0, 1, . . . , 31 are real numbers and {ei }, i = 0, 1, . . . , 31 are imaginary units. The following notations are used:

t1 = a0 +

31 X

ai ei , t2 = b0 +

i=1

31 X

bi ei , t = t1 · t2 = c0 +

i=1

31 X

ci e i .

i=1

Tables 1, 2, 3 and 4 describe the northwest, northeast, southwest and southeastern quadrants of the multiplication table of the imaginary units of trigintaduonions, respectively [15]: The trigintaduonions product can be presented in the matrix-vector multiplication form as: Y32×1 = B32 X32×1

(1)

where X32×1 = [a0 , a1 , . . . , a31 ]T , Y32×1 = [c0 , c1 , . . . , c15 ]T , " B32 =

(0,0)

(0,1)

B16 B16 (1,0) (1,1) B16 B16

# ,

Table 1. North-west quadrant of the table for the trigintaduonions imaginary units multiplication × 1 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

1 e1 1 e1 e1 −1 e2 −e3 e3 e2 e4 −e5 e5 e4 e6 e7 e7 −e6 e8 −e9 e9 e8 e10 e11 e11 −e10 e12 e13 e13 −e12 e14 −e15 e15 e14

e2 e3 e2 e3 e3 −e2 −1 e1 −e1 −1 −e6 −e7 −e7 e6 e4 −e5 e5 e4 −e10 −e11 −e11 e10 e8 −e9 e9 e8 e14 e15 e15 −e14 −e12 e13 −e13 −e12

e4 e5 e6 e4 e5 e6 e5 −e4 −e7 e6 e7 −e4 e7 −e6 e5 −1 e1 e2 −e1 −1 −e3 −e2 e3 −1 −e3 −e2 e1 −e12 −e13 −e14 −e13 e12 e15 −e14 −e15 e12 −e15 e14 −e13 e8 −e9 −e10 e9 e8 e11 e10 −e11 e8 e11 e10 −e9

e7 e7 e6 −e5 −e4 e3 e2 −e1 −1 −e15 −e14 e13 e12 −e11 −e10 e9 e8

e8 e9 e10 e11 e12 e13 e14 e15 e8 e9 e10 e11 e12 e13 e14 e15 e9 −e8 −e11 e10 −e13 e12 e15 −e14 e10 e11 −e8 −e9 −e14 −e15 e12 e13 e11 −e10 e9 −e8 −e15 e14 −e13 e12 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e13 −e12 e15 −e14 e9 −e8 e11 −e10 e14 −e15 −e12 e13 e10 −e11 −e8 e9 e15 e14 −e13 −e12 e11 e10 −e9 −e8 −1 e1 e2 e3 e4 e5 e6 e7 −e1 −1 −e3 e2 −e5 e4 e7 −e6 −e2 e3 −1 −e1 −e6 −e7 e4 e5 −e3 −e2 e1 −1 −e7 e6 −e5 e4 −e4 e5 e6 e7 −1 e1 −e2 −e3 −e5 −e4 e7 −e6 e1 −1 e3 e2 −e6 e7 −e4 e5 e2 −e3 −1 e1 −e7 e6 −e5 −e4 e3 e2 −e1 −1

52

Alexandr Cariow, Galina Cariowa Table 2. North-east quadrant of the table for the trigintaduonions imaginary units multiplication

× 1 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

e16 e17 1 e1 e1 −1 e2 −e3 e3 e2 e4 −e5 e5 e4 e6 e7 e7 −e6 e8 −e9 e9 e8 e10 e11 e11 −e10 e12 e13 e13 −e12 e14 −e15 e15 e14

e18 e19 e2 e3 e3 −e2 −1 e1 −e1 −1 −e6 −e7 −e7 e6 e4 −e5 e5 e4 −e10 −e11 −e11 e10 e8 −e9 e9 e8 e14 e15 e15 −e14 −e12 e13 −e13 −e12

e20 e21 e22 e4 e5 e6 e5 −e4 −e7 e6 e7 −e4 e7 −e6 e5 −1 e1 e2 −e1 −1 −e3 −e2 e3 −1 −e3 −e2 e1 −e12 −e13 −e14 −e13 e12 e15 −e14 −e15 e12 −e15 e14 −e13 e8 −e9 −e10 e9 e8 e11 e10 −e11 e8 e11 e10 −e9

e23 e7 e6 −e5 −e4 e3 e2 −e1 −1 −e15 −e14 e13 e12 −e11 −e10 e9 e8

e24 e25 e26 e27 e28 e29 e30 e31 e8 e9 e10 e11 e12 e13 e14 e15 e9 −e8 −e11 e10 −e13 e12 e15 −e14 e10 e11 −e8 −e9 −e14 −e15 e12 e13 e11 −e10 e9 −e8 −e15 e14 −e13 e12 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e13 −e12 e15 −e14 e9 −e8 e11 −e10 e14 −e15 −e12 e13 e10 −e11 −e8 e9 e15 e14 −e13 −e12 e11 e10 −e9 −e8 −1 e1 e2 e3 e4 e5 e6 e7 −e1 −1 −e3 e2 −e5 e4 e7 −e6 −e2 e3 −1 −e1 −e6 −e7 e4 e5 −e3 −e2 e1 −1 −e7 e6 −e5 e4 −e4 e5 e6 e7 −1 e1 −e2 −e3 −e5 −e4 e7 −e6 e1 −1 e3 e2 −e6 e7 −e4 e5 e2 −e3 −1 e1 −e7 e6 −e5 −e4 e3 e2 −e1 −1

Table 3. South-west quadrant of the table for the trigintaduonions imaginary units multiplication × e16 e17 e18 e19 e20 e21 e22 e23 e24 e25 e26 e27 e28 e29 e30 e31

1 e1 1 e1 e1 −1 e2 −e3 e3 e2 e4 −e5 e5 e4 e6 e7 e7 −e6 e8 −e9 e9 e8 e10 e11 e11 −e10 e12 e13 e13 −e12 e14 −e15 e15 e14

e2 e3 e2 e3 e3 −e2 −1 e1 −e1 −1 −e6 −e7 −e7 e6 e4 −e5 e5 e4 −e10 −e11 −e11 e10 e8 −e9 e9 e8 e14 e15 e15 −e14 −e12 e13 −e13 −e12

e4 e5 e6 e4 e5 e6 e5 −e4 −e7 e6 e7 −e4 e7 −e6 e5 −1 e1 e2 −e1 −1 −e3 −e2 e3 −1 −e3 −e2 e1 −e12 −e13 −e14 −e13 e12 e15 −e14 −e15 e12 −e15 e14 −e13 e8 −e9 −e10 e9 e8 e11 e10 −e11 e8 e11 e10 −e9

e7 e7 e6 −e5 −e4 e3 e2 −e1 −1 −e15 −e14 e13 e12 −e11 −e10 e9 e8

e8 e9 e10 e11 e12 e13 e14 e15 e8 e9 e10 e11 e12 e13 e14 e15 e9 −e8 −e11 e10 −e13 e12 e15 −e14 e10 e11 −e8 −e9 −e14 −e15 e12 e13 e11 −e10 e9 −e8 −e15 e14 −e13 e12 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e13 −e12 e15 −e14 e9 −e8 e11 −e10 e14 −e15 −e12 e13 e10 −e11 −e8 e9 e15 e14 −e13 −e12 e11 e10 −e9 −e8 −1 e1 e2 e3 e4 e5 e6 e7 −e1 −1 −e3 e2 −e5 e4 e7 −e6 −e2 e3 −1 −e1 −e6 −e7 e4 e5 −e3 −e2 e1 −1 −e7 e6 −e5 e4 −e4 e5 e6 e7 −1 e1 −e2 −e3 −e5 −e4 e7 −e6 e1 −1 e3 e2 −e6 e7 −e4 e5 e2 −e3 −1 e1 −e7 e6 −e5 −e4 e3 e2 −e1 −1

Table 4. South-east quadrant of the table for the trigintaduonions imaginary units multiplication × e16 e17 e18 e19 e20 e21 e22 e23 e24 e25 e26 e27 e28 e29 e30 e31

e16 e17 1 e1 e1 −1 e2 −e3 e3 e2 e4 −e5 e5 e4 e6 e7 e7 −e6 e8 −e9 e9 e8 e10 e11 e11 −e10 e12 e13 e13 −e12 e14 −e15 e15 e14

e18 e19 e2 e3 e3 −e2 −1 e1 −e1 −1 −e6 −e7 −e7 e6 e4 −e5 e5 e4 −e10 −e11 −e11 e10 e8 −e9 e9 e8 e14 e15 e15 −e14 −e12 e13 −e13 −e12

e20 e21 e22 e4 e5 e6 e5 −e4 −e7 e6 e7 −e4 e7 −e6 e5 −1 e1 e2 −e1 −1 −e3 −e2 e3 −1 −e3 −e2 e1 −e12 −e13 −e14 −e13 e12 e15 −e14 −e15 e12 −e15 e14 −e13 e8 −e9 −e10 e9 e8 e11 e10 −e11 e8 e11 e10 −e9

e23 e7 e6 −e5 −e4 e3 e2 −e1 −1 −e15 −e14 e13 e12 −e11 −e10 e9 e8

e24 e25 e26 e27 e28 e29 e30 e31 e8 e9 e10 e11 e12 e13 e14 e15 e9 −e8 −e11 e10 −e13 e12 e15 −e14 e10 e11 −e8 −e9 −e14 −e15 e12 e13 e11 −e10 e9 −e8 −e15 e14 −e13 e12 e12 e13 e14 e15 −e8 −e9 −e10 −e11 e13 −e12 e15 −e14 e9 −e8 e11 −e10 e14 −e15 −e12 e13 e10 −e11 −e8 e9 e15 e14 −e13 −e12 e11 e10 −e9 −e8 −1 e1 e2 e3 e4 e5 e6 e7 −e1 −1 −e3 e2 −e5 e4 e7 −e6 −e2 e3 −1 −e1 −e6 −e7 e4 e5 −e3 −e2 e1 −1 −e7 e6 −e5 e4 −e4 e5 e6 e7 −1 e1 −e2 −e3 −e5 −e4 e7 −e6 e1 −1 e3 e2 −e6 e7 −e4 e5 e2 −e3 −1 e1 −e7 e6 −e5 −e4 e3 e2 −e1 −1

53

An algorithm for multiplication of trigintaduonions (0,0)

B16               =             

=

b0 −b1 b1 b0 b2 −b3 b3 b2 b4 −b5 b5 b4 b6 b7 b7 −b6 b8 −b9 b9 b8 b10 b11 b11 −b10 b12 b13 b13 −b12 b14 −b15 b15 b14

(0,1)

B16

=



−b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31

             =             

(1,0)

B16               =             

−b17 −b16 b19 −b18 b21 −b20 −b23 b22 b25 −b24 −b27 b26 −b29 b28 b31 −b30

−b2 −b3 b3 −b2 b0 b1 −b1 b0 −b6 −b7 −b7 b6 b4 −b5 b5 b4 −b10 −b11 −b11 b10 b8 −b9 b9 b8 b14 b15 b15 −b14 −b12 b13 −b13 −b12

−b4 −b5 −b6 b5 −b4 −b7 b6 b7 −b4 b7 −b6 b5 b0 b1 b2 −b1 b0 −b3 −b2 b3 b0 −b3 −b2 b1 −b12 −b13 −b14 −b13 b12 b15 −b14 −b15 b12 −b15 b14 −b13 b8 −b9 −b10 b9 b8 b11 b10 −b11 b8 b11 b10 −b9

−b7 b6 −b5 −b4 b3 b2 −b1 b0 −b15 −b14 b13 b12 −b11 −b10 b9 b8

−b8 −b9 b9 −b8 b10 b11 b11 −b10 b12 b13 b13 −b12 b14 −b15 b15 b14 b0 b1 −b1 b0 −b2 b3 −b3 −b2 −b4 b5 −b5 −b4 −b6 −b7 −b7 b6

−b18 −b19 −b16 b17 b22 b23 −b20 −b21 b26 b27 −b24 −b25 −b30 −b31 b28 b29

−b19 b18 −b17 −b16 b23 −b22 b21 −b20 b27 −b26 b25 −b24 −b31 b30 −b29 b28

−b20 −b21 −b22 −b23 −b16 b17 b18 b19 b28 b29 b30 b31 −b24 −b25 −b26 −b27

−b21 b20 −b23 b22 −b17 −b16 −b19 b18 b29 −b28 b31 −b30 b25 −b24 b27 −b26

−b22 b23 b20 −b21 −b18 b19 −b16 −b17 b30 −b31 −b28 b29 b26 −b27 −b24 b25

−b23 −b22 b21 b20 −b19 −b18 b17 −b16 b31 b30 −b29 −b28 b27 b26 −b25 −b24

−b24 −b25 −b26 −b27 −b28 −b29 −b30 −b31 −b16 b17 b18 b19 b20 b21 b22 b23

−b18 −b19 b16 b17 b22 b23 −b20 −b21 b26 b27 −b24 −b25 −b30 −b31 b28 b29

−b19 b18 −b17 b16 b23 −b22 b21 −b20 b27 −b26 b25 −b24 −b31 b30 −b29 b28

−b20 −b21 −b22 −b23 b16 b17 b18 b19 b28 b29 b30 b31 −b24 −b25 −b26 −b27

−b21 b20 −b23 b22 −b17 b16 −b19 b18 b29 −b28 b31 −b30 b25 −b24 b27 −b26

−b22 b23 b20 −b21 −b18 b19 b16 −b17 b30 −b31 −b28 b29 b26 −b27 −b24 b25

−b23 −b22 b21 b20 −b19 −b18 b17 b16 b31 b30 −b29 −b28 b27 b26 −b25 −b24

−b25 b24 −b27 b26 −b29 b28 b31 −b30 −b17 −b16 −b19 b18 −b21 b20 b23 −b22

−b10 −b11 −b11 b10 −b8 −b9 b9 −b8 b14 b15 b15 −b14 −b12 b13 −b13 −b12 b2 b3 −b3 b2 b0 −b1 b1 b0 b6 b7 b7 −b6 −b4 b5 −b5 −b4

−b12 −b13 −b14 −b13 b12 b15 −b14 −b15 b12 −b15 b14 −b13 −b8 −b9 −b10 b9 −b8 b11 b10 −b11 −b8 b11 b10 −b9 b4 b5 b6 −b5 b4 b7 −b6 −b7 b4 −b7 b6 −b5 b0 −b1 −b2 b1 b0 b3 b2 −b3 b0 b3 b2 −b1

−b15 −b14 b13 b12 −b11 −b10 b9 −b8 b7 −b6 b5 b4 −b3 −b2 b1 b0

              ,             

−b26 b27 b24 −b25 −b30 −b31 b28 b29 −b18 b19 −b16 −b17 −b22 −b23 b20 b21

−b27 −b26 b25 b24 −b31 b30 −b29 b28 −b19 −b18 b17 −b16 −b23 b22 −b21 b20

−b28 b29 b30 b31 b24 −b25 −b26 −b27 −b20 b21 b22 b23 −b16 −b17 −b18 −b19

−b29 −b28 b31 −b30 b25 b24 b27 −b26 −b21 −b20 b23 −b22 b17 −b16 b19 −b18

−b30 −b31 −b28 b29 b26 −b27 b24 b25 −b22 −b23 −b20 b21 b18 −b19 −b16 b17

−b31 b30 −b29 −b28 b27 b26 −b25 b24 −b23 b22 −b21 −b20 b19 b18 −b17 −b16



−b26 b27 b24 −b25 −b30 −b31 b28 b29 −b18 b19 b16 −b17 −b22 −b23 b20 b21

−b27 −b26 b25 b24 −b31 b30 −b29 b28 −b19 −b18 b17 b16 −b23 b22 −b21 b20

−b28 b29 b30 b31 b24 −b25 −b26 −b27 −b20 b21 b22 b23 b16 −b17 −b18 −b19

−b29 −b28 b31 −b30 b25 b24 b27 −b26 −b21 −b20 b23 −b22 b17 b16 b19 −b18

−b30 −b31 −b28 b29 b26 −b27 b24 b25 −b22 −b23 −b20 b21 b18 −b19 b16 b17

−b31 b30 −b29 −b28 b27 b26 −b25 b24 −b23 b22 −b21 −b20 b19 b18 −b17 b16



             ,             

=

b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31

−b17 b16 b19 −b18 b21 −b20 −b23 b22 b25 −b24 −b27 b26 −b29 b28 b31 −b30

−b24 −b25 −b26 −b27 −b28 −b29 −b30 −b31 b16 b17 b18 b19 b20 b21 b22 b23

−b25 b24 −b27 b26 −b29 b28 b31 −b30 −b17 b16 −b19 b18 −b21 b20 b23 −b22

             ,             

54

Alexandr Cariow, Galina Cariowa

(1,1)

B16               =             

=

b0 b1 b2 b3 b4 b5 b6 b7 −b1 b0 −b3 b2 −b5 b4 b7 −b6 −b2 b3 b0 −b1 −b6 −b7 b4 b5 −b3 −b2 b1 b0 −b7 b6 −b5 b4 −b4 b5 b6 b7 b0 −b1 −b2 −b3 −b5 −b4 b7 −b6 b1 b0 b3 −b2 −b6 −b7 −b4 b5 b2 −b3 b0 b1 −b7 b6 −b5 −b4 b3 b2 −b1 b0 −b8 b9 b10 b11 b12 b13 b14 b15 −b9 −b8 b11 −b10 b13 −b12 −b15 b14 −b10 −b11 −b8 b9 b14 b15 −b12 −b13 −b11 b10 −b9 −b8 b15 −b14 b13 −b12 −b12 −b13 −b14 −b15 −b8 b9 b10 b11 −b13 b12 −b15 b14 −b9 −b8 −b11 b10 −b14 b15 b12 −b13 −b10 b11 −b8 −b9 −b15 −b14 b13 b12 −b11 −b10 b9 −b8

b8 b9 b10 b11 b12 b13 b14 b15 −b9 b8 b11 −b10 b13 −b12 −b15 b14 −b10 −b11 b8 b9 b14 b15 −b12 −b13 −b11 b10 −b9 b8 b15 −b14 b13 −b12 −b12 −b13 −b14 −b15 b8 b9 b10 b11 −b13 b12 −b15 b14 −b9 b8 −b11 b10 −b14 b15 b12 −b13 −b10 b11 b8 −b9 −b15 −b14 b13 b12 −b11 −b10 b9 b8 b0 −b1 −b2 −b3 −b4 −b5 −b6 −b7 b1 b0 b3 −b2 b5 −b4 −b7 b6 b2 −b3 b0 b1 b6 b7 −b4 −b5 b3 b2 −b1 b0 b7 −b6 b5 −b4 b4 −b5 −b6 −b7 b0 b1 b2 b3 b5 b4 −b7 b6 −b1 b0 −b3 b2 b6 b7 b4 −b5 −b2 b3 b0 −b1 b7 −b6 b5 b4 −b3 −b2 b1 b0

              .             

The direct computation of matrix-vector product in Eq. (1) requires 1024 real multiplications and 992 real additions. We shall present the algorithm, which reduce multiplicative complexity to 498 real and additive complexity to 943 additions.

2. Synthesis of a fast algorithm for trigintaduonion multiplication The main idea of proposed solution is based on the fact that the original trigintaduonion multiplication matrix B32 can be decomposed as an algebraic sum of symmetric Toeplitz matrix and another matrix which has many zero elements. The Toeplitz matrix is shift-structured and a number of algorithms exist for “fast” matrix-vector multiplication. Such matrix can be diagonalized with the help of Fast Hadamard transform (FHT) and thus vector-matrix product can be computed more efficiently. The same idea was originally used for the multiplication of quaternions in [26, 44]. We have extended this idea in a our previous papers [9, 43] and [11, 12] on the case of octonions and sedenions respectivelly. In this article we further extend this idea to the case of the trigintadounions. ˜ 32×1 = [−c0 , c1 , c2 , . . . , c31 ]T , and X32×1 = [a0 , a1 , . . . , a31 ]T . Let Y Let us multiply the first row of B32 by (−1). This transformation is done in order to present a modified in this manner matrix as an algebraic sum of the block-symmetric Toeplitz-type matrix and some sparse matrix, i.e. matrix containing only small number of non-zero elements. We can easily see that this transformation leads in the future to minimize the computational complexity of the final algorithm. Then we can write ˜ 32×1 = B ˇ 32 X32×1 − 2B ˆ 32 X32×1 Y

where "

ˇ 32 B

# " # ˇ (0,0) B ˇ (0,1) ˆ (0,0) B ˆ (0,1) B B 16 16 16 16 ˆ 32 = = , B , ˇ (1,0) B ˇ (1,1) ˆ (1,0) B ˆ (1,1) B B 16 16 16 16

(2)

55

An algorithm for multiplication of trigintaduonions

ˇ (0,0) = B ˇ (1,1) = B 16 16

              =             

b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15

b1 b0 b3 b2 b5 b4 b7 b6 b9 b8 b11 b10 b13 b12 b15 b14

b2 b3 b0 b1 b6 b7 b4 b5 b10 b11 b8 b9 b14 b15 b12 b13

b3 b2 b1 b0 b7 b6 b5 b4 b11 b10 b9 b8 b15 b14 b13 b12

b4 b5 b6 b7 b0 b1 b2 b3 b12 b13 b14 b15 b8 b9 b10 b11

b5 b4 b7 b6 b1 b0 b3 b2 b13 b12 b15 b14 b9 b8 b11 b10

b6 b7 b4 b5 b2 b3 b0 b1 b14 b15 b12 b13 b10 b11 b8 b9

b7 b6 b5 b4 b3 b2 b1 b0 b15 b14 b13 b12 b11 b10 b9 b8

b8 b9 b10 b11 b12 b13 b14 b15 b0 b1 b2 b3 b4 b5 b6 b7

b9 b8 b11 b10 b13 b12 b15 b14 b1 b0 b3 b2 b5 b4 b7 b6

b10 b11 b8 b9 b14 b15 b12 b13 b2 b3 b0 b1 b6 b7 b4 b5

b11 b10 b9 b8 b15 b14 b13 b12 b3 b2 b1 b0 b7 b6 b5 b4

b12 b13 b14 b15 b8 b9 b10 b11 b4 b5 b6 b7 b0 b1 b2 b3

b13 b12 b15 b14 b9 b8 b11 b10 b5 b4 b7 b6 b1 b0 b3 b2

b14 b15 b12 b13 b10 b11 b8 b9 b6 b7 b4 b5 b2 b3 b0 b1

b15 b14 b13 b12 b11 b10 b9 b8 b7 b6 b5 b4 b3 b2 b1 b0



b28 b29 b30 b31 b24 b25 b26 b27 b20 b21 b22 b23 b16 b17 b18 b19

b29 b28 b31 b30 b25 b24 b27 b26 b21 b20 b23 b22 b17 b16 b19 b18

b30 b31 b28 b29 b26 b27 b24 b25 b22 b23 b20 b21 b18 b19 b16 b17

b31 b30 b29 b28 b27 b26 b25 b24 b23 b22 b21 b20 b19 b18 b17 b16



             ,             

ˇ 32 X32×1 = Y ˇ 32×1 = [y0 , y1 , . . . , y31 ]T , B

ˇ (0,1) = B ˇ (1,0) = B 16 16

              =             

b16 b17 b18 b19 b20 b21 b22 b23 b24 b25 b26 b27 b28 b29 b30 b31

b17 b16 b19 b18 b21 b20 b23 b22 b25 b24 b27 b26 b29 b28 b31 b30

b18 b19 b16 b17 b22 b23 b20 b21 b26 b27 b24 b25 b30 b31 b28 b29

b19 b18 b17 b16 b23 b22 b21 b20 b27 b26 b25 b24 b31 b30 b29 b28

b20 b21 b22 b23 b16 b17 b18 b19 b28 b29 b30 b31 b24 b25 b26 b27

b21 b20 b23 b22 b17 b16 b19 b18 b29 b28 b31 b30 b25 b24 b27 b26

b22 b23 b20 b21 b18 b19 b16 b17 b30 b31 b28 b29 b26 b27 b24 b25

b23 b22 b21 b20 b19 b18 b17 b16 b31 b30 b29 b28 b27 b26 b25 b24

b24 b25 b26 b27 b28 b29 b30 b31 b16 b17 b18 b19 b20 b21 b22 b23

b25 b24 b27 b26 b29 b28 b31 b30 b17 b16 b19 b18 b21 b20 b23 b22

b26 b27 b24 b25 b30 b31 b28 b29 b18 b19 b16 b17 b22 b23 b20 b21

b27 b26 b25 b24 b31 b30 b29 b28 b19 b18 b17 b16 b23 b22 b21 b20

             ,             

56

Alexandr Cariow, Galina Cariowa



ˆ (0,0) B 16

             =              

ˆ (0,1) B 16

             =             

b0 0 0 0 0 b3 0 0 0 b5 0 0 0 0 0 b6 0 b9 0 0 0 0 0 b10 0 0 0 b12 0 b15 0 0

0 0 0 b2 0 0 b1 0 b6 b7 b7 0 0 b5 0 0 b10 b11 b11 0 0 b9 0 0 0 0 0 b14 b12 0 b13 b12

0 0 0 0 b4 b7 0 0 b4 0 b6 0 0 0 0 b1 0 b3 b2 0 0 b3 b2 0 b12 b13 b14 b13 0 0 b14 b15 0 b15 0 b13 0 b9 b10 0 0 0 0 b11 0 0 0 b9

b16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b18 b19 b16 0 0 0 b20 b21 0 0 b24 b25 b30 b31 0 0

b20 b21 b22 b23 b16 0 0 0 0 0 0 0 b24 b25 b26 b27

b17 b16 0 b18 0 b20 b23 0 0 b24 b27 0 b29 0 0 b30

b19 0 b17 b16 0 b22 0 b20 0 b26 0 b24 b31 0 b29 0

b21 0 b23 0 b17 b16 b19 0 0 b28 0 b30 0 b24 0 b26

b22 0 0 b21 b18 0 b16 b17 0 b31 b28 0 0 b27 b24 0

0 0 b5 b4 0 0 b1 0 b15 b14 0 0 b11 b10 0 0 b23 b22 0 0 b19 b18 0 b16 0 0 b29 b28 0 0 b25 b24

0 0 0 0 0 0 0 0 0 b8 b11 0 b13 0 0 b14 0 0 b8 b9 b14 b15 0 0 0 b10 0 b8 b15 0 b13 0 0 0 0 0 b8 b9 b10 b11 0 b12 0 b14 0 b8 0 b10 0 b15 b12 0 0 b11 b8 0 0 0 b13 b12 0 0 b9 b8 0 0 0 0 0 0 0 0 b1 0 b3 0 b5 0 0 b6 b2 0 0 b1 b6 b7 0 0 b3 b2 0 0 b7 0 b5 0 b4 0 0 0 0 b1 b2 b3 b5 b4 0 b 6 0 0 0 b2 b6 b7 b4 0 0 b3 0 0 b7 0 b5 b 4 0 0 b1 0



b24 b25 b26 b27 b28 b29 b30 b31 b16 0 0 0 0 0 0 0



b25 0 b27 0 b29 0 0 b30 b17 b16 b19 0 b21 0 0 b22

b26 0 0 b25 b30 b31 0 0 b18 0 b16 b17 b22 b23 0 0

b27 b26 0 0 b31 0 b29 0 b19 b18 0 b16 b23 0 b21 0

b28 0 0 0 0 b25 b26 b27 b20 0 0 0 b16 b17 b18 b19

b29 b28 0 b30 0 0 0 b26 b21 b20 0 b22 0 b16 0 b18

b30 b31 b28 0 0 b27 0 0 b22 b23 b20 0 0 b19 b16 0

b31 0 b29 b28 0 0 b25 0 b23 0 b21 b20 0 0 b17 b16

             ,             

             ,             

57

An algorithm for multiplication of trigintaduonions



ˆ (1,0) B

             =             

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

b17 0 0 b18 0 b20 b23 0 0 b24 b27 0 b29 0 0 b30

b18 b19 0 0 0 0 b20 b21 0 0 b24 b25 b30 b31 0 0

b19 0 b17 0 0 b22 0 b20 0 b26 0 b24 b31 0 b29 0

b20 b21 b22 b23 0 0 0 0 0 0 0 0 b24 b25 b26 b27

b21 0 b23 0 b17 0 b19 0 0 b28 0 b30 0 b24 0 b26

b22 0 0 b21 b18 0 0 b17 0 b31 b28 0 0 b27 b24 0

b23 b22 0 0 b19 b18 0 0 0 0 b29 b28 0 0 b25 b24

b24 b25 b26 b27 b28 b29 b30 b31 0 0 0 0 0 0 0 0

b25 0 b27 0 b29 0 0 b30 b17 0 b19 0 b21 0 0 b22

b26 0 0 b25 b30 b31 0 0 b18 0 0 b17 b22 b23 0 0

b27 b26 0 0 b31 0 b29 0 b19 b18 0 0 b23 0 b21 0

b28 0 0 0 0 b25 b26 b27 b20 0 0 0 0 b17 b18 b19

b29 b28 0 b30 0 0 0 b26 b21 b20 0 b22 0 0 0 b18

b30 b31 b28 0 0 b27 0 0 b22 b23 b20 0 0 b19 0 0



0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  b1 0 b3 0 b5 0 0 b6 b9 0 0 b10 0 b12 b15   b2 0 0 b b b 0 0 b b 0 0 0 0 b12 1 6 7 10 11   b3 b2 0 0 b7 0 b5 0 b11 0 b9 0 0 b14 0   b4 0 0 0 0 b1 b2 b3 b12 b13 b14 b15 0 0 0   b5 b 4 0 b6 0 0 0 b b 0 b 0 b 0 b 2 13 15 9 11   b6 b 7 b4 0 0 b 0 0 b 0 0 b b 0 0 3 14 13 10   b7 0 b5 b4 0 0 b1 0 b15 b14 0 0 b11 b10 0 (1,1) ˆ B =  b8 0 0 0 0 0 0 0 0 b1 b2 b3 b 4 b5 b6   b9 b8 0 b10 0 b12 b15 0 0 0 0 b2 0 b4 b7   b10 b11 b8 0 0 0 b b 0 b 0 0 0 0 b4 12 13 3   b11 0 b9 b8 0 b14 0 b12 0 0 b1 0 0 b6 0   b12 b13 b14 b15 b8 0 0 0 0 b b b 0 0 0 5 6 7   b13 0 b15 0 b9 b8 b11 0 0 0 b7 0 b1 0 b3   b14 0 0 b13 b10 0 b8 b9 0 0 0 b5 b2 0 0 b15 b14 0 0 b11 b10 0 b8 0 b6 0 0 b3 b2 0 ˇ 32 X32×1 (with Toeplitz-type matrix) can In this case the vector-matrix product B calculated using one of the well-known fast algorithms. ˇ 32 has the following structure: Indeed, it is easy to see that B " # ˇ (0) B ˇ (1) B (0) (0,0) (1,1) (1) (0,1) (1,0) 16 16 ˇ 32 = B = B16 , B16 = B16 = B16 . (1) (0) , B16 = B16 ˇ ˇ B16 B16

The matrix having such a structure can be decomposed as [44]: # " # " #" " IN IN 1 AN + BN 0N IN AN BN 2 2 2 2 2 2 2 = 2 CN = BN AN I N −I N 2 0N BN − AN IN 2

2

2

2

2

2

2

2

b31 0 b29 b28 0 0 b25 0 b23 0 b21 b20 0 0 b17 0



0 0 b13 b12 0 0 b9 0 b7 0 b5 b4 0 0 b1 0



             ,             

             .             

now be

# IN 2 . −I N 2

ˇ 32 X32×1 with the following decomposition: Then we can calculate the product of B ˇ 32 X32×1 = W(0) D(0) W(0) X32×1 B 32 32 32

(3)

58

Alexandr Cariow, Galina Cariowa

where

(0) W32

= H2 ⊗ I16 ,

(0) D32

1 h ˇ (0) ˇ (1)   ˇ (0) ˇ (1) i B16 + B16 ⊕ B16 − B16 , = 2



 1 1 H2 = is the order 2 Hadamard matrix, IN is the order N identity matrix, and “⊗”, 1 −1 “⊕” – denote the Kronecker product and direct sum of two matrices respectively [39]. Let ˘bi = bi + bi+16 and ˜bi = bi − bi+16 , i = 0, 1, 2, . . . , 15. Then

ˇ (0) + B ˇ (1) B 16 16

# " (1) (0) ˇ ˇ B8 B8 , = ˇ (0) ˇ (1) B B 8 8

ˇ (0) − B ˇ (1) B 16 16

˘b4 ˘b5 ˘b6 ˘b7 ˘b0 ˘b1 ˘b2 ˘b3 ˘b12 ˘b13 ˘b14 ˘b15 ˘b8 ˘b9 ˘b10 ˘b11

˘b8 ˘b9 ˘b10 ˘b11 ˘b12 ˘b13 ˘b14 ˘b15 ˘b0 ˘b1 ˘b2 ˘b3 ˘b4 ˘b5 ˘b6 ˘b7

# (3) (2) ˇ ˇ B8 B8 , = ˇ (2) ˇ (3) B B 8 8 "

ˇ (0) + B ˇ (1) = B 16 16

                =              

˘b0 ˘b1 ˘b2 ˘b3 ˘b4 ˘b5 ˘b6 ˘b7 ˘b8 ˘b9 ˘b10 ˘b11 ˘b12 ˘b13 ˘b14 ˘b15

˘b1 ˘b0 ˘b3 ˘b2 ˘b5 ˘b4 ˘b7 ˘b6 ˘b9 ˘b8 ˘b11 ˘b10 ˘b13 ˘b12 ˘b15 ˘b14

˘b2 ˘b3 ˘b0 ˘b1 ˘b6 ˘b7 ˘b4 ˘b5 ˘b10 ˘b11 ˘b8 ˘b9 ˘b14 ˘b15 ˘b12 ˘b13

˘b3 ˘b2 ˘b1 ˘b0 ˘b7 ˘b6 ˘b5 ˘b4 ˘b11 ˘b10 ˘b9 ˘b8 ˘b15 ˘b14 ˘b13 ˘b12

˘b5 ˘b4 ˘b7 ˘b6 ˘b1 ˘b0 ˘b3 ˘b2 ˘b13 ˘b12 ˘b15 ˘b14 ˘b9 ˘b8 ˘b11 ˘b10

˘b6 ˘b7 ˘b4 ˘b5 ˘b2 ˘b3 ˘b0 ˘b1 ˘b14 ˘b15 ˘b12 ˘b13 ˘b10 ˘b11 ˘b8 ˘b9

˘b7 ˘b6 ˘b5 ˘b4 ˘b3 ˘b2 ˘b1 ˘b0 ˘b15 ˘b14 ˘b13 ˘b12 ˘b11 ˘b10 ˘b9 ˘b8

˘b9 ˘b8 ˘b11 ˘b10 ˘b13 ˘b12 ˘b15 ˘b14 ˘b1 ˘b0 ˘b3 ˘b2 ˘b5 ˘b4 ˘b7 ˘b6

˘b10 ˘b11 ˘b8 ˘b9 ˘b14 ˘b15 ˘b12 ˘b13 ˘b2 ˘b3 ˘b0 ˘b1 ˘b6 ˘b7 ˘b4 ˘b5

˘b11 ˘b10 ˘b9 ˘b8 ˘b15 ˘b14 ˘b13 ˘b12 ˘b3 ˘b2 ˘b1 ˘b0 ˘b7 ˘b6 ˘b5 ˘b4

˘b12 ˘b13 ˘b14 ˘b15 ˘b8 ˘b9 ˘b10 ˘b11 ˘b4 ˘b5 ˘b6 ˘b7 ˘b0 ˘b1 ˘b2 ˘b3

˘b13 ˘b12 ˘b15 ˘b14 ˘b9 ˘b8 ˘b11 ˘b10 ˘b5 ˘b4 ˘b7 ˘b6 ˘b1 ˘b0 ˘b3 ˘b2

˘b14 ˘b15 ˘b12 ˘b13 ˘b10 ˘b11 ˘b8 ˘b9 ˘b6 ˘b7 ˘b4 ˘b5 ˘b2 ˘b3 ˘b0 ˘b1

˘b15 ˘b14 ˘b13 ˘b12 ˘b11 ˘b10 ˘b9 ˘b8 ˘b7 ˘b6 ˘b5 ˘b4 ˘b3 ˘b2 ˘b1 ˘b0

                ,              

59

An algorithm for multiplication of trigintaduonions

ˇ (0) − B ˇ (1) = B 16 16

               =              

˜b0 ˜b1 ˜b2 ˜b3 ˜b4 ˜b5 ˜b6 ˜b7 ˜b8 ˜b9 ˜b10 ˜b11 ˜b12 ˜b13 ˜b14 ˜b15

˜b1 ˜b0 ˜b3 ˜b2 ˜b5 ˜b4 ˜b7 ˜b6 ˜b9 ˜b8 ˜b11 ˜b10 ˜b13 ˜b12 ˜b15 ˜b14

˜b2 ˜b3 ˜b0 ˜b1 ˜b6 ˜b7 ˜b4 ˜b5 ˜b10 ˜b11 ˜b8 ˜b9 ˜b14 ˜b15 ˜b12 ˜b13

˜b3 ˜b2 ˜b1 ˜b0 ˜b7 ˜b6 ˜b5 ˜b4 ˜b11 ˜b10 ˜b9 ˜b8 ˜b15 ˜b14 ˜b13 ˜b12

˜b4 ˜b5 ˜b6 ˜b7 ˜b0 ˜b1 ˜b2 ˜b3 ˜b12 ˜b13 ˜b14 ˜b15 ˜b8 ˜b9 ˜b10 ˜b11

˜b5 ˜b4 ˜b7 ˜b6 ˜b1 ˜b0 ˜b3 ˜b2 ˜b13 ˜b12 ˜b15 ˜b14 ˜b9 ˜b8 ˜b11 ˜b10

˜b6 ˜b7 ˜b4 ˜b5 ˜b2 ˜b3 ˜b0 ˜b1 ˜b14 ˜b15 ˜b12 ˜b13 ˜b10 ˜b11 ˜b8 ˜b9

˜b7 ˜b6 ˜b5 ˜b4 ˜b3 ˜b2 ˜b1 ˜b0 ˜b15 ˜b14 ˜b13 ˜b12 ˜b11 ˜b10 ˜b9 ˜b8

˜b8 ˜b9 ˜b10 ˜b11 ˜b12 ˜b13 ˜b14 ˜b15 ˜b0 ˜b1 ˜b2 ˜b3 ˜b4 ˜b5 ˜b6 ˜b7

˜b9 ˜b8 ˜b11 ˜b10 ˜b13 ˜b12 ˜b15 ˜b14 ˜b1 ˜b0 ˜b3 ˜b2 ˜b5 ˜b4 ˜b7 ˜b6

˜b10 ˜b11 ˜b8 ˜b9 ˜b14 ˜b15 ˜b12 ˜b13 ˜b2 ˜b3 ˜b0 ˜b1 ˜b6 ˜b7 ˜b4 ˜b5

˜b11 ˜b10 ˜b9 ˜b8 ˜b15 ˜b14 ˜b13 ˜b12 ˜b3 ˜b2 ˜b1 ˜b0 ˜b7 ˜b6 ˜b5 ˜b4

˜b12 ˜b13 ˜b14 ˜b15 ˜b8 ˜b9 ˜b10 ˜b11 ˜b4 ˜b5 ˜b6 ˜b7 ˜b0 ˜b1 ˜b2 ˜b3

˜b13 ˜b12 ˜b15 ˜b14 ˜b9 ˜b8 ˜b11 ˜b10 ˜b5 ˜b4 ˜b7 ˜b6 ˜b1 ˜b0 ˜b3 ˜b2

˜b14 ˜b15 ˜b12 ˜b13 ˜b10 ˜b11 ˜b8 ˜b9 ˜b6 ˜b7 ˜b4 ˜b5 ˜b2 ˜b3 ˜b0 ˜b1

˜b15 ˜b14 ˜b13 ˜b12 ˜b11 ˜b10 ˜b9 ˜b8 ˜b7 ˜b6 ˜b5 ˜b4 ˜b3 ˜b2 ˜b1 ˜b0

               .              

   (1) (0) (1) (0) ˇ ˇ ˇ ˇ It can be seen that the matrices B16 + B16 and B16 − B16 have the same type of block-structural symmetry that provides “good” decomposition too: 

# (1) (0) ˇ ˇ B8 B8 , = ˇ (0) ˇ (1) B B 8 8

ˇ (0) + B ˇ (1) B 16 16

# (1) (0) ˜ ˜ B8 B8 , = ˜ (0) ˜ (1) B B 8 8 "

"

ˇ (0) − B ˇ (1) B 16 16

where

ˇ (0) B 8

˜ (0) B 8

˘ b0 ˘b1  ˘b  2 ˘ b = ˘3 b4  ˘b5  ˘b6 ˘b7  ˜b0 ˜b  1 ˜b  2 ˜ b = ˜3 b4 ˜ b5  ˜b6 ˜b7

˘b1 ˘b0 ˘b3 ˘b2 ˘b5 ˘b4 ˘b7 ˘b6

˘b2 ˘b3 ˘b0 ˘b1 ˘b6 ˘b7 ˘b4 ˘b5

˘b3 ˘b2 ˘b1 ˘b0 ˘b7 ˘b6 ˘b5 ˘b4

˘b4 ˘b5 ˘b6 ˘b7 ˘b0 ˘b1 ˘b2 ˘b3

˘b5 ˘b4 ˘b7 ˘b6 ˘b1 ˘b0 ˘b3 ˘b2

˘b6 ˘b7 ˘b4 ˘b5 ˘b2 ˘b3 ˘b0 ˘b1

˜b1 ˜b0 ˜b3 ˜b2 ˜b5 ˜b4 ˜b7 ˜b6

˜b2 ˜b3 ˜b0 ˜b1 ˜b6 ˜b7 ˜b4 ˜b5

˜b3 ˜b2 ˜b1 ˜b0 ˜b7 ˜b6 ˜b5 ˜b4

˜b4 ˜b5 ˜b6 ˜b7 ˜b0 ˜b1 ˜b2 ˜b3

˜b5 ˜b4 ˜b7 ˜b6 ˜b1 ˜b0 ˜b3 ˜b2

˜b6 ˜b7 ˜b4 ˜b5 ˜b2 ˜b3 ˜b0 ˜b1

˘b7  ˘b6   ˘b5   ˘b4   , ˘b3    ˘b2   ˘b1  ˘b0  ˜b7 ˜b6   ˜b5   ˜b4   , ˜b3    ˜b2   ˜b1  ˜b0

ˇ (1) B 8

˜ (1) B 8

˘ b8  ˘b9  ˘b  10 ˘ b = ˘11 b12  ˘b13  ˘b14 ˘b15  ˜b8  ˜b  9 ˜b  10 ˜ b = ˜11 b12 ˜ b13  ˜b14 ˜b15

˘b9 ˘b8 ˘b11 ˘b10 ˘b13 ˘b12 ˘b15 ˘b14

˘b10 ˘b11 ˘b8 ˘b9 ˘b14 ˘b15 ˘b12 ˘b13

˘b11 ˘b10 ˘b9 ˘b8 ˘b15 ˘b14 ˘b13 ˘b12

˘b12 ˘b13 ˘b14 ˘b15 ˘b8 ˘b9 ˘b10 ˘b11

˘b13 ˘b12 ˘b15 ˘b14 ˘b9 ˘b8 ˘b11 ˘b10

˘b14 ˘b15 ˘b12 ˘b13 ˘b10 ˘b11 ˘b8 ˘b9

˜b9 ˜b8 ˜b11 ˜b10 ˜b13 ˜b12 ˜b15 ˜b14

˜b10 ˜b11 ˜b8 ˜b9 ˜b14 ˜b15 ˜b12 ˜b13

˜b11 ˜b10 ˜b9 ˜b8 ˜b15 ˜b14 ˜b13 ˜b12

˜b12 ˜b13 ˜b14 ˜b15 ˜b8 ˜b9 ˜b10 ˜b11

˜b13 ˜b12 ˜b15 ˜b14 ˜b9 ˜b8 ˜b11 ˜b10

˜b14 ˜b15 ˜b12 ˜b13 ˜b10 ˜b11 ˜b8 ˜b9

˘b15  ˘b14   ˘b13   ˘b12   , ˘b11    ˘b10   ˘b9  ˘b8  ˜b15 ˜b14   ˜b13   ˜b12   , ˜b11    ˜b10   ˜b9  ˜b8

60

Alexandr Cariow, Galina Cariowa

Therefore, similar to previous we can write: ˇ (0) + B ˇ (1) = W16 D(0) W16 , B ˇ (0) − B ˇ (1) = W16 D(1) W16 , B 16 16 16 16 16 16 h   i 1 ˇ (0) ˇ (1) (0) (0) (1) ˇ ˇ W16 = H2 ⊗ I8 , D16 = B8 + B8 ⊕ B8 − B8 , 2 1 h ˇ (2) ˇ (3)   ˇ (2) ˇ (3) i (1) B8 + B8 ⊕ B8 − B8 . D16 = 2 Combining partial decompositions in a singe procedure we can rewrite (3) as follows: ˇ 32 X32×1 = W(0) W(1) D(1) W(1) W(0) X32×1 B 32 32 32 32 32 where (1) W32

= I2 ⊗ W16 = I2 ⊗ H2 ⊗ I8 ,

(1) D32

(4)

" # 1 D(0) 0 16 16 , = 4 016 D(1) 16

ˇ (0) + B ˇ (1) = B 8 8

˘ b0 + ˘b8  ˘b1 + ˘b9  ˘b + ˘b 10  2 ˘ ˘ b + b = ˘3 ˘11 b4 + b12  ˘b5 + ˘b13  ˘b6 + ˘b14 ˘b7 + ˘b15

˘b1 + ˘b9 ˘b0 + ˘b8 ˘b3 + ˘b11 ˘b2 + ˘b10 ˘b5 + ˘b13 ˘b4 + ˘b12 ˘b7 + ˘b15 ˘b6 + ˘b14

˘b2 + ˘b10 ˘b3 + ˘b11 ˘b0 + ˘b8 ˘b1 + ˘b9 ˘b6 + ˘b14 ˘b7 + ˘b15 ˘b4 + ˘b12 ˘b5 + ˘b13

˘b3 + ˘b11 ˘b2 + ˘b10 ˘b1 + ˘b9 ˘b0 + ˘b8 ˘b7 + ˘b15 ˘b6 + ˘b14 ˘b5 + ˘b13 ˘b4 + ˘b12

˘b4 + ˘b12 ˘b5 + ˘b13 ˘b6 + ˘b14 ˘b7 + ˘b15 ˘b0 + ˘b8 ˘b1 + ˘b9 ˘b2 + ˘b10 ˘b3 + ˘b11

˘b5 + ˘b13 ˘b4 + ˘b12 ˘b7 + ˘b15 ˘b6 + ˘b14 ˘b1 + ˘b9 ˘b0 + ˘b8 ˘b3 + ˘b11 ˘b2 + ˘b10

˘b6 + ˘b14 ˘b7 + ˘b15 ˘b4 + ˘b12 ˘b5 + ˘b13 ˘b2 + ˘b10 ˘b3 + ˘b11 ˘b0 + ˘b8 ˘b1 + ˘b9

˘b7 + ˘b15  ˘b6 + ˘b14   ˘b5 + ˘b13   ˘b4 + ˘b12   , ˘b3 + ˘b11    ˘b2 + ˘b10   ˘b1 + ˘b9  ˘b0 + ˘b8

˘b1 − ˘b9 ˘b0 − ˘b8 ˘b3 − ˘b11 ˘b2 − ˘b10 ˘b5 − ˘b13 ˘b4 − ˘b12 ˘b7 − ˘b15 ˘b6 − ˘b14

˘b2 − ˘b10 ˘b3 − ˘b11 ˘b0 − ˘b8 ˘b1 − ˘b9 ˘b6 − ˘b14 ˘b7 − ˘b15 ˘b4 − ˘b12 ˘b5 − ˘b13

˘b3 − ˘b11 ˘b2 − ˘b10 ˘b1 − ˘b9 ˘b0 − ˘b8 ˘b7 − ˘b15 ˘b6 − ˘b14 ˘b5 − ˘b13 ˘b4 − ˘b12

˘b4 − ˘b12 ˘b5 − ˘b13 ˘b6 − ˘b14 ˘b7 − ˘b15 ˘b0 − ˘b8 ˘b1 − ˘b9 ˘b2 − ˘b10 ˘b3 − ˘b11

˘b5 − ˘b13 ˘b4 − ˘b12 ˘b7 − ˘b15 ˘b6 − ˘b14 ˘b1 − ˘b9 ˘b0 − ˘b8 ˘b3 − ˘b11 ˘b2 − ˘b10

˘b6 − ˘b14 ˘b7 − ˘b15 ˘b4 − ˘b12 ˘b5 − ˘b13 ˘b2 − ˘b10 ˘b3 − ˘b11 ˘b0 − ˘b8 ˘b1 − ˘b9

˘b7 − ˘b15  ˘b6 − ˘b14   ˘b5 − ˘b13   ˘b4 − ˘b12   , ˘b3 − ˘b11    ˘b2 − ˘b10   ˘b1 − ˘b9  ˘b0 − ˘b8

˜b1 + ˜b9 ˜b0 + ˜b8 ˜b3 + ˜b11 ˜b2 + ˜b10 ˜b5 + ˜b13 ˜b4 + ˜b12 ˜b7 + ˜b15 ˜b6 + ˜b14

˜b2 + ˜b10 ˜b3 + ˜b11 ˜b0 + ˜b8 ˜b1 + ˜b9 ˜b6 + ˜b14 ˜b7 + ˜b15 ˜b4 + ˜b12 ˜b5 + ˜b13

˜b3 + ˜b11 ˜b2 + ˜b10 ˜b1 + ˜b9 ˜b0 + ˜b8 ˜b7 + ˜b15 ˜b6 + ˜b14 ˜b5 + ˜b13 ˜b4 + ˜b12

˜b4 + ˜b12 ˜b5 + ˜b13 ˜b6 + ˜b14 ˜b7 + ˜b15 ˜b0 + ˜b8 ˜b1 + ˜b9 ˜b2 + ˜b10 ˜b3 + ˜b11

˜b5 + ˜b13 ˜b4 + ˜b12 ˜b7 + ˜b15 ˜b6 + ˜b14 ˜b1 + ˜b9 ˜b0 + ˜b8 ˜b3 + ˜b11 ˜b2 + ˜b10

˜b6 + ˜b14 ˜b7 + ˜b15 ˜b4 + ˜b12 ˜b5 + ˜b13 ˜b2 + ˜b10 ˜b3 + ˜b11 ˜b0 + ˜b8 ˜b1 + ˜b9

 ˜b7 + ˜b15 ˜b6 + ˜b14   ˜b5 + ˜b13   ˜b4 + ˜b12   , ˜b3 + ˜b11   ˜b2 + ˜b10    ˜b1 + ˜b9  ˜b0 + ˜b8

ˇ (0) − B ˇ (1) = B 8 8

˘ b0 − ˘b8  ˘b1 − ˘b9  ˘b − ˘b 10  2 ˘ ˘ b − b = ˘3 ˘11 b4 − b12  ˘b5 − ˘b13  ˘b6 − ˘b14 ˘b7 − ˘b15 ˇ (2) + B ˇ (3) = B 8 8

˜b0 + ˜b8  ˜b + ˜b 9  1 ˜b + ˜b 10  2 ˜ b3 + ˜b11 = ˜ b4 + ˜b12 ˜ b5 + ˜b13  ˜b6 + ˜b14 ˜b7 + ˜b15 

61

An algorithm for multiplication of trigintaduonions

ˇ (2) − B ˇ (3) = B 8 8

˜b0 − ˜b8  ˜b − ˜b 9  1 ˜b − ˜b 10  2 ˜ ˜ b − b = ˜3 ˜11 b4 − b12 ˜ b5 − ˜b13  ˜b6 − ˜b14 ˜b7 − ˜b15 

˜b1 − ˜b9 ˜b0 − ˜b8 ˜b3 − ˜b11 ˜b2 − ˜b10 ˜b5 − ˜b13 ˜b4 − ˜b12 ˜b7 − ˜b15 ˜b6 − ˜b14

˜b2 − ˜b10 ˜b3 − ˜b11 ˜b0 − ˜b8 ˜b1 − ˜b9 ˜b6 − ˜b14 ˜b7 − ˜b15 ˜b4 − ˜b12 ˜b5 − ˜b13

˜b3 − ˜b11 ˜b2 − ˜b10 ˜b1 − ˜b9 ˜b0 − ˜b8 ˜b7 − ˜b15 ˜b6 − ˜b14 ˜b5 − ˜b13 ˜b4 − ˜b12

˜b4 − ˜b12 ˜b5 − ˜b13 ˜b6 − ˜b14 ˜b7 − ˜b15 ˜b0 − ˜b8 ˜b1 − ˜b9 ˜b2 − ˜b10 ˜b3 − ˜b11

˜b5 − ˜b13 ˜b4 − ˜b12 ˜b7 − ˜b15 ˜b6 − ˜b14 ˜b1 − ˜b9 ˜b0 − ˜b8 ˜b3 − ˜b11 ˜b2 − ˜b10

˜b6 − ˜b14 ˜b7 − ˜b15 ˜b4 − ˜b12 ˜b5 − ˜b13 ˜b2 − ˜b10 ˜b3 − ˜b11 ˜b0 − ˜b8 ˜b1 − ˜b9

 ˜b7 − ˜b15 ˜b6 − ˜b14   ˜b5 − ˜b13   ˜b4 − ˜b12   . ˜b3 − ˜b11   ˜b2 − ˜b10    ˜b1 − ˜b9  ˜b0 − ˜b8

ˇ (0) + B ˇ (1) , B ˇ (0) − B ˇ (1) , B ˇ (2) + B ˇ (3) and B ˇ (2) − B ˇ (3) It is easy to see that the matrices B 8 8 8 8 8 8 8 8 possess a structures that provide effective decomposition again: " # " # (0) (1) (2) (3) ˇ ˇ ˇ ˇ ˇ (0) + B ˇ (1) = B4 B4 , B ˇ (0) − B ˇ (1) = B4 B4 , B 8 8 8 8 (1) (0) ˇ ˇ ˇ (3) B ˇ (2) B4 B4 B 4 4 # # " " (6) (7) (4) (5) ˇ ˇ ˇ ˇ B B B B 4 4 4 4 ˇ (2) − B ˇ (3) = ˇ (2) + B ˇ (3) = , , B B 8 8 8 8 ˇ (7) B ˇ (6) ˇ (5) B ˇ (4) B B 4 4 4 4   ˘b0 + ˘b8 ˘b1 + ˘b9 ˘b2 + ˘b10 ˘b3 + ˘b11  ˘ b1 + ˘b9 ˘b0 + ˘b8 ˘b3 + ˘b11 ˘b2 + ˘b10  ˇ (0) =  B ,  4 ˘b2 + ˘b10 ˘b3 + ˘b11 ˘b0 + ˘b8 ˘b1 + ˘b9  ˘b3 + ˘b11 ˘b2 + ˘b10 ˘b1 + ˘b9 ˘b0 + ˘b8   ˘b4 + ˘b12 ˘b5 + ˘b13 ˘b6 + ˘b14 ˘b7 + ˘b15  ˘ b5 + ˘b13 ˘b4 + ˘b12 ˘b7 + ˘b15 ˘b6 + ˘b14  ˇ (1) =  B , ˘ 4 b6 + ˘b14 ˘b7 + ˘b15 ˘b4 + ˘b12 ˘b5 + ˘b13  ˘b7 + ˘b15 ˘b6 + ˘b14 ˘b5 + ˘b13 ˘b4 + ˘b12   ˘b0 − ˘b8 ˘b1 − ˘b9 ˘b2 − ˘b10 ˘b3 − ˘b11  ˘  b1 − ˘b9 ˘b0 − ˘b8 ˘b3 − ˘b11 ˘b2 − ˘b10  (2) ˇ B4 = ˘ , b2 − ˘b10 ˘b3 − ˘b11 ˘b0 − ˘b8 ˘b1 − ˘b9  ˘b3 − ˘b11 ˘b2 − ˘b10 ˘b1 − ˘b9 ˘b0 − ˘b8   ˘b4 − ˘b12 ˘b5 − ˘b13 ˘b6 − ˘b14 ˘b7 − ˘b15 ˘  b5 − ˘b13 ˘b4 − ˘b12 ˘b7 − ˘b15 ˘b6 − ˘b14  ˇ (3) =  B  , 4 ˘b6 − ˘b14 ˘b7 − ˘b15 ˘b4 − ˘b12 ˘b5 − ˘b13  ˘b7 − ˘b15 ˘b6 − ˘b14 ˘b5 − ˘b13 ˘b4 − ˘b12  ˜ b0 + ˜b8 ˜b1 + ˜b9 ˜b2 + ˜b10 ˜b3 + ˜b11 ˜ ˜ ˜ ˜ ˜ ˜ ˜  ˜ ˇ (4) =  b1 + b9 b0 + b8 b3 + b11 b2 + b10  , B 4 ˜b2 + ˜b10 ˜b3 + ˜b11 ˜b0 + ˜b8 ˜b1 + ˜b9  ˜b3 + ˜b11 ˜b2 + ˜b10 ˜b1 + ˜b9 ˜b0 + ˜b8 ˜  b4 + ˜b12 ˜b5 + ˜b13 ˜b6 + ˜b14 ˜b7 + ˜b15 ˜ ˜ ˜ ˜ ˜ ˜ ˜  ˜ ˇ (5) = b5 + b13 b4 + b12 b7 + b15 b6 + b14  , B 4 ˜b6 + ˜b14 ˜b7 + ˜b15 ˜b4 + ˜b12 ˜b5 + ˜b13  ˜b7 + ˜b15 ˜b6 + ˜b14 ˜b5 + ˜b13 ˜b4 + ˜b12

62

Alexandr Cariow, Galina Cariowa

ˇ (6) B 4

ˇ (7) B 4

˜ b0 − ˜b8  ˜b1 − ˜b9 = ˜b2 − ˜b10 ˜b3 − ˜b11 ˜ b4 − ˜b12 ˜b5 − ˜b13 = ˜b6 − ˜b14 ˜b7 − ˜b15

˜b1 − ˜b9 ˜b0 − ˜b8 ˜b3 − ˜b11 ˜b2 − ˜b10

˜b2 − ˜b10 ˜b3 − ˜b11 ˜b0 − ˜b8 ˜b1 − ˜b9

˜b5 − ˜b13 ˜b4 − ˜b12 ˜b7 − ˜b15 ˜b6 − ˜b14

˜b6 − ˜b14 ˜b7 − ˜b15 ˜b4 − ˜b12 ˜b5 − ˜b13

˜b3 − ˜b11  ˜b2 − ˜b10   ˜b1 − ˜b9  , ˜b0 − ˜b8 ˜b7 − ˜b15  ˜b6 − ˜b14   ˜b5 − ˜b13  . ˜b4 − ˜b12

Then we can write: ˇ (0) + B ˇ (1) = W8 D(0) W8 , B 8 8 8

ˇ (0) − B ˇ (1) = W8 D(1) W8 , B 8 8 8

ˇ (2) + B ˇ (3) = W8 D(2) W8 , B 8 8 8

ˇ (2) − B ˇ (3) = W8 D(3) W8 , B 8 8 8

W8 = H2 ⊗ I4 , i i   h (0) ˇ (1) , D(1) = 1 B ˇ (3) , ˇ (1) ⊕ B ˇ (0) − B ˇ (3) ⊕ B ˇ (2) − B ˇ (0) + B ˇ (2) + B D8 = B 8 4 4 4 4 4 4 4 4 2 2 1 h ˇ (4) ˇ (5)   ˇ (4) ˇ (5) i 1 h ˇ (6) ˇ (7)   ˇ (6) ˇ (7) i (2) (3) D8 = , D8 = . B4 + B4 ⊕ B4 − B4 B4 + B4 ⊕ B4 − B4 2 2 Combining partial decompositions in a single procedure again we can rewrite (4) as follows: ˇ 32 X32×1 = W(0) W(1) W(2) D(2) W(2) W(1) W(0) X32×1 B (5) 32 32 32 32 32 32 32 1 h





where

  1 (2) (0) (1) (2) (3) D32 = diag D8 , D8 , D8 , D8 . 8 ˇ (0) + B ˇ (1) , B ˇ (0) − B ˇ (1) , B ˇ (2) + B ˇ (3) , B ˇ (2) − B ˇ (3) , B ˇ (4) + B ˇ (5) , B ˇ (4) − B ˇ (5) , Matrices B 4 4 4 4 4 4 4 4 4 4 4 4 ˇ (7) possess mentioned structures. ˇ (6) − B ˇ (7) , B ˇ (6) + B B 4 4 4 4 (2)

W32 = I4 ⊗ W8 ,

ˇ (0) + B ˇ (1) = B 4 4  ˘b0 + ˘b8 + ˘b4 + ˘b12 ˘b1 + ˘b9 + ˘b5 + ˘b13 ˘ ˘b + ˘b + ˘b + ˘b  b + ˘b + ˘b + ˘b = ˘ 1 ˘ 9 ˘5 ˘13 ˘ 0 ˘ 8 ˘4 ˘12 b2 + b10 + b6 + b14 b3 + b11 + b7 + b15 ˘b3 + ˘b11 + ˘b7 + ˘b15 ˘b2 + ˘b10 + ˘b6 + ˘b14

ˇ (0) − B ˇ (1) = B 4 4  ˘b0 + ˘b8 − ˘b4 − ˘b12 ˘  b + ˘b − ˘b − ˘b = ˘ 1 ˘ 9 ˘5 ˘13 b2 + b10 − b6 − b14 ˘b3 + ˘b11 − ˘b7 − ˘b15

˘b2 + ˘b10 + ˘b6 + ˘b14 ˘b3 + ˘b11 + ˘b7 + ˘b15 ˘b0 + ˘b8 + ˘b4 + ˘b12 ˘b1 + ˘b9 + ˘b5 + ˘b13

 ˘b3 + ˘b11 + ˘b7 + ˘b15 ˘b2 + ˘b10 + ˘b6 + ˘b14   = ˘b1 + ˘b9 + ˘b5 + ˘b13   ˘b0 + ˘b8 + ˘b4 + ˘b12 " # ˇ (0) B ˇ (1) B 2 2 = , ˇ (1) B ˇ (0) B 2 2

 ˘b1 + ˘b9 − ˘b5 − ˘b13 ˘b2 + ˘b10 − ˘b6 − ˘b14 ˘b3 + ˘b11 − ˘b7 − ˘b15 ˘b0 + ˘b8 − ˘b4 − ˘b12 ˘b3 + ˘b11 − ˘b7 − ˘b15 ˘b2 + ˘b10 − ˘b6 − ˘b14   = ˘b3 + ˘b11 − ˘b7 − ˘b15 ˘b0 + ˘b8 − ˘b4 − ˘b12 ˘b1 + ˘b9 − ˘b5 − ˘b13   ˘b2 + ˘b10 − ˘b6 − ˘b14 ˘b1 + ˘b9 − ˘b5 − ˘b13 ˘b0 + ˘b8 − ˘b4 − ˘b12 # " ˇ (2) B ˇ (3) B 2 2 = , ˇ (3) B ˇ (2) B 2 2

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An algorithm for multiplication of trigintaduonions

ˇ (2) + B ˇ (3) = B 4 4  ˘b0 − ˘b8 + ˘b4 − ˘b12 ˘  b − ˘b + ˘b − ˘b = ˘ 1 ˘ 9 ˘5 ˘13 b2 − b10 + b6 − b14 ˘b3 − ˘b11 + ˘b7 − ˘b15

˘b1 − ˘b9 + ˘b5 − ˘b13 ˘b0 − ˘b8 + ˘b4 − ˘b12 ˘b3 − ˘b11 + ˘b7 − ˘b15 ˘b2 − ˘b10 + ˘b6 − ˘b14

˘b2 − ˘b10 + ˘b6 − ˘b14 ˘b3 − ˘b11 + ˘b7 − ˘b15 ˘b0 − ˘b8 + ˘b4 − ˘b12 ˘b1 − ˘b9 + ˘b5 − ˘b13

 ˘b3 − ˘b11 + ˘b7 − ˘b15 ˘b2 − ˘b10 + ˘b6 − ˘b14   = ˘b1 − ˘b9 + ˘b5 − ˘b13   ˘b0 − ˘b8 + ˘b4 − ˘b12 "

# ˇ (4) B ˇ (5) B 2 2 = , ˇ (5) B ˇ (4) B 2 2 ˇ (2) − B ˇ (3) = B 4 4   ˘b0 − ˘b8 − ˘b4 + ˘b12 ˘b1 − ˘b9 − ˘b5 + ˘b13 ˘b2 − ˘b10 − ˘b6 + ˘b14 ˘b3 − ˘b11 − ˘b7 + ˘b15 ˘ ˘b − ˘b − ˘b + ˘b ˘b − ˘b − ˘b + ˘b ˘b − ˘b − ˘b + ˘b    b − ˘b − ˘b + ˘b = ˘ 1 ˘ 9 ˘5 ˘13 ˘ 0 ˘ 8 ˘4 ˘12 ˘3 ˘11 ˘ 7 ˘ 15 ˘2 ˘10 ˘ 6 ˘ 14  = b2 − b10 − b6 + b14 b3 − b11 − b7 + b15 b0 − b8 − b4 + b12 b1 − b9 − b5 + b13  ˘b3 − ˘b11 − ˘b7 + ˘b15 ˘b2 − ˘b10 − ˘b6 + ˘b14 ˘b1 − ˘b9 − ˘b5 + ˘b13 ˘b0 − ˘b8 − ˘b4 + ˘b12 # " (7) (6) ˇ ˇ B2 B2 , = ˇ (7) B ˇ (6) B 2 2 ˇ (4) + B ˇ (5) = B 4 4 ˜ b0 + ˜b8 + ˜b4 + ˜b12  ˜b1 + ˜b9 + ˜b5 + ˜b13 = ˜b2 + ˜b10 + ˜b6 + ˜b14 ˜b3 + ˜b11 + ˜b7 + ˜b15

˜b1 + ˜b9 + ˜b5 + ˜b13 ˜b0 + ˜b8 + ˜b4 + ˜b12 ˜b3 + ˜b11 + ˜b7 + ˜b15 ˜b2 + ˜b10 + ˜b6 + ˜b14

˜b2 + ˜b10 + ˜b6 + ˜b14 ˜b3 + ˜b11 + ˜b7 + ˜b15 ˜b0 + ˜b8 + ˜b4 + ˜b12 ˜b1 + ˜b9 + ˜b5 + ˜b13

˜b3 + ˜b11 + ˜b7 + ˜b15  ˜b2 + ˜b10 + ˜b6 + ˜b14   ˜b1 + ˜b9 + ˜b5 + ˜b13  = ˜b0 + ˜b8 + ˜b4 + ˜b12 # " ˇ (8) B ˇ (9) B 2 2 , = ˇ (9) B ˇ (8) B 2 2

ˇ (4) − B ˇ (5) = B 4 4 ˜ b0 + ˜b8 − ˜b4 − ˜b12  ˜b1 + ˜b9 − ˜b5 − ˜b13 = ˜b2 + ˜b10 − ˜b6 − ˜b14 ˜b3 + ˜b11 − ˜b7 − ˜b15

˜b1 + ˜b9 − ˜b5 − ˜b13 ˜b0 + ˜b8 − ˜b4 − ˜b12 ˜b3 + ˜b11 − ˜b7 − ˜b15 ˜b2 + ˜b10 − ˜b6 − ˜b14

˜b2 + ˜b10 − ˜b6 − ˜b14 ˜b3 + ˜b11 − ˜b7 − ˜b15 ˜b0 + ˜b8 − ˜b4 − ˜b12 ˜b1 + ˜b9 − ˜b5 − ˜b13

˜b3 + ˜b11 − ˜b7 − ˜b15  ˜b2 + ˜b10 − ˜b6 − ˜b14   ˜b1 + ˜b9 − ˜b5 − ˜b13  = ˜b0 + ˜b8 − ˜b4 − ˜b12 " # (10) (11) ˇ ˇ B2 B2 = , (11) ˇ ˇ (10) B2 B 2

ˇ (6) + B ˇ (7) = B 4 4 ˜ b0 − ˜b8 + ˜b4 − ˜b12 ˜  b1 − ˜b9 + ˜b5 − ˜b13 = ˜b2 − ˜b10 + ˜b6 − ˜b14 ˜b3 − ˜b11 + ˜b7 − ˜b15

˜b1 − ˜b9 + ˜b5 − ˜b13 ˜b2 − ˜b10 + ˜b6 − ˜b14 ˜b3 − ˜b11 + ˜b7 − ˜b15  ˜b0 − ˜b8 + ˜b4 − ˜b12 ˜b3 − ˜b11 + ˜b7 − ˜b15 ˜b2 − ˜b10 + ˜b6 − ˜b14   ˜b3 − ˜b11 + ˜b7 − ˜b15 ˜b0 − ˜b8 + ˜b4 − ˜b12 ˜b1 − ˜b9 + ˜b5 − ˜b13  = ˜b2 − ˜b10 + ˜b6 − ˜b14 ˜b1 − ˜b9 + ˜b5 − ˜b13 ˜b0 − ˜b8 + ˜b4 − ˜b12 # " ˇ (12) B ˇ (13) B 2 2 = , ˇ (13) B ˇ (12) B 2 2

64

Alexandr Cariow, Galina Cariowa

ˇ (6) − B ˇ (7) = B 4 4 ˜ b0 − ˜b8 − ˜b4 + ˜b12  ˜b1 − ˜b9 − ˜b5 + ˜b13 = ˜b2 − ˜b10 − ˜b6 + ˜b14 ˜b3 − ˜b11 − ˜b7 + ˜b15

where

˜b1 − ˜b9 − ˜b5 + ˜b13 ˜b0 − ˜b8 − ˜b4 + ˜b12 ˜b3 − ˜b11 − ˜b7 + ˜b15 ˜b2 − ˜b10 − ˜b6 + ˜b14

˜b2 − ˜b10 − ˜b6 + ˜b14 ˜b3 − ˜b11 − ˜b7 + ˜b15 ˜b0 − ˜b8 − ˜b4 + ˜b12 ˜b1 − ˜b9 − ˜b5 + ˜b13

 ˘b + ˘b + ˘b + ˘b (0) ˇ B2 = ˘0 ˘8 ˘4 ˘12 b1 + b9 + b5 + b13  ˘b + ˘b + ˘b + ˘b (1) ˇ B2 = ˘2 ˘10 ˘6 ˘14 b3 + b11 + b7 + b15  ˘b + ˘b − ˘b − ˘b (2) ˇ B2 = ˘0 ˘8 ˘4 ˘12 b1 + b9 − b5 − b13  ˘b + ˘b − ˘b − ˘b (3) ˇ B2 = ˘2 ˘10 ˘6 ˘14 b3 + b11 − b7 − b15  ˘ ˘ ˘ ˘ ˇ (4) = b0 − b8 + b4 − b12 B 2 ˘b1 − ˘b9 + ˘b5 − ˘b13  ˘ ˘ ˘ ˘ (5) ˇ = b2 − b10 + b6 − b14 B 2 ˘b3 − ˘b11 + ˘b7 − ˘b15  ˘ ˘ ˘ ˘ (6) ˇ = b0 − b8 − b4 + b12 B 2 ˘b1 − ˘b9 − ˘b5 + ˘b13  ˘ ˘ ˘ ˘ (7) ˇ = b2 − b10 − b6 + b14 B 2 ˘b3 − ˘b11 − ˘b7 + ˘b15  ˜b + ˜b + ˜b + ˜b (8) ˇ B2 = ˜0 ˜8 ˜4 ˜12 b1 + b9 + b5 + b13  ˜b + ˜b + ˜b + ˜b (9) ˇ B2 = ˜2 ˜10 ˜6 ˜14 b3 + b11 + b7 + b15  ˜ ˜ ˜ ˜ ˇ (10) = b0 + b8 − b4 − b12 B 2 ˜b1 + ˜b9 − ˜b5 − ˜b13  ˜b + ˜b − ˜b − ˜b (11) ˇ B2 = ˜2 ˜10 ˜6 ˜14 b3 + b11 − b7 − b15  ˜b − ˜b + ˜b − ˜b (12) ˇ B2 = ˜0 ˜8 ˜4 ˜12 b1 − b9 + b5 − b13  ˜ ˜ ˜ ˜ ˇ (13) = b2 − b10 + b6 − b14 B 2 ˜b3 − ˜b11 + ˜b7 − ˜b15

˜b3 − ˜b11 − ˜b7 + ˜b15  ˜b2 − ˜b10 − ˜b6 + ˜b14   ˜b1 − ˜b9 − ˜b5 + ˜b13  = ˜b0 − ˜b8 − ˜b4 + ˜b12 # " ˇ (14) B ˇ (15) B 2 2 , = ˇ (15) B ˇ (14) B 2 2

 ˘b1 + ˘b9 + ˘b5 + ˘b13 ˘b0 + ˘b8 + ˘b4 + ˘b12 ,  ˘b3 + ˘b11 + ˘b7 + ˘b15 ˘b2 + ˘b10 + ˘b6 + ˘b14 ,  ˘b1 + ˘b9 − ˘b5 − ˘b13 ˘b0 + ˘b8 − ˘b4 − ˘b12 ,  ˘b3 + ˘b11 − ˘b7 − ˘b15 ˘b2 + ˘b10 − ˘b6 − ˘b14 ,  ˘b1 − ˘b9 + ˘b5 − ˘b13 ˘b0 − ˘b8 + ˘b4 − ˘b12 ,  ˘b3 − ˘b11 + ˘b7 − ˘b15 ˘b2 − ˘b10 + ˘b6 − ˘b14 ,  ˘b1 − ˘b9 − ˘b5 + ˘b13 ˘b0 − ˘b8 − ˘b4 + ˘b12 ,  ˘b3 − ˘b11 − ˘b7 + ˘b15 ˘b2 − ˘b10 − ˘b6 + ˘b14 ,  ˜b1 + ˜b9 + ˜b5 + ˜b13 ˜b0 + ˜b8 + ˜b4 + ˜b12 ,  ˜b3 + ˜b11 + ˜b7 + ˜b15 ˜b2 + ˜b10 + ˜b6 + ˜b14 ,  ˜b1 + ˜b9 − ˜b5 − ˜b13 ˜b0 + ˜b8 − ˜b4 − ˜b12 ,  ˜b3 + ˜b11 − ˜b7 − ˜b15 ˜b2 + ˜b10 − ˜b6 − ˜b14 ,  ˜b1 − ˜b9 + ˜b5 − ˜b13 ˜b0 − ˜b8 + ˜b4 − ˜b12 ,  ˜b3 − ˜b11 + ˜b7 − ˜b15 ˜b2 − ˜b10 + ˜b6 − ˜b14 ,

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An algorithm for multiplication of trigintaduonions

 ˜b − ˜b − ˜b + ˜b (14) ˇ B2 = ˜0 ˜8 ˜4 ˜12 b1 − b9 − b5 + b13  ˜ ˜ ˜ ˜ ˇ (15) = b2 − b10 − b6 + b14 B 2 ˜b3 − ˜b11 − ˜b7 + ˜b15

 ˜b1 − ˜b9 − ˜b5 + ˜b13 ˜b0 − ˜b8 − ˜b4 + ˜b12 ,  ˜b3 − ˜b11 − ˜b7 + ˜b15 ˜b2 − ˜b10 − ˜b6 + ˜b14 .

Then we have the following decompositions: ˇ (0) + B ˇ (1) = W4 D(0) W4 , B 4 4 4

ˇ (0) − B ˇ (1) = W4 D(1) W4 , B 4 4 4

ˇ (2) + B ˇ (3) = W4 D(2) W4 , B 4 4 4

ˇ (2) − B ˇ (3) = W4 D(3) W4 , B 4 4 4

ˇ (4) + B ˇ (1) = W4 D(4) W4 , B 4 5 4

ˇ (4) − B ˇ (1) = W4 D(5) W4 , B 4 5 4

ˇ (6) + B ˇ (7) = W4 D(6) W4 , B ˇ (6) − B ˇ (7) = W4 D(7) W4 , B 4 4 4 4 4 4 h   i 1 ˇ (0) ˇ (1) (0) ˇ (0) − B ˇ (1) , D4 = B2 + B2 ⊕ B 2 2 2 1 h ˇ (2) ˇ (3)   ˇ (2) ˇ (3) i (1) B4 + B4 ⊕ B4 − B4 , D4 = 2 1 h ˇ (4) ˇ (5)   ˇ (4) ˇ (5) i (2) D4 = B2 + B2 ⊕ B2 − B2 , 2 1 h ˇ (6) ˇ (7)   ˇ (6) ˇ (7) i (3) , B4 + B4 ⊕ B4 − B4 D4 = 2 1 h ˇ (8) ˇ (9)   ˇ (8) ˇ (9) i (4) D4 = , B2 + B2 ⊕ B2 − B2 2 1 h ˇ (10) ˇ (11)   ˇ (10) ˇ (11) i (5) , ⊕ B4 − B4 B4 + B4 D4 = 2 1 h ˇ (12) ˇ (13)   ˇ (12) ˇ (13) i (6) D4 = , ⊕ B2 − B2 B2 + B2 2 1 h ˇ (14) ˇ (15)   ˇ (14) ˇ (15) i (7) D4 = B4 + B4 ⊕ B4 − B4 , 2 W4 = H2 ⊗ I2 . Combining partial decompositions in a common procedure we can rewrite (5) as follows: ˇ 32 X32×1 = W(0) W(1) W(2) W(3) D(3) W(3) W(2) W(1) W(0) X32×1 B 32 32 32 32 32 32 32 32 32

(6)

where (3)

W32 = I8 ⊗ W4 , (3)

D32 =

  1 (0) (1) (2) (3) (4) (5) (6) (7) diag D4 , D4 , D4 , D4 , D4 , D4 , D4 , D4 16

and ˇ (0) + B ˇ (1) = B 2 2

 ˘b + ˘b + ˘b + ˘b + ˘b + ˘b + ˘b + ˘b = ˘0 ˘8 ˘4 ˘12 ˘2 ˘10 ˘6 ˘14 b1 + b9 + b5 + b13 + b3 + b11 + b7 + b15

 ˘b1 + ˘b9 + ˘b5 + ˘b13 + ˘b3 + ˘b11 + ˘b7 + ˘b15 ˘b0 + ˘b8 + ˘b4 + ˘b12 + ˘b2 + ˘b10 + ˘b6 + ˘b14 ,

66

Alexandr Cariow, Galina Cariowa

ˇ (0) − B ˇ (1) = B 2 2  ˘b0 + ˘b8 + ˘b4 + ˘b12 − ˘b2 − ˘b10 − ˘b6 − ˘b14 = ˘ b1 + ˘b9 + ˘b5 + ˘b13 − ˘b3 − ˘b11 − ˘b7 − ˘b15

 ˘b1 + ˘b9 + ˘b5 + ˘b13 − ˘b3 − ˘b11 − ˘b7 − ˘b15 ˘b0 + ˘b8 + ˘b4 + ˘b12 − ˘b2 − ˘b10 − ˘b6 − ˘b14 ,

ˇ (2) + B ˇ (3) = B 2 2  ˘b0 + ˘b8 − ˘b4 − ˘b12 + ˘b2 + ˘b10 − ˘b6 − ˘b14 = ˘ b1 + ˘b9 − ˘b5 − ˘b13 + ˘b3 + ˘b11 − ˘b7 − ˘b15

 ˘b1 + ˘b9 − ˘b5 − ˘b13 + ˘b3 + ˘b11 − ˘b7 − ˘b15 ˘b0 + ˘b8 − ˘b4 − ˘b12 + ˘b2 + ˘b10 − ˘b6 − ˘b14 ,

ˇ (2) − B ˇ (3) = B 2 2   ˘b0 + ˘b8 − ˘b4 − ˘b12 − ˘b2 − ˘b10 + ˘b6 + ˘b14 ˘b1 + ˘b9 − ˘b5 − ˘b13 − ˘b3 − ˘b11 + ˘b7 + ˘b15 , = ˘ b1 + ˘b9 − ˘b5 − ˘b13 − ˘b3 − ˘b11 + ˘b7 + ˘b15 ˘b0 + ˘b8 − ˘b4 − ˘b12 − ˘b2 − ˘b10 + ˘b6 + ˘b14 ˇ (4) + B ˇ (5) = B 2 2   ˘b0 − ˘b8 + ˘b4 − ˘b12 + ˘b2 − ˘b10 + ˘b6 − ˘b14 ˘b1 − ˘b9 + ˘b5 − ˘b13 + ˘b3 − ˘b11 + ˘b7 − ˘b15 , = ˘ b1 − ˘b9 + ˘b5 − ˘b13 + ˘b3 − ˘b11 + ˘b7 − ˘b15 ˘b0 − ˘b8 + ˘b4 − ˘b12 + ˘b2 − ˘b10 + ˘b6 − ˘b14 ˇ (4) − B ˇ (5) = B 2 2  ˘b0 − ˘b8 + ˘b4 − ˘b12 − ˘b2 + ˘b10 − ˘b6 + ˘b14 = ˘ b1 − ˘b9 + ˘b5 − ˘b13 − ˘b3 + ˘b11 − ˘b7 + ˘b15

 ˘b1 − ˘b9 + ˘b5 − ˘b13 − ˘b3 + ˘b11 − ˘b7 + ˘b15 ˘b0 − ˘b8 + ˘b4 − ˘b12 − ˘b2 + ˘b10 − ˘b6 + ˘b14 ,

ˇ (7) = ˇ (6) + B B 2 2  ˘b0 − ˘b8 − ˘b4 + ˘b12 + ˘b2 − ˘b10 − ˘b6 + ˘b14 = ˘ b1 − ˘b9 − ˘b5 + ˘b13 + ˘b3 − ˘b11 − ˘b7 + ˘b15

 ˘b1 − ˘b9 − ˘b5 + ˘b13 + ˘b3 − ˘b11 − ˘b7 + ˘b15 ˘b0 − ˘b8 − ˘b4 + ˘b12 + ˘b2 − ˘b10 − ˘b6 + ˘b14 ,

ˇ (7) = ˇ (6) − B B 2 2  ˘b0 − ˘b8 − ˘b4 + ˘b12 − ˘b2 + ˘b10 + ˘b6 − ˘b14 = ˘ b1 − ˘b9 − ˘b5 + ˘b13 − ˘b3 + ˘b11 + ˘b7 − ˘b15

 ˘b1 − ˘b9 − ˘b5 + ˘b13 − ˘b3 + ˘b11 + ˘b7 − ˘b15 ˘b0 − ˘b8 − ˘b4 + ˘b12 − ˘b2 + ˘b10 + ˘b6 − ˘b14 ,

ˇ (8) + B ˇ (9) = B 2 2

  ˜b0 + ˜b8 + ˜b4 + ˜b12 + ˜b2 + ˜b10 + ˜b6 + ˜b14 ˜b1 + ˜b9 + ˜b5 + ˜b13 + ˜b3 + ˜b11 + ˜b7 + ˜b15 , = ˜ b1 + ˜b9 + ˜b5 + ˜b13 + ˜b3 + ˜b11 + ˜b7 + ˜b15 ˜b0 + ˜b8 + ˜b4 + ˜b12 + ˜b2 + ˜b10 + ˜b6 + ˜b14 ˇ (8) − B ˇ (9) = B 2 2  ˜b0 + ˜b8 + ˜b4 + ˜b12 − ˜b2 − ˜b10 − ˜b6 − ˜b14 = ˜ b1 + ˜b9 + ˜b5 + ˜b13 − ˜b3 − ˜b11 − ˜b7 − ˜b15

 ˜b1 + ˜b9 + ˜b5 + ˜b13 − ˜b3 − ˜b11 − ˜b7 − ˜b15 ˜b0 + ˜b8 + ˜b4 + ˜b12 − ˜b2 − ˜b10 − ˜b6 − ˜b14 ,

ˇ (10) + B ˇ (11) = B 2 2  ˜b0 + ˜b8 − ˜b4 − ˜b12 + ˜b2 + ˜b10 − ˜b6 − ˜b14 = ˜ b1 + ˜b9 − ˜b5 − ˜b13 + ˜b3 + ˜b11 − ˜b7 − ˜b15

 ˜b1 + ˜b9 − ˜b5 − ˜b13 + ˜b3 + ˜b11 − ˜b7 − ˜b15 ˜b0 + ˜b8 − ˜b4 − ˜b12 + ˜b2 + ˜b10 − ˜b6 − ˜b14 ,

ˇ (10) − B ˇ (11) = B 2 2  ˜b0 + ˜b8 − ˜b4 − ˜b12 − ˜b2 − ˜b10 + ˜b6 + ˜b14 = ˜ b1 + ˜b9 − ˜b5 − ˜b13 − ˜b3 − ˜b11 + ˜b7 + ˜b15

 ˜b1 + ˜b9 − ˜b5 − ˜b13 − ˜b3 − ˜b11 + ˜b7 + ˜b15 ˜b0 + ˜b8 − ˜b4 − ˜b12 − ˜b2 − ˜b10 + ˜b6 + ˜b14 ,

67

An algorithm for multiplication of trigintaduonions

ˇ (12) + B ˇ (13) = B 2 2  ˜b0 − ˜b8 + ˜b4 − ˜b12 + ˜b2 − ˜b10 + ˜b6 − ˜b14 = ˜ b1 − ˜b9 + ˜b5 − ˜b13 + ˜b3 − ˜b11 + ˜b7 − ˜b15

 ˜b1 − ˜b9 + ˜b5 − ˜b13 + ˜b3 − ˜b11 + ˜b7 − ˜b15 ˜b0 − ˜b8 + ˜b4 − ˜b12 + ˜b2 − ˜b10 + ˜b6 − ˜b14 ,

ˇ (12) − B ˇ (13) = B 2 2   ˜b0 − ˜b8 + ˜b4 − ˜b12 − ˜b2 + ˜b10 − ˜b6 + ˜b14 ˜b1 − ˜b9 + ˜b5 − ˜b13 − ˜b3 + ˜b11 − ˜b7 + ˜b15 = ˜ , b1 − ˜b9 + ˜b5 − ˜b13 − ˜b3 + ˜b11 − ˜b7 + ˜b15 ˜b0 − ˜b8 + ˜b4 − ˜b12 − ˜b2 + ˜b10 − ˜b6 + ˜b14 ˇ (14) + B ˇ (15) = B 2 2  ˜b0 − ˜b8 − ˜b4 + ˜b12 + ˜b2 − ˜b10 − ˜b6 + ˜b14 = ˜ b1 − ˜b9 − ˜b5 + ˜b13 + ˜b3 − ˜b11 − ˜b7 + ˜b15

 ˜b1 − ˜b9 − ˜b5 + ˜b13 + ˜b3 − ˜b11 − ˜b7 + ˜b15 ˜b0 − ˜b8 − ˜b4 + ˜b12 + ˜b2 − ˜b10 − ˜b6 + ˜b14 ,

ˇ (14) − B ˇ (15) = B 2 2  ˜b0 − ˜b8 − ˜b4 + ˜b12 − ˜b2 + ˜b10 + ˜b6 − ˜b14 = ˜ b1 − ˜b9 − ˜b5 + ˜b13 − ˜b3 + ˜b11 + ˜b7 − ˜b15

 ˜b1 − ˜b9 − ˜b5 + ˜b13 − ˜b3 + ˜b11 + ˜b7 − ˜b15 ˜b0 − ˜b8 − ˜b4 + ˜b12 − ˜b2 + ˜b10 + ˜b6 − ˜b14 .

Introduce the following notation: σ0 = ˘b0 + ˘b8 + ˘b4 + ˘b12 + ˘b2 + ˘b10 + ˘b6 + ˘b14 ,

σ1 = ˘b1 + ˘b9 + ˘b5 + ˘b13 + ˘b3 + ˘b11 + ˘b7 + ˘b15 ,

σ2 = ˘b0 + ˘b8 + ˘b4 + ˘b12 − ˘b2 − ˘b10 − ˘b6 − ˘b14 ,

σ3 = ˘b1 + ˘b9 + ˘b5 + ˘b13 − ˘b3 − ˘b11 − ˘b7 − ˘b15 ,

σ4 = ˘b0 + ˘b8 − ˘b4 − ˘b12 + ˘b2 + ˘b10 − ˘b6 − ˘b14 ,

σ5 = ˘b1 + ˘b9 − ˘b5 − ˘b13 + ˘b3 + ˘b11 − ˘b7 − ˘b15 ,

σ6 = ˘b0 + ˘b8 − ˘b4 − ˘b12 − ˘b2 − ˘b10 + ˘b6 + ˘b14 ,

σ7 = ˘b1 + ˘b9 − ˘b5 − ˘b13 − ˘b3 − ˘b11 + ˘b7 + ˘b15 ,

σ8 = ˘b0 − ˘b8 + ˘b4 − ˘b12 + ˘b2 − ˘b10 + ˘b6 − ˘b14 ,

σ9 = ˘b1 − ˘b9 + ˘b5 − ˘b13 + ˘b3 − ˘b11 + ˘b7 − ˘b15 ,

σ10 = ˘b0 − ˘b8 + ˘b4 − ˘b12 − ˘b2 + ˘b10 − ˘b6 + ˘b14 ,

σ11 = ˘b1 − ˘b9 + ˘b5 − ˘b13 − ˘b3 + ˘b11 − ˘b7 + ˘b15 ,

σ12 = ˘b0 − ˘b8 − ˘b4 + ˘b12 + ˘b2 − ˘b10 − ˘b6 + ˘b14 ,

σ13 = ˘b1 − ˘b9 − ˘b5 + ˘b13 + ˘b3 − ˘b11 − ˘b7 + ˘b15 ,

σ14 = ˘b0 − ˘b8 − ˘b4 + ˘b12 − ˘b2 + ˘b10 + ˘b6 − ˘b14 ,

σ15 = ˘b1 − ˘b9 − ˘b5 + ˘b13 − ˘b3 + ˘b11 + ˘b7 − ˘b15 ,

σ16 = ˜b0 + ˜b8 + ˜b4 + ˜b12 + ˜b2 + ˜b10 + ˜b6 + ˜b14 ,

σ17 = ˜b1 + ˜b9 + ˜b5 + ˜b13 + ˜b3 + ˜b11 + ˜b7 + ˜b15 ,

σ18 = ˜b0 + ˜b8 + ˜b4 + ˜b12 − ˜b2 − ˜b10 − ˜b6 − ˜b14 ,

σ19 = ˜b1 + ˜b9 + ˜b5 + ˜b13 − ˜b3 − ˜b11 − ˜b7 − ˜b15 ,

σ20 = ˜b0 + ˜b8 − ˜b4 − ˜b12 + ˜b2 + ˜b10 − ˜b6 − ˜b14 ,

σ21 = ˜b1 + ˜b9 − ˜b5 − ˜b13 + ˜b3 + ˜b11 − ˜b7 − ˜b15 ,

σ22 = ˜b0 + ˜b8 − ˜b4 − ˜b12 − ˜b2 − ˜b10 + ˜b6 + ˜b14 ,

σ23 = ˜b1 + ˜b9 − ˜b5 − ˜b13 − ˜b3 − ˜b11 + ˜b7 + ˜b15 ,

σ24 = ˜b0 − ˜b8 + ˜b4 − ˜b12 + ˜b2 − ˜b10 + ˜b6 − ˜b14 ,

σ25 = ˜b1 − ˜b9 + ˜b5 − ˜b13 + ˜b3 − ˜b11 + ˜b7 − ˜b15 ,

σ26 = ˜b0 − ˜b8 + ˜b4 − ˜b12 − ˜b2 + ˜b10 − ˜b6 + ˜b14 ,

σ27 = ˜b1 − ˜b9 + ˜b5 − ˜b13 − ˜b3 + ˜b11 − ˜b7 + ˜b15 ,

σ28 = ˜b0 − ˜b8 − ˜b4 + ˜b12 + ˜b2 − ˜b10 − ˜b6 + ˜b14 ,

σ29 = ˜b1 − ˜b9 − ˜b5 + ˜b13 + ˜b3 − ˜b11 − ˜b7 + ˜b15 ,

σ30 = ˜b0 − ˜b8 − ˜b4 + ˜b12 − ˜b2 + ˜b10 + ˜b6 − ˜b14 ,

σ31 = ˜b1 − ˜b9 − ˜b5 + ˜b13 − ˜b3 + ˜b11 + ˜b7 − ˜b15 .

Then

 σ (0) (1) ˇ ˇ B2 + B2 = 0 σ1  ˇ (2) + B ˇ (3) = σ4 B 2 2 σ5  ˇ (4) + B ˇ (5) = σ8 B 2 2 σ9

  σ1 σ (0) (1) ˇ ˇ , B2 − B2 = 2 σ0 σ3   σ5 ˇ (2) − B ˇ (3) = σ6 , B 2 2 σ4 σ7   σ9 ˇ (4) − B ˇ (5) = σ10 , B 2 2 σ8 σ11

 σ3 , σ2  σ7 , σ6  σ11 , σ10

68

Alexandr Cariow, Galina Cariowa

ˇ (6) B 2

+

ˇ (7) B 2

ˇ (8) + B ˇ (9) B 2 2 ˇ (10) + B ˇ (11) B 2 2 ˇ (12) + B ˇ (13) B 2 2 ˇ (14) + B ˇ (15) B 2 2



 σ12 σ13 = , σ13 σ12   σ16 σ17 = , σ17 σ16   σ20 σ21 = , σ21 σ20   σ24 σ25 = , σ25 σ24   σ28 σ29 = , σ29 σ28

  σ14 σ15 − = , σ15 σ14   σ18 σ19 (8) (9) ˇ ˇ B2 − B2 = , σ19 σ18   σ22 σ23 (10) (11) ˇ ˇ B2 − B2 = , σ23 σ22   σ26 σ27 (12) (13) ˇ ˇ B2 − B2 = , σ27 σ26   σ σ (14) (15) 30 31 ˇ ˇ B −B = . 2 2 σ31 σ30

ˇ (6) B 2

ˇ (7) B 2

Therefore we have the following decompositions: ˇ (0) + B ˇ (1) = H2 D(0) H2 , B 2 2 2

ˇ (0) − B ˇ (1) = H2 D(1) H2 , B 2 2 2

ˇ (3) = H2 D(2) H2 , ˇ (2) + B B 2 2 2

ˇ (3) = H2 D(3) H2 , ˇ (2) − B B 2 2 2

ˇ (4) + B ˇ (5) = H2 D(4) H2 , B 2 2 2

ˇ (4) − B ˇ (5) = H2 D(5) H2 , B 2 2 2

ˇ (6) + B ˇ (7) = H2 D(6) H2 , B 2 2 2

ˇ (6) − B ˇ (7) = H2 D(7) H2 , B 2 2 2

ˇ (8) + B ˇ (9) = H2 D(8) H2 , B 2 2 2

ˇ (8) − B ˇ (9) = H2 D(9) H2 , B 2 2 2

ˇ (11) = H2 D(10) H2 , ˇ (10) + B B 2 2 2

ˇ (11) = H2 D(11) H2 , ˇ (10) − B B 2 2 2

ˇ (12) + B ˇ (13) = H2 D(12) H2 , B 2 2 2

ˇ (12) − B ˇ (13) = H2 D(13) H2 , B 2 2 2

ˇ (14) + B ˇ (15) = H2 D(14) H2 , B ˇ (14) − B ˇ (15) = H2 D(15) H2 , B 2 2 2 2 2 2     1 σ0 + σ1 1 σ2 + σ3 0 0 (0) (1) , D2 = , D2 = 0 σ0 − σ1 0 σ2 − σ3 2 2     1 σ4 + σ5 1 σ6 + σ7 0 0 (2) (3) D2 = , D2 = , 0 σ4 − σ5 0 σ6 − σ7 2 2     1 σ8 + σ9 1 σ10 + σ11 0 0 (5) (4) , D2 = , D2 = 0 σ8 − σ9 0 σ10 − σ11 2 2     1 σ12 + σ13 1 σ14 + σ15 0 0 (6) (7) D2 = , D2 = , 0 σ12 − σ13 0 σ14 − σ15 2 2     1 σ16 + σ17 1 σ18 + σ19 0 0 (8) (9) D2 = , D2 = , 0 σ16 − σ17 0 σ18 − σ19 2 2     1 σ20 + σ21 1 σ22 + σ23 0 0 (10) (11) D2 = , D2 = , 0 σ20 − σ21 0 σ22 − σ23 2 2     1 σ24 + σ25 1 σ26 + σ27 0 0 (12) (13) D2 = , D2 = , 0 σ24 − σ25 0 σ26 − σ27 2 2     1 σ28 + σ29 1 σ30 + σ31 0 0 (14) (15) D2 = , D2 = , 0 σ28 − σ29 0 σ30 − σ31 2 2

69

An algorithm for multiplication of trigintaduonions

Combining partial decompositions in a final procedure we can rewrite (6) as: ˇ 32 X32×1 = W(0) W(1) W(2) W(3) W(4) D(4) W(4) W(3) W(2) W(1) W(0) X32×1 B 32 32 32 32 32 32 32 32 32 32 32

(7)

where (4)

W32 = I16 ⊗ H2 , (4)

D32 and

! 15 M 1 (l) D2 , = diag (s0 , s1 , . . . , s15 ) = diag 16 l=0

l = 0, 1, . . . , 15

1 1 1 (σ0 + σ1 ) , s1 = (σ0 − σ1 ) , s2 = (σ2 + σ3 ) , 32 32 32 1 1 1 (σ2 − σ3 ) , s4 = (σ4 + σ5 ) , s5 = (σ4 − σ5 ) , s3 = 32 32 32 1 1 1 s6 = (σ6 + σ7 ) , s7 = (σ6 − σ7 ) , s8 = (σ8 + σ9 ) , 32 32 32 1 1 1 (σ8 − σ9 ) , s10 = (σ10 + σ11 ) , s11 = (σ10 − σ11 ) , s9 = 32 32 32 1 1 1 s12 = (σ12 + σ13 ) , s13 = (σ12 − σ13 ) , s14 = (σ14 + σ15 ) , 32 32 32 1 1 1 s15 = (σ14 − σ15 ) , s16 = (σ16 + σ17 ) , s17 = (σ16 − σ17 ) , 32 32 32 1 1 1 s18 = (σ18 + σ19 ) , s19 = (σ18 − σ19 ) , s20 = (σ20 + σ21 ) , 32 32 32 1 1 1 s21 = (σ20 − σ21 ) , s22 = (σ22 + σ23 ) , s23 = (σ22 − σ23 ) , 32 32 32 1 1 1 (σ24 + σ25 ) , s25 = (σ24 − σ25 ) , s26 = (σ26 + σ27 ) , s24 = 32 32 32 1 1 1 s27 = (σ26 − σ27 ) , s28 = (σ28 + σ29 ) , s29 = (σ28 − σ29 ) , 32 32 32 1 1 s30 = (σ30 + σ31 ) , s31 = (σ30 − σ31 ) . 32 32 s0 =

(4)

It is easy to see that the elements {sl } of the matrix D32 can be calculated using the following vector-matrix procedure: S32×1 =

1 (5) (4) (3) (2) (1) (0) W32 W32 W32 W32 W32 W32 B32×1 32

(8)

where S32×1 = [s0 , s1 , . . . , s31 ]T

and B32×1 = [b0 , b1 , . . . , b31 ]T .

ˆ 32 X32×1 cannot be reduced Unfortunately the computational complexity of the product 2B and this product is calculated in the usual way, without any tricks. Combining the calculations for both matrices in a single procedure we finally obtain: Y32×1 =

70

Alexandr Cariow, Galina Cariowa (0)

(1)

(2)

(3)

(4)

(4)

(3)

(2)

(1)

(0)

= Σ32×64 W64 W64 W64 W64 W64 S64 W64 W64 W64 W64 W64 P64×32 X32×1 where Σ32×64 = (¯ 11×2 ⊗ I32 , ) ,

P64×32 = (12×1 ⊗ I32 ) , 12×1 = [1 (0)

1]T ,

¯ 11×2 = [1 − 1] ,

(0)

(1)

W64 = W32 ⊕ I32 ,

(1)

W64 = W32 ⊕ I32 ,

(3)

(3)

W64 = W32 ⊕ I32 , (0) W32

(1) W32

H2

x1

H2

x2

H2

x3 x4

H2

x5 x6

H2 H2

x7

H2

x8

H2

x9

H2

H2

x11

H2

x12 x13

H2

H2

x14

H2

x15 x16

H2 H2

x17 x18

H2

x19

H2

x20 x21 x22 x23

H2

H2 H2

H2

x24

H2

x25 x26 x27

H2

H2

H2

H2

x28 x29 x30 x31

H2

H2 H2

H2

s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31

(0)

(2)

(2)

W64 = W32 ⊕ I32 ,

ˆ 32 . S64 = D32 ⊕ 2B

( 2) (3) (4) (4) (3) ( 2) W32 W32 W32 D32 W32 W32 W32

x0

x10

(0)

W64 = W32 ⊕ I32 ,

(0) W32

(1) W32

y0

H2 H2

y1 y2

H2

y3

H2

y4

H2

y5

H2

y6

H2

y7 y8

H2

y9

H2

y10

H2

H2

y11 y12

H2

y13

H2 H2

y14

H2

y15 y16

H2 H2

y17

H2

y19 y 20

H2 H2

y18

H2

H2

H2

y 22 y 23 y 24

H2 H2

y 25

H2

y 26

H2

y 27 y 28

H2 H2

y 21

H2 H2

ˇ 32 by vector X32×1 multiplication algorithm. Figure 1. Data flow diagram ( for matrix B

Fig. 1. Data flow diagram for matrix B 32 by vector X 32×1 multiplication algorithm.

y 29 y30 y31

71

An algorithm for multiplication of trigintaduonions

ˇ 32 by vector X32×1 fast multiplication alFig. 1 shows a data flow diagram of for matrix B gorithm. The circles in this and follows figures show the operation of multiplication by a value inscribed inside a circle. In turn, the rectangles indicate the matrix-vector multiplications with matrices inscribed inside rectangles. In this paper the data flow diagrams are oriented from left to right. Straight lines in the figure denote the operation of data transfer. We use the solid lines without any arrows, so as not to clutter up the presented diagrams. Fig. 2 shows data flow diagram describing the process of calculating elements of the vector S32×1 in accordance with the procedure (8) and Fig. 3 shows a data flow diagram for trigintaduonion multiplication rationalized algorithm. In Fig. 3, the points where lines converge denote summation. As follows from Fig. 2, calculation of elements of diagonal b0

H2

b1

H2

b2

H2

b3 b4

H2

b5 b6

H2 H2

b7

H2

b8

H2

b9 b10

H2

H2

b11

H2

b12 b13

H2

H2

b14

H2

b15 b16

H2 H2

b17 b18

H2

b19

H2

b20 b21 b22 b23

H2

H2 H2

H2

b24

H2

b25 b26 b27

H2

H2

H2

H2

b28 b29 b30 b31

H2

H2 H2

H2

1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32

s0 s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s12 s13 s14 s15 s16 s17 s18 s19 s20 s21 s22 s23 s24 s25 s26 s27 s28 s29 s30 s31

Fig. 2. diagram describing the process of calculating elements of the vectorofSthe accordance 32×1 in Figure 2.The Thesignal signal diagram describing the process of calculating elements vector S32×1with in the procedure (8). accordance with the procedure (8).

72

Alexandr Cariow, Galina Cariowa

x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x 10 x11 x 12 x13 x 14 x15 x 16 x 17 x18 x 19 x 20 x 21 x 22 x 23 x 24 x 25 x 26 x 27 x 28 x 29 x 30 x 31

(0) (4) (3) ( 2) (1) (0) ( 2) (4) (1) (3) D32 W32 W32 W32 W32 W32 W32 W32 W32 W32 W32

) 2B32

−c0 c1 c2 c3 c4 c5 c6 c7 c8 c9 c10 c11 c12 c13 c14 c15 c16 c17 c18 c19 c 20 c 21 c 22 c 23 c 24 c 25 c 26 c 27 c 28 c 29 c30 c31

Fig. 3. Data flow diagram trigintaduonion multiplication algorithm.multiplication algorithm. Figure 3. for Data flow diagram for trigintaduonion 3. Evaluation of computational complexity We calculate how many real multiplications (excluding multiplications by power of two) and additions are required, and compare this with the number required for a direct evaluation of matrix-vector product in equation D (1). number performing of real multiplications required using proposed algorithm 498. Thuspower using proposed matrix requires only trivial multiplications by the is negative of two. 32The algorithm the number of real multiplications to implement the trigintaduonions product is reduced. The number Such operations may be implemented using convention right-shift operations, which have of real additions required using our algorithm is 943. We observe that the direct computation of trigintaduonion

simple realization and hence may be neglected during computational complexity estimation.

An algorithm for multiplication of trigintaduonions

73

3. Evaluation of computational complexity We calculate how many real multiplications (excluding multiplications by power of two) and additions are required, and compare this with the number required for a direct evaluation of matrix-vector product in equation (1). The number of real multiplications required using proposed algorithm is 498. Thus using proposed algorithm the number of real multiplications to implement the trigintaduonions product is reduced. The number of real additions required using our algorithm is 943. We observe that the direct computation of trigintaduonion product requires 49 additions more, than the proposed algorithm. Thus, our proposed algorithms saves 526 multiplications and 49 additions compared with direct method. Nonetheless, the total number of arithmetic operations for proposed algorithm is more than 20% less than that of the direct evaluation.

4. Conclusions In this paper, we have proposed an effective algorithm to multiply two trigintaduonions with reduced computational complexity. As a result of rationalization the number of multiplications required to calculate the trigintaduonion product is reduced from 1024 to 498. In addition, the number of additions has reduced by 49 compared with the naive algorithm. As a result, the total number of arithmetic operations required to calculate the product of two trigintaduonions is reduced by 575. It should be noted that in some practical applications, one of the trigintaduonions contain constants. In this case elements of the diagonal matrix D32 can be precomputed and stored in the computation units memory ahead of time. Then the number of additions in the proposed algorithm can be further reduced by 160.

References [1] R. Abłamowicz (ed.), Clifford Algebras – Applications to Mathematics, Physics, and Engineering, PIM 34, Birkhauser, Basel 2004. [2] D. Alfsmann, H. G. G¨ockler, S. J. Sangwine, and T. A. Ell. Hypercomplex Algebras in Digital Signal Processing: Benefits and Drawbacks (Tutorial). Proc. EURASIP 15th European Signal Processing Conference (EUSIPCO 2007), Pozna´n, Poland, (2007), pp. 1322–1326. [3] E. Bayro-Corrochano. Multi-resolution image analysis using the quaternion wavelet transform, Numerical Algorithms, vol. 39, No 1–3, (2005), pp. 35–55. [4] N. Le Bihan and J. Mars. Singular value decomposition of quaternion matrices: A new tool for vector-sensor signal processing, Signal Process., vol. 84, no. 7, (2004), pp. 1177–1199. [5] N. L. Bihan, S. J. Sangwine. Quaternion principal component analysis of color images. In: IEEE International Conference on Image Processing (ICIP 2003), vol. 1, Barcelona (Spain), (2003), pp. 809–812. [6] S. Buchholz and N. Le Bihan. Polarized signal classification by complex and quaternionic multi-layer perceptrons, Int. J. Neural Syst., vol. 18, no. 2, (2008), pp. 75–85. [7] T. B¨ulow and G. Sommer. Hypercomplex signals – a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Sign. Proc., vol. SP–49, no. 11, (2001), pp. 2844–2852. [8] R. Calderbank, S. Das, N. Al-Dhahir and S. Diggavi. Construction And Analysis Of A New Quaternionic Space-Time Code For 4 Transmit Antennas, Communications In Information And Systems, vol. 5, No. 1, (2005), pp. 97–122. [9] A. Cariow, G. Cariowa, Algorithm for multiplying two octonions, Radioelectronics and Communications Systems (Allerton Press, Inc. USA), vol. 55, No 10, (2012), pp. 464–473.

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Alexandr Cariow, Galina Cariowa

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