Derivation of a low multiplicative complexity algorithm for multiplying hyperbolic octonions

Abstract. We present an efficient algorithm to multiply two hyperbolic (countercomplex) octonions. The direct multiplication of two hyperbolic octonions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the hyperbolic octonions with 26 real multiplications and 92 real additions. During synthesis of the discussed algorithm we use the fact that product of two hyperbolic octonions may be represented as a matrix–vector product. The matrix multiplicand that participates in the product calculating has unique structural properties that allow performing its advantageous factorization. Namely this factorization leads to significant reducing of the computational complexity of hyperbolic octonions multiplication.